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ISBN 978-91-7485-431-2ISSN 1651-4238

Address: P.O. Box 883, SE-721 23 Västerås. SwedenAddress: P.O. Box 325, SE-631 05 Eskilstuna. SwedenE-mail: [email protected] Web: www.mdh.se

Extreme points of the Vandermonde determinant and phenomenological modelling with power exponential functionsKarl Lundengård

Mälardalen University Doctoral Dissertation 293

Mälardalen University Press DissertationsNo. 293

EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND PHENOMENOLOGICAL

MODELLING WITH POWER EXPONENTIAL FUNCTIONS

Karl Lundengård

2019

School of Education, Culture and Communication


EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND PHENOMENOLOGICAL

MODELLING WITH POWER EXPONENTIAL FUNCTIONS

Karl Lundengård

2019

School of Education, Culture and Communication

111

Copyright © Karl Lundengård, 2019ISBN 978-91-7485-431-2ISSN 1651-4238Printed by E-Print AB, Stockholm, Sweden

Copyright © Karl Lundengård, 2019ISBN 978-91-7485-431-2ISSN 1651-4238Printed by E-Print AB, Stockholm, Sweden

222


EXTREME POINTS OF THE VANDERMONDE DETERMINANT ANDPHENOMENOLOGICAL MODELLING WITH POWER EXPONENTIAL FUNCTIONS

Karl Lundengård

Akademisk avhandling

som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematikvid Akademin för utbildning, kultur och kommunikation kommer att offentligen

försvaras torsdagen den 26 september 2019, 13.15 i Delta, Mälardalens högskola, Västerås.

Fakultetsopponent: Professor Palle Jorgensen, University of Iowa

Akademin för utbildning, kultur och kommunikation


EXTREME POINTS OF THE VANDERMONDE DETERMINANT ANDPHENOMENOLOGICAL MODELLING WITH POWER EXPONENTIAL FUNCTIONS

Karl Lundengård

Akademisk avhandling

som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematikvid Akademin för utbildning, kultur och kommunikation kommer att offentligen

försvaras torsdagen den 26 september 2019, 13.15 i Delta, Mälardalens högskola, Västerås.

Fakultetsopponent: Professor Palle Jorgensen, University of Iowa

Akademin för utbildning, kultur och kommunikation

333

AbstractThis thesis discusses two topics, finding the extreme points of the Vandermonde determinant on various surfaces and phenomenological modelling using power-exponential functions. The relation between these two problems is that they are both related to methods for curve-fitting. Two applications of the mathematical models and methods are also discussed, modelling of electrostatic discharge currents for use in electromagnetic compatibility and modelling of mortality rates for humans. Both the construction and evaluation of models is discussed.

In the first chapter the basic theory for later chapters is introduced. First the Vandermonde matrix, a matrix whose rows (or columns) consists of monomials of sequential powers, its history and some of its properties are discussed. Next, some considerations and typical methods for a common class of curve fitting problems are presented, as well as how to analyse and evaluate the resulting fit. In preparation for the later parts of the thesis the topics of electromagnetic compatibility and mortality rate modelling are briefly introduced.

The second chapter discusses some techniques for finding the extreme points for the determinant of the Vandermonde matrix on various surfaces including spheres, ellipsoids and cylinders. The discussion focuses on low dimensions, but some results are given for arbitrary (finite) dimensions.

In the third chapter a particular model called the p-peaked Analytically Extended Function (AEF) is introduced and fitted to data taken either from a standard for electromagnetic compatibility or experimental measurements. The discussion here is entirely focused on currents originating from lightning or electrostatic discharges.

The fourth chapter consists of a comparison of several different methods for modelling mortality rates, including a model constructed in a similar way to the AEF found in the third chapter. The models are compared with respect to how well they can be fitted to estimated mortality rate for several countries and several years and the results when using the fitted models for mortality rate forecasting is also compared.

ISBN 978-91-7485-431-2ISSN 1651-4238

AbstractThis thesis discusses two topics, finding the extreme points of the Vandermonde determinant on various surfaces and phenomenological modelling using power-exponential functions. The relation between these two problems is that they are both related to methods for curve-fitting. Two applications of the mathematical models and methods are also discussed, modelling of electrostatic discharge currents for use in electromagnetic compatibility and modelling of mortality rates for humans. Both the construction and evaluation of models is discussed.

In the first chapter the basic theory for later chapters is introduced. First the Vandermonde matrix, a matrix whose rows (or columns) consists of monomials of sequential powers, its history and some of its properties are discussed. Next, some considerations and typical methods for a common class of curve fitting problems are presented, as well as how to analyse and evaluate the resulting fit. In preparation for the later parts of the thesis the topics of electromagnetic compatibility and mortality rate modelling are briefly introduced.

The second chapter discusses some techniques for finding the extreme points for the determinant of the Vandermonde matrix on various surfaces including spheres, ellipsoids and cylinders. The discussion focuses on low dimensions, but some results are given for arbitrary (finite) dimensions.

In the third chapter a particular model called the p-peaked Analytically Extended Function (AEF) is introduced and fitted to data taken either from a standard for electromagnetic compatibility or experimental measurements. The discussion here is entirely focused on currents originating from lightning or electrostatic discharges.

The fourth chapter consists of a comparison of several different methods for modelling mortality rates, including a model constructed in a similar way to the AEF found in the third chapter. The models are compared with respect to how well they can be fitted to estimated mortality rate for several countries and several years and the results when using the fitted models for mortality rate forecasting is also compared.

ISBN 978-91-7485-431-2ISSN 1651-4238

444

Acknowledgements

Many thanks to all my coauthors and supervisors. My main supervisor, Pro-fessor Sergei Silvestrov, introduced me to the Vandermonde matrix and fre-quently suggested new problems and research directions throughout my timeas a doctoral student. I have learned many lessons about mathematics andacademia from him and my co-supervisor Professor Anatoliy Malyarenko.My other co-supervisor Dr. Milica Rancic played a crucial role and she isa role model with regards to conscientiousness, work ethic, communicationand patience. I have learned invaluable lessons about interdisciplinary re-search, communication and time and resource management from her. I alsowant to thank Dr. Vesna Javor for her regular input that improved theresearch on electromagnetic compatibility considerably.

Cooperating with other doctoral students was very valuable. JonasOsterberg and Asaph Keikara Muhumuza (with support from his super-visors Dr. John M. Mango and Dr. Godwin Kakuba) made importantcontributions to the research on the Vandermonde determinant and SamyaSuleiman’s understanding of mortality rate forecasting and other aspects ofactuarial mathematics was necessary for the work to progress.

I am also glad that I had the opportunity to take part in the supervisionof talented master students Andromachi Boulogari and Belinda Strass anduse the foundations they laid in their degree projects for further research.

Many thanks to all my coworkers at Malardalen University, especiallyto Dr. Christopher Engstrom, Dr. Johan Richter and Docent Linus Carls-son for managing the bachelor’s and master’s programmes in Engineeringmathematics together with me.

Perhaps most importantly, I thank my family for all the support, en-couragement and assistance you have given me. A special mention to mysister for help with translating from 18th century French, it is perfectly un-derstandable that you decided to move to the other side of the Earth afterthat. I will wonder my whole life how my father, whose entire mathematicscareer consisted of unsuccessfully solving a single problem on the blackboardin 9th grade, would have reacted to this dissertation if he were still with us.Fortunately my mother continues to be an endless source of support andencouragement. I am continually surprised and delighted over how much ofher work ethics, sense of quality and unhealthy work habits I seem to haveinherited from her.

Without the ideas, requests, remarks, questions, encouragements andpatience of those around me this work would not have been completed.

Karl Lundengard Vasteras, September, 2019

3

Acknowledgements

Many thanks to all my coauthors and supervisors. My main supervisor, Pro-fessor Sergei Silvestrov, introduced me to the Vandermonde matrix and fre-quently suggested new problems and research directions throughout my timeas a doctoral student. I have learned many lessons about mathematics andacademia from him and my co-supervisor Professor Anatoliy Malyarenko.My other co-supervisor Dr. Milica Rancic played a crucial role and she isa role model with regards to conscientiousness, work ethic, communicationand patience. I have learned invaluable lessons about interdisciplinary re-search, communication and time and resource management from her. I alsowant to thank Dr. Vesna Javor for her regular input that improved theresearch on electromagnetic compatibility considerably.

Cooperating with other doctoral students was very valuable. JonasOsterberg and Asaph Keikara Muhumuza (with support from his super-visors Dr. John M. Mango and Dr. Godwin Kakuba) made importantcontributions to the research on the Vandermonde determinant and SamyaSuleiman’s understanding of mortality rate forecasting and other aspects ofactuarial mathematics was necessary for the work to progress.

I am also glad that I had the opportunity to take part in the supervisionof talented master students Andromachi Boulogari and Belinda Strass anduse the foundations they laid in their degree projects for further research.

Many thanks to all my coworkers at Malardalen University, especiallyto Dr. Christopher Engstrom, Dr. Johan Richter and Docent Linus Carls-son for managing the bachelor’s and master’s programmes in Engineeringmathematics together with me.

Perhaps most importantly, I thank my family for all the support, en-couragement and assistance you have given me. A special mention to mysister for help with translating from 18th century French, it is perfectly un-derstandable that you decided to move to the other side of the Earth afterthat. I will wonder my whole life how my father, whose entire mathematicscareer consisted of unsuccessfully solving a single problem on the blackboardin 9th grade, would have reacted to this dissertation if he were still with us.Fortunately my mother continues to be an endless source of support andencouragement. I am continually surprised and delighted over how much ofher work ethics, sense of quality and unhealthy work habits I seem to haveinherited from her.

Without the ideas, requests, remarks, questions, encouragements andpatience of those around me this work would not have been completed.

Karl Lundengard Vasteras, September, 2019

3

5

Extreme points of the Vandermonde determinant andphenomenological modelling with power-exponential functions

Popularvetenskaplig sammanfattning

Det finns manga foreteelser i varlden som det ar onskvart att beskriva meden matematisk modell. I basta fall kan modellen harledas ifran lampliggrundlaggande teori men ibland ar det inte mojligt att gora det, antingendarfor att det inte finns nagon val utvecklad teori eller for att den teori somfinns kraver information som inte ar tillganglig. I detta fall sa behovs enmodell som, i nagon man, stammer overens med teori och empiriska observa-tioner men som inte ar harledd fran den grundlaggande teorin. Sadana mod-eller kallas for fenomenologiska modeller. I denna avhandling konstruerasfenomenologiska modeller av tva olika fenomen, strommen i elektrostatiskaurladdningar och dodsrisk.

Elektrostatiska urladdningar sker nar laddning snabbt flodar fran ettobjekt till ett annat. Valbekanta exempel ar blixtnedslag eller sma stotarorsakade av statisk elektricitet. For ingenjorer ar det viktigt att kunnabeskriva denna typ av elektriska strommar for att se till att elektroniskasystem inte ar for kansliga for elektromagnetisk paverkan utifran och att deinte stor andra system da de anvands.

Dodsrisken beskriver sannolikheten for dod vid en viss alder. Den kananvandas for att uppskatta livskvaliteten i ett land eller andra demografiskaeller forsakringsrelaterade andamal.

En egenskap hos bade elektrostatiska urladdningar och dodsrisk somkan vara utmanande att modellera ar omraden dar en brant okning foljsav en langsam sankning. Sadana monster forekommer ofta i elektrostatiskaurladdningar och i manga lander okar dodsrisken kraftigt vid overgangenfran barn till vuxen och forandras sedan langsamt fram till tidig medelalder.

I denna avhandling anvands en matematisk funktion som kallas poten-sexponentialfunktionen som en byggsten for att konstruera fenomenologiskamodeller av strommen i elektrostatiska urladdningar samt dodsrisk utifranempiriska data for respektive fenomen. For elektrostatiska urladdningarforeslas en metod som kan konstruera modeller med olika noggrannhet ochkomplexitet. For dodsrisker foreslas nagra enkla modeller som sedan jamforsmed tidigare foreslagna modeller.

I avhandlingen diskuteras ocksa extrempunkterna hos Vandermonde de-terminanten. Detta ar ett matematiskt problem som forekommer inom fleraolika omraden men for avhandlingen ar den mest relevanta tillampningenatt extrempunkterna kan hjalpa till att valja lampliga data att anvanda narman konstruerar modeller med hjalp av en teknik som kallas for optimaldesign. Nagra allmanna resultat for hur extrempunkterna kan hittas pa di-verse ytor, t.ex. sfarer och kuber, presenteras och det ges exempel pa hurresultaten kan tillampas.

4


Popularvetenskaplig sammanfattning

Det finns manga foreteelser i varlden som det ar onskvart att beskriva meden matematisk modell. I basta fall kan modellen harledas ifran lampliggrundlaggande teori men ibland ar det inte mojligt att gora det, antingendarfor att det inte finns nagon val utvecklad teori eller for att den teori somfinns kraver information som inte ar tillganglig. I detta fall sa behovs enmodell som, i nagon man, stammer overens med teori och empiriska observa-tioner men som inte ar harledd fran den grundlaggande teorin. Sadana mod-eller kallas for fenomenologiska modeller. I denna avhandling konstruerasfenomenologiska modeller av tva olika fenomen, strommen i elektrostatiskaurladdningar och dodsrisk.

Elektrostatiska urladdningar sker nar laddning snabbt flodar fran ettobjekt till ett annat. Valbekanta exempel ar blixtnedslag eller sma stotarorsakade av statisk elektricitet. For ingenjorer ar det viktigt att kunnabeskriva denna typ av elektriska strommar for att se till att elektroniskasystem inte ar for kansliga for elektromagnetisk paverkan utifran och att deinte stor andra system da de anvands.

Dodsrisken beskriver sannolikheten for dod vid en viss alder. Den kananvandas for att uppskatta livskvaliteten i ett land eller andra demografiskaeller forsakringsrelaterade andamal.

En egenskap hos bade elektrostatiska urladdningar och dodsrisk somkan vara utmanande att modellera ar omraden dar en brant okning foljsav en langsam sankning. Sadana monster forekommer ofta i elektrostatiskaurladdningar och i manga lander okar dodsrisken kraftigt vid overgangenfran barn till vuxen och forandras sedan langsamt fram till tidig medelalder.

I denna avhandling anvands en matematisk funktion som kallas poten-sexponentialfunktionen som en byggsten for att konstruera fenomenologiskamodeller av strommen i elektrostatiska urladdningar samt dodsrisk utifranempiriska data for respektive fenomen. For elektrostatiska urladdningarforeslas en metod som kan konstruera modeller med olika noggrannhet ochkomplexitet. For dodsrisker foreslas nagra enkla modeller som sedan jamforsmed tidigare foreslagna modeller.

I avhandlingen diskuteras ocksa extrempunkterna hos Vandermonde de-terminanten. Detta ar ett matematiskt problem som forekommer inom fleraolika omraden men for avhandlingen ar den mest relevanta tillampningenatt extrempunkterna kan hjalpa till att valja lampliga data att anvanda narman konstruerar modeller med hjalp av en teknik som kallas for optimaldesign. Nagra allmanna resultat for hur extrempunkterna kan hittas pa di-verse ytor, t.ex. sfarer och kuber, presenteras och det ges exempel pa hurresultaten kan tillampas.

4

6

Popular science summary

There are many phenomena in the world that it is desirable to describe usinga mathematical model. Ideally the mathematical model is derived from theappropriate fundamental theory but sometimes this is not feasible, eitherbecause the fundamental theory is not well understood or because the theoryrequires a lot of information to be applicable. In these cases it is necessaryto create a model that, to some degree, matches the fundamental theoryand the empirical observations, but is not derived from the fundamentaltheory. Such models are called phenomenological models. In this thesisphenomenological models are constructed for two phenomena, electrostaticdischarge currents and mortality rates.

Electrostatic discharge currents are rapid flows of electric charge fromone object to another. Well-known examples are lightning strikes or smallelectric chocks caused by static electricity. Describing such currents is im-portant when engineers want to ensure that electronic systems are not dis-turbed too much by external electromagnetic disturbances or disturbs othersystems when used.

Mortality rate describes the probability of a dying at certain age. It canbe used to assess the quality of life in a country or for other demographicalor actuarial purposes.

For electrostatic discharge currents and mortality rates an importantfeature that can be challenging to model is a steep increase followed by aslower decrease. This pattern is often observed in electrostatic dischargecurrents and in many countries the mortality rate increases rapidly in thetransition from childhood to adulthood and then changes slowly until thebeginning of middle age.

In this thesis a mathematical function called the power-exponential func-tion is used as a building block to construct phenomenological models ofelectrostatic discharge currents and mortality rates based on empirical datafor the respective phenomena. For electrostatic discharge currents a method-ology for constructing models with different accuracy and complexity is pro-posed. For the mortality rates a few simple models are suggested and com-pared to previously suggested models.

The thesis also discusses the extreme points of the Vandermonde deter-minant. This is a mathematical problem that appears in many areas butfor this thesis the most relevant application is that it helps choosing theappropriate data to use when constructing a model using a technique calledoptimal design. Some general results for finding the extreme points of theVandermonde determinant on various surfaces, e.g. spheres or cubes, andapplications of these results are discussed.

5

Popular science summary

There are many phenomena in the world that it is desirable to describe usinga mathematical model. Ideally the mathematical model is derived from theappropriate fundamental theory but sometimes this is not feasible, eitherbecause the fundamental theory is not well understood or because the theoryrequires a lot of information to be applicable. In these cases it is necessaryto create a model that, to some degree, matches the fundamental theoryand the empirical observations, but is not derived from the fundamentaltheory. Such models are called phenomenological models. In this thesisphenomenological models are constructed for two phenomena, electrostaticdischarge currents and mortality rates.

Electrostatic discharge currents are rapid flows of electric charge fromone object to another. Well-known examples are lightning strikes or smallelectric chocks caused by static electricity. Describing such currents is im-portant when engineers want to ensure that electronic systems are not dis-turbed too much by external electromagnetic disturbances or disturbs othersystems when used.

Mortality rate describes the probability of a dying at certain age. It canbe used to assess the quality of life in a country or for other demographicalor actuarial purposes.

For electrostatic discharge currents and mortality rates an importantfeature that can be challenging to model is a steep increase followed by aslower decrease. This pattern is often observed in electrostatic dischargecurrents and in many countries the mortality rate increases rapidly in thetransition from childhood to adulthood and then changes slowly until thebeginning of middle age.

In this thesis a mathematical function called the power-exponential func-tion is used as a building block to construct phenomenological models ofelectrostatic discharge currents and mortality rates based on empirical datafor the respective phenomena. For electrostatic discharge currents a method-ology for constructing models with different accuracy and complexity is pro-posed. For the mortality rates a few simple models are suggested and com-pared to previously suggested models.

The thesis also discusses the extreme points of the Vandermonde deter-minant. This is a mathematical problem that appears in many areas butfor this thesis the most relevant application is that it helps choosing theappropriate data to use when constructing a model using a technique calledoptimal design. Some general results for finding the extreme points of theVandermonde determinant on various surfaces, e.g. spheres or cubes, andapplications of these results are discussed.

5

7

Notation

Matrix and vector notation

v, M - Bold, roman lower- and uppercase lettersdenote vectors and matrices respectively.

Mi,j - Element on the ith row and jth column of M.

M·,j , Mi,· - Column (row) vector containing all elementsfrom the jth column (ith row) of M.

[aij ]nmij - n×m matrix with element aij in

the ith row and jth column.

Vnm, Vn = Vnn - n×m Vandermonde matrix.

Gnm, Gn = Gnn - n×m generalized Vandermonde matrix.

Anm, An = Ann - n×m alternant matrix.

Standard sets

Z, N, R, C - Sets of all integers, natural numbers (including 0),real numbers and complex numbers.

Snp , Sn = Sn2 - The n-dimensional sphere defined by the p - norm,

Snp (r) =

x ∈ Rn+1

∣∣∣∣ n+1∑k=1

|xk|p = r

.

Ck[K] - All functions on K with continuous kth derivative.

Special functions

Definitions can be found in standard texts.Suggested sources use notation consistent with thesis.

Hn, P(α,β)n - Hermite and Jacobi polynomials, see [2].

Γ(x), γ(x, y), ψ(x) - The Gamma-, incomplete Gamma andDigamma functions, see [2].

2F2(a, b; c;x) - The hypergeometric function, see [2].

Gm,np,q

(z

∣∣∣∣ab)

- The Meijer G-function, see [236].

Ei(x) - The exponential integral, see [2].

6

Notation

Matrix and vector notation

v, M - Bold, roman lower- and uppercase lettersdenote vectors and matrices respectively.

Mi,j - Element on the ith row and jth column of M.

M·,j , Mi,· - Column (row) vector containing all elementsfrom the jth column (ith row) of M.

[aij ]nmij - n×m matrix with element aij in

the ith row and jth column.

Vnm, Vn = Vnn - n×m Vandermonde matrix.

Gnm, Gn = Gnn - n×m generalized Vandermonde matrix.

Anm, An = Ann - n×m alternant matrix.

Standard sets

Z, N, R, C - Sets of all integers, natural numbers (including 0),real numbers and complex numbers.

Snp , Sn = Sn2 - The n-dimensional sphere defined by the p - norm,

Snp (r) =

x ∈ Rn+1

∣∣∣∣ n+1∑k=1

|xk|p = r

.

Ck[K] - All functions on K with continuous kth derivative.

Special functions

Definitions can be found in standard texts.Suggested sources use notation consistent with thesis.

Hn, P(α,β)n - Hermite and Jacobi polynomials, see [2].

Γ(x), γ(x, y), ψ(x) - The Gamma-, incomplete Gamma andDigamma functions, see [2].

2F2(a, b; c;x) - The hypergeometric function, see [2].

Gm,np,q

(z

∣∣∣∣ab)

- The Meijer G-function, see [236].

Ei(x) - The exponential integral, see [2].

6

8

Probability theory and statistics

Pr[A] - Probability of event A.

Pr[A|B] - Conditional probability of event A given B.

EX [Y ] - Expected value of quantity Y with respect to X.

Var(X) - Variance of X.

AIC - Akaike Information Criterion, see Definition 1.14.

AICC - Second order correction of the AIC, see Remark 1.9.

I(f, g) - Kullback–Leibler divergence, see Definition 1.15.

Mortality rate

Sx(∆x) - Survival function, see Definition 1.19.

Tx - Remaining lifetime for an individual of age x.

µ(x) - Mortality rate at age x, see Definition 1.20.

mx,t - Central mortality rate at age x, year t, see page 66.

Other

df

dx= f ′(x) - Derivative of the function f with respect to x.

dkf

dxk= f (k)(x) - kth derivative of the function f with respect to x.

∂f

∂x= f ′(x) - Partial derivative of the function f with respect to x.

ab - Rising factorial ab = a(a+ 1) · · · (a+ b− 1).

7

Probability theory and statistics

Pr[A] - Probability of event A.

Pr[A|B] - Conditional probability of event A given B.

EX [Y ] - Expected value of quantity Y with respect to X.

Var(X) - Variance of X.

AIC - Akaike Information Criterion, see Definition 1.14.

AICC - Second order correction of the AIC, see Remark 1.9.

I(f, g) - Kullback–Leibler divergence, see Definition 1.15.

Mortality rate

Sx(∆x) - Survival function, see Definition 1.19.

Tx - Remaining lifetime for an individual of age x.

µ(x) - Mortality rate at age x, see Definition 1.20.

mx,t - Central mortality rate at age x, year t, see page 66.

Other

df

dx= f ′(x) - Derivative of the function f with respect to x.

dkf

dxk= f (k)(x) - kth derivative of the function f with respect to x.

∂f

∂x= f ′(x) - Partial derivative of the function f with respect to x.

ab - Rising factorial ab = a(a+ 1) · · · (a+ b− 1).

7

9

10

Contents

List of Papers 13

1 Introduction 15

1.1 The Vandermonde matrix . . . . . . . . . . . . . . . . . . . . 19

1.1.1 Who was Vandermonde? . . . . . . . . . . . . . . . . . 19

1.1.2 The Vandermonde determinant . . . . . . . . . . . . . 21

1.1.3 Inverse of the Vandermonde matrix . . . . . . . . . . . 25

1.1.4 The alternant matrix . . . . . . . . . . . . . . . . . . . 26

1.1.5 The generalized Vandermonde matrix . . . . . . . . . 29

1.1.6 The Vandermonde determinant in systems withCoulombian interactions . . . . . . . . . . . . . . . . . 30

1.1.7 The Vandermonde determinant in random matrixtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.2 Curve fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.2.1 Linear interpolation . . . . . . . . . . . . . . . . . . . 37

1.2.2 Generalized divided differences and interpolation . . . 42

1.2.3 Least squares fitting . . . . . . . . . . . . . . . . . . . 45

1.2.4 Linear least squares fitting . . . . . . . . . . . . . . . 45

1.2.5 Non-linear least squares fitting . . . . . . . . . . . . . 46

1.2.6 The Marquardt least squares method . . . . . . . . . . 47

1.3 Analysing how well a curve fits . . . . . . . . . . . . . . . . . 50

1.3.1 Regression . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.3.2 Quantile-Quantile plots . . . . . . . . . . . . . . . . . 52

9

Contents

List of Papers 13

1 Introduction 15

1.1 The Vandermonde matrix . . . . . . . . . . . . . . . . . . . . 19

1.1.1 Who was Vandermonde? . . . . . . . . . . . . . . . . . 19

1.1.2 The Vandermonde determinant . . . . . . . . . . . . . 21

1.1.3 Inverse of the Vandermonde matrix . . . . . . . . . . . 25

1.1.4 The alternant matrix . . . . . . . . . . . . . . . . . . . 26

1.1.5 The generalized Vandermonde matrix . . . . . . . . . 29

1.1.6 The Vandermonde determinant in systems withCoulombian interactions . . . . . . . . . . . . . . . . . 30

1.1.7 The Vandermonde determinant in random matrixtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.2 Curve fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.2.1 Linear interpolation . . . . . . . . . . . . . . . . . . . 37

1.2.2 Generalized divided differences and interpolation . . . 42

1.2.3 Least squares fitting . . . . . . . . . . . . . . . . . . . 45

1.2.4 Linear least squares fitting . . . . . . . . . . . . . . . 45

1.2.5 Non-linear least squares fitting . . . . . . . . . . . . . 46

1.2.6 The Marquardt least squares method . . . . . . . . . . 47

1.3 Analysing how well a curve fits . . . . . . . . . . . . . . . . . 50

1.3.1 Regression . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.3.2 Quantile-Quantile plots . . . . . . . . . . . . . . . . . 52

9

11


1.3.3 The Akaike information criterion . . . . . . . . . . . . 53

1.4 D-optimal experiment design . . . . . . . . . . . . . . . . . . 57

1.5 Electromagnetic compatibility andelectrostatic discharge currents . . . . . . . . . . . . . . . . . 60

1.5.1 Electrostatic discharge modelling . . . . . . . . . . . . 62

1.6 Modelling mortality rates . . . . . . . . . . . . . . . . . . . . 65

1.6.1 Lee–Carter method for forecasting . . . . . . . . . . . 68

1.7 Summaries of papers . . . . . . . . . . . . . . . . . . . . . . . 71

2 Extreme points of the Vandermonde determinant 75

2.1 Extreme points of the Vandermonde determinant and relateddeterminants on various surfaces in three dimensions . . . . . 77

2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions . . . . . . . . . . . . . . . 77

2.1.2 Extreme points of the Vandermonde determinant onthe three-dimensional unit sphere . . . . . . . . . . . . 81

2.1.3 Optimisation using Grobner bases . . . . . . . . . . . 82

2.1.4 Extreme points on the ellipsoid in three dimensions . 83

2.1.5 Extreme points on the cylinder in three dimensions . . 85

2.1.6 Optimizing the Vandermonde determinant on a sur-face defined by a homogeneous polynomial . . . . . . . 87

2.2 Extreme points of the Vandermonde determinant on the sphere 89

2.2.1 The extreme points on the sphere given by roots of apolynomial . . . . . . . . . . . . . . . . . . . . . . . . 89

2.2.2 Further visual exploration on the sphere . . . . . . . . 96

2.3 Extreme points of the Vandermonde determinant on somesurfaces implicitly defined by a univariate polynomial . . . . 103

2.3.1 Critical points on surfaces given by a first degree uni-variate polynomial . . . . . . . . . . . . . . . . . . . . 104

2.3.2 Critical points on surfaces given by a second degreeunivariate polynomial . . . . . . . . . . . . . . . . . . 105

2.3.3 Critical points on the sphere defined by a p-norm . . . 107

2.3.4 The case p = 4 and n = 4 . . . . . . . . . . . . . . . . 107

10


1.3.3 The Akaike information criterion . . . . . . . . . . . . 53

1.4 D-optimal experiment design . . . . . . . . . . . . . . . . . . 57

1.5 Electromagnetic compatibility andelectrostatic discharge currents . . . . . . . . . . . . . . . . . 60

1.5.1 Electrostatic discharge modelling . . . . . . . . . . . . 62

1.6 Modelling mortality rates . . . . . . . . . . . . . . . . . . . . 65

1.6.1 Lee–Carter method for forecasting . . . . . . . . . . . 68

1.7 Summaries of papers . . . . . . . . . . . . . . . . . . . . . . . 71

2 Extreme points of the Vandermonde determinant 75

2.1 Extreme points of the Vandermonde determinant and relateddeterminants on various surfaces in three dimensions . . . . . 77

2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions . . . . . . . . . . . . . . . 77

2.1.2 Extreme points of the Vandermonde determinant onthe three-dimensional unit sphere . . . . . . . . . . . . 81

2.1.3 Optimisation using Grobner bases . . . . . . . . . . . 82

2.1.4 Extreme points on the ellipsoid in three dimensions . 83

2.1.5 Extreme points on the cylinder in three dimensions . . 85

2.1.6 Optimizing the Vandermonde determinant on a sur-face defined by a homogeneous polynomial . . . . . . . 87

2.2 Extreme points of the Vandermonde determinant on the sphere 89

2.2.1 The extreme points on the sphere given by roots of apolynomial . . . . . . . . . . . . . . . . . . . . . . . . 89

2.2.2 Further visual exploration on the sphere . . . . . . . . 96

2.3 Extreme points of the Vandermonde determinant on somesurfaces implicitly defined by a univariate polynomial . . . . 103

2.3.1 Critical points on surfaces given by a first degree uni-variate polynomial . . . . . . . . . . . . . . . . . . . . 104

2.3.2 Critical points on surfaces given by a second degreeunivariate polynomial . . . . . . . . . . . . . . . . . . 105

2.3.3 Critical points on the sphere defined by a p-norm . . . 107

2.3.4 The case p = 4 and n = 4 . . . . . . . . . . . . . . . . 107

10

12

CONTENTS

2.3.5 Some results for even n and p . . . . . . . . . . . . . . 110

2.3.6 Some results for cubes and intersections of planes . . . 118

2.3.7 Optimising the probability density function of theeigenvalues of the Wishart matrix . . . . . . . . . . . 120

3 Approximation of electrostatic discharge currents using theanalytically extended function 123

3.1 The analytically extended function (AEF) . . . . . . . . . . . 125

3.1.1 The p-peak analytically extended function . . . . . . . 126

3.2 Approximation of lightning discharge current functions . . . . 133

3.2.1 Fitting the AEF . . . . . . . . . . . . . . . . . . . . . 133

3.2.2 Estimating parameters for underdetermined systems . 134

3.2.3 Fitting with data points as well as charge flow andspecific energy conditions . . . . . . . . . . . . . . . . 135

3.2.4 Calculating the η-parameters from the β-parameters . 138

3.2.5 Explicit formulas for a single-peak AEF . . . . . . . . 139

3.2.6 Fitting to lightning discharge currents . . . . . . . . . 140

3.3 Approximation of electrostatic discharge currents using theAEF by interpolation on a D-optimal design . . . . . . . . . 143

3.3.1 D-optimal approximation for exponents given by aclass of arithmetic sequences . . . . . . . . . . . . . . 145

3.3.2 D-optimal interpolation on the rising part . . . . . . . 146

3.3.3 D-optimal interpolation on the decaying part . . . . . 148

3.3.4 Examples of models from applications and experiments 150

3.3.5 Modelling of ESD currents . . . . . . . . . . . . . . . 150

3.3.6 Modelling of lightning discharge currents . . . . . . . 152

3.3.7 Summary of ESD modelling . . . . . . . . . . . . . . . 159

4 Comparison of models of mortality rate 161

4.1 Modelling and forecasting mortality rates . . . . . . . . . . . 162

4.2 Overview of models . . . . . . . . . . . . . . . . . . . . . . . . 162

4.3 Power-exponential mortality rate models . . . . . . . . . . . . 164

11

CONTENTS

2.3.5 Some results for even n and p . . . . . . . . . . . . . . 110

2.3.6 Some results for cubes and intersections of planes . . . 118

2.3.7 Optimising the probability density function of theeigenvalues of the Wishart matrix . . . . . . . . . . . 120

3 Approximation of electrostatic discharge currents using theanalytically extended function 123

3.1 The analytically extended function (AEF) . . . . . . . . . . . 125

3.1.1 The p-peak analytically extended function . . . . . . . 126

3.2 Approximation of lightning discharge current functions . . . . 133

3.2.1 Fitting the AEF . . . . . . . . . . . . . . . . . . . . . 133

3.2.2 Estimating parameters for underdetermined systems . 134

3.2.3 Fitting with data points as well as charge flow andspecific energy conditions . . . . . . . . . . . . . . . . 135

3.2.4 Calculating the η-parameters from the β-parameters . 138

3.2.5 Explicit formulas for a single-peak AEF . . . . . . . . 139

3.2.6 Fitting to lightning discharge currents . . . . . . . . . 140

3.3 Approximation of electrostatic discharge currents using theAEF by interpolation on a D-optimal design . . . . . . . . . 143

3.3.1 D-optimal approximation for exponents given by aclass of arithmetic sequences . . . . . . . . . . . . . . 145

3.3.2 D-optimal interpolation on the rising part . . . . . . . 146

3.3.3 D-optimal interpolation on the decaying part . . . . . 148

3.3.4 Examples of models from applications and experiments 150

3.3.5 Modelling of ESD currents . . . . . . . . . . . . . . . 150

3.3.6 Modelling of lightning discharge currents . . . . . . . 152

3.3.7 Summary of ESD modelling . . . . . . . . . . . . . . . 159

4 Comparison of models of mortality rate 161

4.1 Modelling and forecasting mortality rates . . . . . . . . . . . 162

4.2 Overview of models . . . . . . . . . . . . . . . . . . . . . . . . 162

4.3 Power-exponential mortality rate models . . . . . . . . . . . . 164

11

13


4.3.1 Multiple humps . . . . . . . . . . . . . . . . . . . . . . 165

4.3.2 Single hump model . . . . . . . . . . . . . . . . . . . . 165

4.3.3 Split power-exponential model . . . . . . . . . . . . . 166

4.3.4 Adjusted power-exponential model . . . . . . . . . . . 166

4.4 Fitting and comparing models . . . . . . . . . . . . . . . . . . 167

4.4.1 Some comments on fitting . . . . . . . . . . . . . . . . 168

4.4.2 Results and discussion . . . . . . . . . . . . . . . . . . 174

4.5 Comparison of parametric models appliedto mortality rate forecasting . . . . . . . . . . . . . . . . . . . 178

4.5.1 Comparison of models . . . . . . . . . . . . . . . . . . 180

4.5.2 Results, discussion and further work . . . . . . . . . . 180

References 185

Index 209

List of Figures 211

List of Tables 215

List of Definitions 216

List of Theorems 217

List of Lemmas 218

12


4.3.1 Multiple humps . . . . . . . . . . . . . . . . . . . . . . 165

4.3.2 Single hump model . . . . . . . . . . . . . . . . . . . . 165

4.3.3 Split power-exponential model . . . . . . . . . . . . . 166

4.3.4 Adjusted power-exponential model . . . . . . . . . . . 166

4.4 Fitting and comparing models . . . . . . . . . . . . . . . . . . 167

4.4.1 Some comments on fitting . . . . . . . . . . . . . . . . 168

4.4.2 Results and discussion . . . . . . . . . . . . . . . . . . 174

4.5 Comparison of parametric models appliedto mortality rate forecasting . . . . . . . . . . . . . . . . . . . 178

4.5.1 Comparison of models . . . . . . . . . . . . . . . . . . 180

4.5.2 Results, discussion and further work . . . . . . . . . . 180

References 185

Index 209

List of Figures 211

List of Tables 215

List of Definitions 216

List of Theorems 217

List of Lemmas 218

12

14

List of Papers

Paper A Karl Lundengard, Jonas Osterberg and Sergei Silvestrov.Extreme points of the Vandermonde determinant on the sphere andsome limits involving the generalized Vandermonde determinant.Accepted for publication in Algebraic structures and Applications.SPAS2017, Vasteras and Stockholm, Sweden, October 4 – 6, 2017,Sergei Silvestrov, Anatoliy Malyarenko, Milica Rancic (Eds),Springer International Publishing, 2019.

Paper B Karl Lundengard, Jonas Osterberg and Sergei Silvestrov.Optimization of the determinant of the Vandermonde matrixon the sphere and related surfaces.Methodology and Computing in Applied Probability, Volume 20,Issue 4, pages 1417 – 1428, 2018.

Paper C Asaph Keikara Muhumuza, Karl Lundengard, Jonas Osterberg,Sergei Silvestrov, John Magero Mango and Godwin Kakuba.Extreme points of the Vandermonde determinant on surfacesimplicitly determined by a univariate polynomial.Accepted for publication in Algebraic structures and Applications.SPAS2017, Vasteras and Stockholm, Sweden, October 4 – 6, 2017,Sergei Silvestrov, Anatoliy Malyarenko, Milica Rancic (Eds),Springer International Publishing, 2019.

Paper D Asaph Keikara Muhumuza, Karl Lundengard, Jonas Osterberg,Sergei Silvestrov, John Magero Mango and Godwin Kakuba.Optimization of the Wishart joint eigenvalue probability densitydistribution based on the Vandermonde determinant.Accepted for publication in Algebraic structures and Applications.SPAS2017, Vasteras and Stockholm, Sweden, October 4 – 6, 2017,Sergei Silvestrov, Anatoliy Malyarenko, Milica Rancic (Eds),Springer International Publishing, 2019.

Paper E Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.On some properties of the multi-peaked analytically extendedfunction for approximation of lightning discharge currents.Chapter 10 in Engineering Mathematics I: Electromagnetics, FluidMechanics, Material Physics and Financial Engineering,Volume 178 of Springer Proceedings in Mathematics & Statistics,Sergei Silvestrov and Milica Rancic (Eds),Springer International Publishing, pages 151–176, 2016.

List of Papers






15

Paper F Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.Estimation of parameters for the multi-peaked AEF currentfunctions.Methodology and Computing in Applied Probability, Volume 19,Issue 4, pages 1107 – 1121, 2017.

Paper G Karl Lundengard, Milica Rancic, Vesna Javor and Sergei Silvestrov.Electrostatic discharge currents representation using theanalytically extended function with p peaks by interpolation on aD-optimal design.Facta Universitatis Series: Electronics and Energetics, Volume 32,Issue 1, pages 25 – 49, 2019.

Paper H Karl Lundengard, Milica Rancic and Sergei Silvestrov.Modelling mortality rates using power-exponential functions.Submitted to journal, 2019.

Paper I Andromachi Boulougari, Karl Lundengard, Milica Rancic,

Sergei Silvestrov, Belinda Strass and Samya Suleiman.

Application of a power-exponential function based model to

mortality rates forecasting.

Communications in Statistics: Case Studies, Data Analysis and

Applications, Volume 5, Issue 1, pages 3 – 10, 2019.

Parts of the thesis have been presented at the following international conferences:

• ASMDA 2015 - 16th Applied Stochastic Models and Data Analysis In-ternational Conference with 4th Demographics 2015 Workshop, Piraeus,Greece, June 30 – July 4, 2015.

• SPLITECH 2017 - 2nd International Multidisciplinary Conference onComputer and Energy Science, Split, Croatia, July 12 – 14, 2017.

• EMC+SIPI 2017 - IEEE International Symposium on ElectromagneticCompatibility, Signal and Power Integrity, Washington DC, USA, August7 – 11, 2017.

• SPAS 2017 - International Conference on Stochastic Processes and Alge-braic Structures, Vasteras, Sweden, October 4 – 6, 2017.

• SMTDA 2018 - 5th Stochastic Modelling Techniques and Data AnalysisInternational Conference, Chania, Crete, Greece, June 12 – 15, 2018.

• IWAP 2018 - 9th International Workshop on Applied Probability, Bu-

dapest, Hungary, June 18–21, 2018.

Summaries of papers A-I with a brief description of the thesis authors contributions

to each paper can be found in Section 1.7.

14










Parts of the thesis have been presented at the following international conferences:

• ASMDA 2015 - 16th Applied Stochastic Models and Data Analysis In-ternational Conference with 4th Demographics 2015 Workshop, Piraeus,Greece, June 30 – July 4, 2015.

• SPLITECH 2017 - 2nd International Multidisciplinary Conference onComputer and Energy Science, Split, Croatia, July 12 – 14, 2017.

• EMC+SIPI 2017 - IEEE International Symposium on ElectromagneticCompatibility, Signal and Power Integrity, Washington DC, USA, August7 – 11, 2017.

• SPAS 2017 - International Conference on Stochastic Processes and Alge-braic Structures, Vasteras, Sweden, October 4 – 6, 2017.

• SMTDA 2018 - 5th Stochastic Modelling Techniques and Data AnalysisInternational Conference, Chania, Crete, Greece, June 12 – 15, 2018.

• IWAP 2018 - 9th International Workshop on Applied Probability, Bu-

dapest, Hungary, June 18–21, 2018.

Summaries of papers A-I with a brief description of the thesis authors contributions

to each paper can be found in Section 1.7.

14

16

Chapter 1

Introduction

This chapter is partially based on Papers D, E, H, and I










Chapter 1

Introduction

This chapter is partially based on Papers D, E, H, and I










17

18

INTRODUCTION

Two topics are discussed in this thesis, finding the extreme points of theVandermonde determinant and phenomenological modelling using power-exponential functions. Several of the methods and approaches that arediscussed are also applied to modelling of electrical current for use in elec-tromagnetic compatibility, or to modelling of mortality rate of humans foractuarial or demographical purposes. The topics are related since the ex-treme points of the Vandermonde determinant is relevant for certain curvefitting problems that can appear in the construction of the phenomenologi-cal models. An overview of the major relations between the different partsof the thesis are illustrated in Figure 1.1. The relations are of many kinds,common definitions and dependent results, conceptual connections as wellas similarities in proof techniques and problem formulations.

This thesis is based on the nine papers listed on pages 13–14. Thecontents of the papers have been rearranged (and in some cases parts havebeen omitted) to avoid repetition and improve cohesion, but the original textand structure of the papers have been largely preserved. Significant partsof Chapters 1-3 have also appeared in [180]. If a section is based on a paperthis is specified at the beginning of the section and unless otherwise specifiedany subsections are from the same source. A section that is based on a papercontains text from the paper that is unchanged except for modifications tocorrect misprints and ensure consistency within the thesis.

Chapter 1 introduces concepts used in later chapters. The Vandermondematrix, its history, applications, generalizations and some of its proper-ties are introduced in Section 1.1. Section 1.2 discusses a few different ap-proaches to curve fitting. Section 1.3 discusses a few methods for evaluatingthe result. Basic optimal design is discussed in Section 1.4. Sections 1.5 and1.6 introduce electromagnetic compatibility and mortality rate modelling.

Chapter 2 discusses the optimisation of the Vandermonde determinantover various surfaces. First the extreme points on a few different surfaces inthree dimensions are examined, see Section 2.1. In Section 2.2 the determi-nant is optimised on the sphere in higher dimensions and some results forsurfaces defined by a univariate polynomial are discussed in Section 2.3.

Chapter 3 discusses fitting a piecewise non-linear regression model todata. The particular model is introduced in Section 3.1 and a general frame-work for fitting it to data using the Marquardt least squares method is de-scribed in Sections 3.2.1–3.2.5. The framework is then applied to lightningdischarge currents in Section 3.2.6. An alternate curve fitting method basedon D-optimal interpolation (found analogously to the results in Section 2.2)is described and applied to electrostatic discharge currents in Section 3.3.

Chapter 4 compares several different mathematical models of mortalityrate for humans. The comparison is done by fitting the models to centralmortality rates from several different countries and then analysing how wellthe model fits and what happens when the results of the fitting is used formortality rate forecasting (using the so called Lee–Carter method).

17

INTRODUCTION

Two topics are discussed in this thesis, finding the extreme points of theVandermonde determinant and phenomenological modelling using power-exponential functions. Several of the methods and approaches that arediscussed are also applied to modelling of electrical current for use in elec-tromagnetic compatibility, or to modelling of mortality rate of humans foractuarial or demographical purposes. The topics are related since the ex-treme points of the Vandermonde determinant is relevant for certain curvefitting problems that can appear in the construction of the phenomenologi-cal models. An overview of the major relations between the different partsof the thesis are illustrated in Figure 1.1. The relations are of many kinds,common definitions and dependent results, conceptual connections as wellas similarities in proof techniques and problem formulations.

This thesis is based on the nine papers listed on pages 13–14. Thecontents of the papers have been rearranged (and in some cases parts havebeen omitted) to avoid repetition and improve cohesion, but the original textand structure of the papers have been largely preserved. Significant partsof Chapters 1-3 have also appeared in [180]. If a section is based on a paperthis is specified at the beginning of the section and unless otherwise specifiedany subsections are from the same source. A section that is based on a papercontains text from the paper that is unchanged except for modifications tocorrect misprints and ensure consistency within the thesis.

Chapter 1 introduces concepts used in later chapters. The Vandermondematrix, its history, applications, generalizations and some of its proper-ties are introduced in Section 1.1. Section 1.2 discusses a few different ap-proaches to curve fitting. Section 1.3 discusses a few methods for evaluatingthe result. Basic optimal design is discussed in Section 1.4. Sections 1.5 and1.6 introduce electromagnetic compatibility and mortality rate modelling.

Chapter 2 discusses the optimisation of the Vandermonde determinantover various surfaces. First the extreme points on a few different surfaces inthree dimensions are examined, see Section 2.1. In Section 2.2 the determi-nant is optimised on the sphere in higher dimensions and some results forsurfaces defined by a univariate polynomial are discussed in Section 2.3.

Chapter 3 discusses fitting a piecewise non-linear regression model todata. The particular model is introduced in Section 3.1 and a general frame-work for fitting it to data using the Marquardt least squares method is de-scribed in Sections 3.2.1–3.2.5. The framework is then applied to lightningdischarge currents in Section 3.2.6. An alternate curve fitting method basedon D-optimal interpolation (found analogously to the results in Section 2.2)is described and applied to electrostatic discharge currents in Section 3.3.

Chapter 4 compares several different mathematical models of mortalityrate for humans. The comparison is done by fitting the models to centralmortality rates from several different countries and then analysing how wellthe model fits and what happens when the results of the fitting is used formortality rate forecasting (using the so called Lee–Carter method).

17

19


Curve fitting

Linear interpolationSection 1.2.1

Least squaresmethod

Section 1.2.3

Non-linearleast squares fitting

Section 1.2.5

D-optimal designSection 1.4

Linearleast squares fitting

Section 1.2.4

The Marquardtleast squares method

Section 1.2.6

Extreme points ofthe Vandermonde

determinant

Vandermonde matrixSection 1.1

Extreme points onvarious surfaces in 3D

Section 2.1

Optimizationon a sphereSection 2.2

Optimization on asurface defined by a

univariate polynomialSection 2.3

Phenomenological modelling withpower-exponential functions

Power exponential function

ElectromagneticcompatibilitySection 1.5

The AnalyticallyExtended Function

Section 3.1

Lightning dischargecurrent modelling

Section 3.2

Interpolation ona D-optimal design

Section 3.3

Evaluation ofcurve fit

Section 1.3

Mortality ratemodellingSection 1.6

Mortality ratemodels fitted

to dataSection 4.1

Mortality ratemodels appliedto forecasting

Section 4.5

Figure 1.1: Illustration of the most significant connections in the thesis.

18


Curve fitting

Linear interpolationSection 1.2.1

Least squaresmethod

Section 1.2.3

Non-linearleast squares fitting

Section 1.2.5

D-optimal designSection 1.4

Linearleast squares fitting

Section 1.2.4

The Marquardtleast squares method

Section 1.2.6

Extreme points ofthe Vandermonde

determinant

Vandermonde matrixSection 1.1

Extreme points onvarious surfaces in 3D

Section 2.1

Optimizationon a sphereSection 2.2

Optimization on asurface defined by a

univariate polynomialSection 2.3

Phenomenological modelling withpower-exponential functions

Power exponential function

ElectromagneticcompatibilitySection 1.5

The AnalyticallyExtended Function

Section 3.1

Lightning dischargecurrent modelling

Section 3.2

Interpolation ona D-optimal design

Section 3.3

Evaluation ofcurve fit

Section 1.3

Mortality ratemodellingSection 1.6

Mortality ratemodels fitted

to dataSection 4.1

Mortality ratemodels appliedto forecasting

Section 4.5

Figure 1.1: Illustration of the most significant connections in the thesis.

18

20

1.1. THE VANDERMONDE MATRIX

1.1 The Vandermonde matrix

The Vandermonde matrix is a well-known matrix with a very special formthat appears in many different circumstances, a few examples are polynomialinterpolation (see Sections 1.2.1 and 1.2.2), least squares curve fitting (seeSection 1.2.3), optimal experiment design (see Section 1.4), constructionof error-detecting and error-correcting codes (see [31, 124, 242] as well asmore recent work such as [28]), determining if a market with a finite set oftraded assets is complete [62], calculation of the discrete Fourier transform[241] and related transforms such as the fractional discrete Fourier transform[215], the quantum Fourier transform [70], and the Vandermonde transform[11, 12], solving systems of differential equations with constant coefficients[213], various problems in mathematical physics [283], nuclear physics [51],and quantum physics [249, 271], systems of Coulombian interactions (seeSection 1.1.6) and describing properties of the Fisher information matrix ofstationary stochastic processes [158] and in various places in random matrixtheory (see Sections 1.1.7 and 2.3.7).

In this section we will review some of the basic properties of the Van-dermonde matrix, starting with its definition.

Definition 1.1. A Vandermonde matrix is an n×m matrix of the form

Vmn(xn) =[xi−1j

]m,ni,j

=

1 1 · · · 1x1 x2 · · · xn...

.... . .

...

xm−11 xm−1

2 · · · xm−1n

(1)

where xi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, the notationVn = Vnm will be used.

Remark 1.1. Note that in the literature the term Vandermonde matrix isoften used for the transpose of the matrix given above.

1.1.1 Who was Vandermonde?

The matrix is named after Alexandre Theophile Vandermonde (1735–1796)who had a varied career that began with law studies and some success asa concert violinist, transitioned into work in science and mathematics inthe beginning of the 1770s that gradually turned into administrative andleadership positions at various Parisian institutions as well as work in politicsand economics in the end of the 1780s [86]. His entire mathematical careerconsisted of four published papers, first presented to the French Academyof Sciences in 1770 and 1771 and published a few years later.

The first paper, Memoire sur la resolution des equations [279], discussessome properties of the roots of polynomial equations, more specifically for-mulas for the sum of the roots and a sum of symmetric functions of the pow-

19


1.1 The Vandermonde matrix

The Vandermonde matrix is a well-known matrix with a very special formthat appears in many different circumstances, a few examples are polynomialinterpolation (see Sections 1.2.1 and 1.2.2), least squares curve fitting (seeSection 1.2.3), optimal experiment design (see Section 1.4), constructionof error-detecting and error-correcting codes (see [31, 124, 242] as well asmore recent work such as [28]), determining if a market with a finite set oftraded assets is complete [62], calculation of the discrete Fourier transform[241] and related transforms such as the fractional discrete Fourier transform[215], the quantum Fourier transform [70], and the Vandermonde transform[11, 12], solving systems of differential equations with constant coefficients[213], various problems in mathematical physics [283], nuclear physics [51],and quantum physics [249, 271], systems of Coulombian interactions (seeSection 1.1.6) and describing properties of the Fisher information matrix ofstationary stochastic processes [158] and in various places in random matrixtheory (see Sections 1.1.7 and 2.3.7).

In this section we will review some of the basic properties of the Van-dermonde matrix, starting with its definition.

Definition 1.1. A Vandermonde matrix is an n×m matrix of the form

Vmn(xn) =[xi−1j

]m,ni,j

=

1 1 · · · 1x1 x2 · · · xn...

.... . .

...

xm−11 xm−1

2 · · · xm−1n

(1)

where xi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, the notationVn = Vnm will be used.

Remark 1.1. Note that in the literature the term Vandermonde matrix isoften used for the transpose of the matrix given above.

1.1.1 Who was Vandermonde?

The matrix is named after Alexandre Theophile Vandermonde (1735–1796)who had a varied career that began with law studies and some success asa concert violinist, transitioned into work in science and mathematics inthe beginning of the 1770s that gradually turned into administrative andleadership positions at various Parisian institutions as well as work in politicsand economics in the end of the 1780s [86]. His entire mathematical careerconsisted of four published papers, first presented to the French Academyof Sciences in 1770 and 1771 and published a few years later.

The first paper, Memoire sur la resolution des equations [279], discussessome properties of the roots of polynomial equations, more specifically for-mulas for the sum of the roots and a sum of symmetric functions of the pow-

19

21


ers of the roots. This paper has been mentioned as important since it con-tains some of the fundamental ideas of group theory (see for instance [168]),but generally this work is overshadowed by the works of the contempo-rary Joseph Louis Lagrange (1736–1813) [166]. He also notices the equalitya2b+ b2c+ ac2 − a2c− ab2 − bc2 = (a− b)(a− c)(b− c), which is a specialcase of the formula for the determinant of the Vandermonde matrix, butthis connection is not discussed in the paper.

The second paper, Remarques sur des problemes de situation [280], dis-cusses the problem of the knight’s tour (what sequence of moves allows aknight to visit all squares on a chessboard exactly once). This paper is con-sidered the first mathematical paper that uses the basic ideas of what is nowcalled knot theory [237].

The third paper, Memoire sur des irrationnelles de differents ordres avecune application au cercle [281], is a paper on combinatorics and the mostwell-known result from the paper is the Chu–Vandermonde identity,

n∑k=1

k∏j=1

r + 1− jj

n−k∏j=1

s+ 1− jj

=

n∏j=1

r + s+ 1− jj

,

where r, s ∈ R and n ∈ Z. The identity was first found by Chu Shih-Chieh(ca 1260 – ca 1320, traditional chinese: 朱世傑

)in 1303 in The precious

mirror of the four elements(四元玉

)and was rediscovered (apparently

independently) by Vandermonde [8, 223].In the fourth paper Memoire sur l’elimination [282] Vandermonde dis-

cusses some ideas for what we today call determinants, which are functionsthat can tell us if a linear equation system has a unique solution or not.The paper predates the modern definitions of determinants but Vander-monde discusses a general method for solving linear equation systems usingalternating functions, which has strong relation to determinants. He alsonotices that exchanging exponents for indices in a class of expressions fromhis first paper will give a class of expressions that he discusses in his fourthpaper [300]. This relation is mirrored in the relationship between the deter-minant of the Vandermonde matrix and the determinant of a general matrixdescribed in Theorem 1.3.

While Vandermonde’s papers can be said to contain many importantideas they do not bring any of them to maturity and he is therefore usu-ally considered a minor scientist and mathematician compared to well-known contemporary mathematicians such as Etienne Bezout (1730–1783)and Pierre-Simon de Laplace (1749–1827) or scientists such as the chemistAntoine Lavoisier (1743–1794) that he worked with for some time after hismathematical career. The Vandermonde matrix does not appear in any ofVandermonde’s published works, which is not surprising considering thatthe modern matrix concept did not really take shape until almost a hundredyears later in the works of Sylvester and Cayley [43, 268]. It is therefore

20


ers of the roots. This paper has been mentioned as important since it con-tains some of the fundamental ideas of group theory (see for instance [168]),but generally this work is overshadowed by the works of the contempo-rary Joseph Louis Lagrange (1736–1813) [166]. He also notices the equalitya2b+ b2c+ ac2 − a2c− ab2 − bc2 = (a− b)(a− c)(b− c), which is a specialcase of the formula for the determinant of the Vandermonde matrix, butthis connection is not discussed in the paper.

The second paper, Remarques sur des problemes de situation [280], dis-cusses the problem of the knight’s tour (what sequence of moves allows aknight to visit all squares on a chessboard exactly once). This paper is con-sidered the first mathematical paper that uses the basic ideas of what is nowcalled knot theory [237].

The third paper, Memoire sur des irrationnelles de differents ordres avecune application au cercle [281], is a paper on combinatorics and the mostwell-known result from the paper is the Chu–Vandermonde identity,

n∑k=1

k∏j=1

r + 1− jj

n−k∏j=1

s+ 1− jj

=

n∏j=1

r + s+ 1− jj

,

where r, s ∈ R and n ∈ Z. The identity was first found by Chu Shih-Chieh(ca 1260 – ca 1320, traditional chinese: 朱世傑

)in 1303 in The precious

mirror of the four elements(四元玉

)and was rediscovered (apparently

independently) by Vandermonde [8, 223].In the fourth paper Memoire sur l’elimination [282] Vandermonde dis-

cusses some ideas for what we today call determinants, which are functionsthat can tell us if a linear equation system has a unique solution or not.The paper predates the modern definitions of determinants but Vander-monde discusses a general method for solving linear equation systems usingalternating functions, which has strong relation to determinants. He alsonotices that exchanging exponents for indices in a class of expressions fromhis first paper will give a class of expressions that he discusses in his fourthpaper [300]. This relation is mirrored in the relationship between the deter-minant of the Vandermonde matrix and the determinant of a general matrixdescribed in Theorem 1.3.

While Vandermonde’s papers can be said to contain many importantideas they do not bring any of them to maturity and he is therefore usu-ally considered a minor scientist and mathematician compared to well-known contemporary mathematicians such as Etienne Bezout (1730–1783)and Pierre-Simon de Laplace (1749–1827) or scientists such as the chemistAntoine Lavoisier (1743–1794) that he worked with for some time after hismathematical career. The Vandermonde matrix does not appear in any ofVandermonde’s published works, which is not surprising considering thatthe modern matrix concept did not really take shape until almost a hundredyears later in the works of Sylvester and Cayley [43, 268]. It is therefore

20

22


strange that the Vandermonde matrix was named after him, a thoroughdiscussion on this can be found in [300], but a possible reason is the simpleformula for the determinant that Vandermonde briefly discusses in his fourthpaper can be generalized to a Vandermonde matrix of any size. One of themain reasons that the Vandermonde matrix has become known is that ithas an exceptionally simple expression for its determinant that in turn hasa surprisingly fundamental relation to the determinant of a general matrix.We will be taking a closer look at the determinant of the Vandermonde ma-trix and related matrices several times in this thesis so the next section willintroduce it and some of its properties.

1.1.2 The Vandermonde determinant

Often it is not the Vandermonde matrix itself that is useful, instead it is themultivariate polynomial given by its determinant that is examined and used.The determinant of the Vandermonde matrix is usually called the Vander-monde determinant (or Vandermonde polynomial or Vandermondian [283])and can be written using an exceptionally simple formula. But before wediscuss the Vandermonde determinant we will disuss the general determi-nant.

Definition 1.2. The determinant is a function of square matrices over afield F to the field F, det : Mn×n(F) → F such that if we consider thedeterminant as a function of the columns

det(M) = det(M·,1,M·,2, . . . ,M·,n)

of the matrix the determinant must have the following properties

• The determinant must be multilinear

det(M·,1, . . . , aM·,k + bN·,k, . . . ,M·,n)

= a det(M·,1, . . . ,M·,k, . . . ,M·,n) + bdet(M·,1, . . . ,N·,k, . . . ,M·,n).

• The determinant must be alternating, that is if M·,i = M·,j for somei 6= j then det(M) = 0.

• If I is the identity matrix then det(I) = 1.

Remark 1.2. Defining the multilinear and alternating properties from therows of the matrix will give the same determinant. The name of the alter-nating property comes from the fact that it combined with multilinearityimplies that switching places between two columns changes the sign of thedeterminant.

This definition of the determinant is quite abstract but it is sufficient todefine a unique function.

21


strange that the Vandermonde matrix was named after him, a thoroughdiscussion on this can be found in [300], but a possible reason is the simpleformula for the determinant that Vandermonde briefly discusses in his fourthpaper can be generalized to a Vandermonde matrix of any size. One of themain reasons that the Vandermonde matrix has become known is that ithas an exceptionally simple expression for its determinant that in turn hasa surprisingly fundamental relation to the determinant of a general matrix.We will be taking a closer look at the determinant of the Vandermonde ma-trix and related matrices several times in this thesis so the next section willintroduce it and some of its properties.

1.1.2 The Vandermonde determinant

Often it is not the Vandermonde matrix itself that is useful, instead it is themultivariate polynomial given by its determinant that is examined and used.The determinant of the Vandermonde matrix is usually called the Vander-monde determinant (or Vandermonde polynomial or Vandermondian [283])and can be written using an exceptionally simple formula. But before wediscuss the Vandermonde determinant we will disuss the general determi-nant.

Definition 1.2. The determinant is a function of square matrices over afield F to the field F, det : Mn×n(F) → F such that if we consider thedeterminant as a function of the columns

det(M) = det(M·,1,M·,2, . . . ,M·,n)

of the matrix the determinant must have the following properties

• The determinant must be multilinear

det(M·,1, . . . , aM·,k + bN·,k, . . . ,M·,n)

= a det(M·,1, . . . ,M·,k, . . . ,M·,n) + bdet(M·,1, . . . ,N·,k, . . . ,M·,n).

• The determinant must be alternating, that is if M·,i = M·,j for somei 6= j then det(M) = 0.

• If I is the identity matrix then det(I) = 1.

Remark 1.2. Defining the multilinear and alternating properties from therows of the matrix will give the same determinant. The name of the alter-nating property comes from the fact that it combined with multilinearityimplies that switching places between two columns changes the sign of thedeterminant.

This definition of the determinant is quite abstract but it is sufficient todefine a unique function.

21

23


Theorem 1.1 (Leibniz formula for determinants). A standard result fromlinear algebra says that the determinant is unique and that it is given by thefollowing formula

det(M) =∑σ∈Sn

(−1)I(σ)n∏i=1

mi,σ(i) (2)

where Sn is the set of all permutations of the set 1, 2, . . . , n, that is all liststhat contain the numbers 1, 2, . . . , n exactly once, and if σ is a permutationthen σ(i) is the ith element of that permutation.

Remark 1.3. Often formula (2) is used immediately as the definition ofthe determinant of a matrix, see for instance [9]. The formula is usuallyattributed to Gottfried Wilhem Leibniz (1646–1716), probably due to aletter that he wrote to Guillaume de l’Hopital (1661–1704) in 1693 where hedescribes a method of solving linear equation systems that is closely relatedto Cramer’s rule [218], the particular letter was published in [173] and atranslation can be found in [263].

The determinant has several uses and interpretations, for example

• If det(M) 6= 0 then the vectors corresponding to the columns (orrows) are linearly independent. Compare this to the properties of theWronskian matrix described on page 27.

• If the columns (or rows) of M are interpreted as sides defining ann-dimensional parallelepiped the absolute value of det(M) will give thevolume of this parallelepiped. Compare this to the interpretation ofD-optimality on page 58. The sign of the determinant is also importantwhen considering the orientation of the surface which is highly relevantin geometric algebra and integration over several variables, see [123,246] for examples in geometric algebra, physics and analysis.

We will now discuss the Vandermonde determinant specifically.

Theorem 1.2. The Vandermonde determinant, vn(x1, . . . , xn), is given by

vn(x1, . . . , xn) = det(Vn(x1, . . . , xn)) =∏

1≤i<j≤n(xj − xi).

A simple way to phrase Theorem 1.2 is that the Vandermonde determi-nant is the product of all differences of the values that define the elements(note that this does not give the sign of the determinant).

There are several proofs of Theorem 1.2. Many are based on a com-bination of using elementary row or column operations together with in-duction [160, 209] but there are also several proofs using combinatorial [19]or graph-based techniques [110, 238]. Here we will present a simple proof,that dates back to some very early results on determinants [42] and has aninteresting connection to the general concept of a determinant.

22


Theorem 1.1 (Leibniz formula for determinants). A standard result fromlinear algebra says that the determinant is unique and that it is given by thefollowing formula

det(M) =∑σ∈Sn

(−1)I(σ)n∏i=1

mi,σ(i) (2)

where Sn is the set of all permutations of the set 1, 2, . . . , n, that is all liststhat contain the numbers 1, 2, . . . , n exactly once, and if σ is a permutationthen σ(i) is the ith element of that permutation.

Remark 1.3. Often formula (2) is used immediately as the definition ofthe determinant of a matrix, see for instance [9]. The formula is usuallyattributed to Gottfried Wilhem Leibniz (1646–1716), probably due to aletter that he wrote to Guillaume de l’Hopital (1661–1704) in 1693 where hedescribes a method of solving linear equation systems that is closely relatedto Cramer’s rule [218], the particular letter was published in [173] and atranslation can be found in [263].

The determinant has several uses and interpretations, for example

• If det(M) 6= 0 then the vectors corresponding to the columns (orrows) are linearly independent. Compare this to the properties of theWronskian matrix described on page 27.

• If the columns (or rows) of M are interpreted as sides defining ann-dimensional parallelepiped the absolute value of det(M) will give thevolume of this parallelepiped. Compare this to the interpretation ofD-optimality on page 58. The sign of the determinant is also importantwhen considering the orientation of the surface which is highly relevantin geometric algebra and integration over several variables, see [123,246] for examples in geometric algebra, physics and analysis.

We will now discuss the Vandermonde determinant specifically.

Theorem 1.2. The Vandermonde determinant, vn(x1, . . . , xn), is given by

vn(x1, . . . , xn) = det(Vn(x1, . . . , xn)) =∏


A simple way to phrase Theorem 1.2 is that the Vandermonde determi-nant is the product of all differences of the values that define the elements(note that this does not give the sign of the determinant).

There are several proofs of Theorem 1.2. Many are based on a com-bination of using elementary row or column operations together with in-duction [160, 209] but there are also several proofs using combinatorial [19]or graph-based techniques [110, 238]. Here we will present a simple proof,that dates back to some very early results on determinants [42] and has aninteresting connection to the general concept of a determinant.

22

24


Proof of Theorem 1.2. There are many versions of this proof, see for exam-ple [18,36,42,126], with focus on different aspects of the proof. Here we willprovide a fairly concise version that still makes all the steps of the proofclear. We start by only considering one of the variables xk, which gives asingle variable function vn(xk). From the general expression for the deter-minant, expression (2) it is clear that vn(xk) must be a polynomial of degreen in xk. We also know that if we let xk = xi for any 1 ≤ i ≤ n, i 6= k, thedeterminant will be equal to zero since the corresponding matrix will havetwo identical columns. Thus if vn(xi) = 0 we can write

vn(xk) = P (xk)n∏i=1i 6=k

(xk − xi)

where P (xk) is a polynomial. If we repeat this argument for all the variables,and ensure that no roots appear twice in the factorization, we get

vn(x1, . . . , xn) = Pn(x1, . . . , xn)

n−1∏i=1

(xn − xi)

= Pn−1(x1, . . . , xn)

n−2∏i=1

(xn−1 − xi)n−1∏i=1

(xn − xi)

= P0(x1, . . . , xn)(x2 − x1)(x3 − x2)(x3 − x1) · · ·n−1∏i=1

(xn − xi)

and since this factorization has each xk appear as a root n times we canconclude that

vn(x1, . . . , xn) = det(Vn(x1, . . . , xn)) = C∏

1≤i<j≤n(xj − xi)

where C is a constant. From Leibniz formula, expression (2), we can seethat the coefficient in front of any term of the form xnk must be 1, thusC = 1, which concludes the proof.

Theorem 1.3. There is a relationship between the exponents of the expandedVandermonde determinant and the indices in the expression for a generaldeterminant, more specifically(

n∏i=1

xi

)vn(x1, . . . , xn) =

(n∏i=1

xi

) ∏1≤i<j≤n

(xj − xi)

=∑σ∈Sn

(−1)I(σ)n∏i=1

xiσ(i). (3)

Clearly replacing xji with xi,j in (3) gives (2).

23


Proof of Theorem 1.2. There are many versions of this proof, see for exam-ple [18,36,42,126], with focus on different aspects of the proof. Here we willprovide a fairly concise version that still makes all the steps of the proofclear. We start by only considering one of the variables xk, which gives asingle variable function vn(xk). From the general expression for the deter-minant, expression (2) it is clear that vn(xk) must be a polynomial of degreen in xk. We also know that if we let xk = xi for any 1 ≤ i ≤ n, i 6= k, thedeterminant will be equal to zero since the corresponding matrix will havetwo identical columns. Thus if vn(xi) = 0 we can write

vn(xk) = P (xk)n∏i=1i 6=k

(xk − xi)

where P (xk) is a polynomial. If we repeat this argument for all the variables,and ensure that no roots appear twice in the factorization, we get

vn(x1, . . . , xn) = Pn(x1, . . . , xn)

n−1∏i=1

(xn − xi)

= Pn−1(x1, . . . , xn)

n−2∏i=1

(xn−1 − xi)n−1∏i=1

(xn − xi)

= P0(x1, . . . , xn)(x2 − x1)(x3 − x2)(x3 − x1) · · ·n−1∏i=1

(xn − xi)

and since this factorization has each xk appear as a root n times we canconclude that

vn(x1, . . . , xn) = det(Vn(x1, . . . , xn)) = C∏

1≤i<j≤n(xj − xi)

where C is a constant. From Leibniz formula, expression (2), we can seethat the coefficient in front of any term of the form xnk must be 1, thusC = 1, which concludes the proof.

Theorem 1.3. There is a relationship between the exponents of the expandedVandermonde determinant and the indices in the expression for a generaldeterminant, more specifically(

n∏i=1

xi

)vn(x1, . . . , xn) =

(n∏i=1

xi

) ∏1≤i<j≤n

(xj − xi)

=∑σ∈Sn

(−1)I(σ)n∏i=1

xiσ(i). (3)

Clearly replacing xji with xi,j in (3) gives (2).

23

25


Proof. We will prove this theorem by showing that replacing exponents withindices will give a function that by Definition 1.2 is a determinant. InDefinition 1.2 we interpreted the determinant as a function of the columns ofthe matrix, for the Vandermonde determinant this corresponds to a functionof the xi since they define the columns. Here we will interpret each part ofDefinition 1.2 as a statement about the xi and then show how it is impliedby the Vandermonde determinant.

• Alternating: The alternating property is easy to interpret in termsof the xi since if xi = xj for some i 6= j then we have two identicalcolumns. Consider the product form of the Vandermonde determinantgiven in Theorem 1.2. Switching places between xi and xj with i < j inthe Vandermonde determinant is equal to switching sign in all factorsthat contain either xi or xj as well as xk with i ≤ k ≤ j. There willbe j − i− 1 factors that contain xi and satisfy i < k ≤ j and j − i− 1factors that contain xj satisfy i ≤ k < j and one factor (xi−xj). Thismeans that in total we will change sign in 2(j − i) − 1 factors whichmeans the sign of the whole product will change.

• Multilinearity: If we denote the left hand side in (3) with w

w =

(n∏k=1

xk

)vn(x1, . . . , xn)

then multiplying the kth column by a scalar can be interpreted asfollows

M·,k → aM·,k ⇔ w =

n∑i=1

xikci →n∑i=1

axikci

and addition of columns as

M·,k →M·,k + N·,k ⇔ w =n∑i=1

xikci →n∑i=1

(xik + yik)ci

and multilinearity follows immediately from this.

• det(I) = 1: For the identity matrix we have

xi,j =

1 i = j

0 i 6= j

which for the expanded Vandermonde determinant corresponds to thetransformation

xji →

1 i = j

0 i 6= j

24


Proof. We will prove this theorem by showing that replacing exponents withindices will give a function that by Definition 1.2 is a determinant. InDefinition 1.2 we interpreted the determinant as a function of the columns ofthe matrix, for the Vandermonde determinant this corresponds to a functionof the xi since they define the columns. Here we will interpret each part ofDefinition 1.2 as a statement about the xi and then show how it is impliedby the Vandermonde determinant.

• Alternating: The alternating property is easy to interpret in termsof the xi since if xi = xj for some i 6= j then we have two identicalcolumns. Consider the product form of the Vandermonde determinantgiven in Theorem 1.2. Switching places between xi and xj with i < j inthe Vandermonde determinant is equal to switching sign in all factorsthat contain either xi or xj as well as xk with i ≤ k ≤ j. There willbe j − i− 1 factors that contain xi and satisfy i < k ≤ j and j − i− 1factors that contain xj satisfy i ≤ k < j and one factor (xi−xj). Thismeans that in total we will change sign in 2(j − i) − 1 factors whichmeans the sign of the whole product will change.

• Multilinearity: If we denote the left hand side in (3) with w

w =

(n∏k=1

xk

)vn(x1, . . . , xn)

then multiplying the kth column by a scalar can be interpreted asfollows

M·,k → aM·,k ⇔ w =

n∑i=1

xikci →n∑i=1

axikci

and addition of columns as

M·,k →M·,k + N·,k ⇔ w =n∑i=1

xikci →n∑i=1

(xik + yik)ci

and multilinearity follows immediately from this.

• det(I) = 1: For the identity matrix we have

xi,j =

1 i = j

0 i 6= j

which for the expanded Vandermonde determinant corresponds to thetransformation

xji →

1 i = j

0 i 6= j

24

26


when expanding the Vandermonde determinant we get

vn(x1, . . . , xn) = vn−1(x1, . . . , xn−1)

n−1∏i=1

(xn − xi)

= xn−1n vn−1(x1, . . . , xn−1) + P (n)

= xn−1n vn−2(x1, . . . , xn−2)

n−2∏i=1

(xn−1 − xi) + P (n)

= xn−1n xn−2

n−1vn−2(x1, . . . , xn−2) + P (n, n− 1)

=

n∏k=1

xk−1k + P (n, n− 1, . . . , 1)

where P (I), I ⊂ Z>0, does not contain any terms of the form xk−1k

for all k ∈ I. Thus applying the transformation corresponding to theidentity matrix we get(

n∏i=1

xi

)vn(x1, . . . , xn) =

n∏k=1

xkk + P (n, . . . , 1)→n∏k=1

1 + 0 = 1.

Thus if we take the right hand side in equation (3) and exchange exponentsfor indices we get a determinant be Definition 1.2 and since the determinantis unique by Theorem 1.1 and xi,j = xji in the Vandermonde matrix thismust be equal to(

n∏i=1

xi

)vn(x1, . . . , xn) =

∑σ∈Sn

(−1)I(σ)n∏i=1

xiσ(i).

1.1.3 Inverse of the Vandermonde matrix

The inverse for the Vandermonde matrix has been known for a long time, es-pecially since the solution to a Lagrange interpolation problems (see Section1.2.1) gives the inverse indirectly. Here we will only give a short overviewof the work on expressing the inverse as an explicit matrix.

An explicit expression for the inverse matrix has been known since atleast the end of the 1950s, see [199].

Theorem 1.4. The elements of the inverse of an n-dimensional Vander-monde matrix V can be calculated by

(V−1n

)ij

=(−1)j−1σn−j,in∏k=1k 6=i

(xk − xi)(4)

25


when expanding the Vandermonde determinant we get

vn(x1, . . . , xn) = vn−1(x1, . . . , xn−1)

n−1∏i=1

(xn − xi)

= xn−1n vn−1(x1, . . . , xn−1) + P (n)

= xn−1n vn−2(x1, . . . , xn−2)

n−2∏i=1

(xn−1 − xi) + P (n)

= xn−1n xn−2

n−1vn−2(x1, . . . , xn−2) + P (n, n− 1)

=

n∏k=1

xk−1k + P (n, n− 1, . . . , 1)

where P (I), I ⊂ Z>0, does not contain any terms of the form xk−1k

for all k ∈ I. Thus applying the transformation corresponding to theidentity matrix we get(

n∏i=1

xi

)vn(x1, . . . , xn) =

n∏k=1

xkk + P (n, . . . , 1)→n∏k=1

1 + 0 = 1.

Thus if we take the right hand side in equation (3) and exchange exponentsfor indices we get a determinant be Definition 1.2 and since the determinantis unique by Theorem 1.1 and xi,j = xji in the Vandermonde matrix thismust be equal to(

n∏i=1

xi

)vn(x1, . . . , xn) =

∑σ∈Sn

(−1)I(σ)n∏i=1

xiσ(i).

1.1.3 Inverse of the Vandermonde matrix

The inverse for the Vandermonde matrix has been known for a long time, es-pecially since the solution to a Lagrange interpolation problems (see Section1.2.1) gives the inverse indirectly. Here we will only give a short overviewof the work on expressing the inverse as an explicit matrix.

An explicit expression for the inverse matrix has been known since atleast the end of the 1950s, see [199].

Theorem 1.4. The elements of the inverse of an n-dimensional Vander-monde matrix V can be calculated by

(V−1n

)ij

=(−1)j−1σn−j,in∏k=1k 6=i

(xk − xi)(4)

25

27


where σj,i is the j:th elementary symmetric polynomial with variable xi setto zero.

σj,i =∑

1≤m1<m2<... <mj≤n

j∏k=1

xmk(1− δmk,i) , δa,b =

1 , a = b0 , a 6= b

(5)

We will not give the proof of this theorem here, but the general outlineof a proof will be given in Section 1.2.1.

In the literature there are many cases where the inverse is instead writtenas a product of several simpler matrices, usually triangular or diagonal [214,225,226,277]. There is also a lot of literature that takes a more algorithmicapproach and tries to find fast ways of computing the elements, classicalexamples include the Parker–Traub algorithm [274] and the Bjorck–Pereyraalgorithm [23], and more recent results can be found in [84].

1.1.4 The alternant matrix

Many generalizations of the Vandermonde matrix have been proposed andstudied in the literature. An early generalization is the alternant matrixwhich is a matrix that exchanges the powers in the Vandermonde matrixwith other functions [219].

Definition 1.3. An alternant matrix is a matrix of the form

Amn(fm; xn) = [fi(xj)]m,ni,j =

f1(x1) f1(x2) · · · f1(xn)f2(x1) f2(x2) · · · f2(xn)

......

. . ....

fm(x1) fm(x2) · · · fm(xn)

(6)

where fi : F → F where F is a field. If the matrix is square, n = m, thenotation An = Anm will be used.

Remark 1.4. Someties the alternant matrix is used as an alternative namefor the Vandermonde matrix or the Vandermonde matrix multiplied by adiagonal matrix [276].

There are several special cases of alternant matrices that are useful orinteresting in various mathematical fields:

Interpolation and curve fitting

Just like the Vandermonde matrix can be used for polynomial interpolationthe alternant matrix can be used to describe interpolation with other setsof function, see Section 1.2.1 and 1.2.2, as well as approximate curve fitting,for example using the least squares method described in Section 1.2.3.

26


where σj,i is the j:th elementary symmetric polynomial with variable xi setto zero.

σj,i =∑

1≤m1<m2<... <mj≤n

j∏k=1

xmk(1− δmk,i) , δa,b =

1 , a = b0 , a 6= b

(5)

We will not give the proof of this theorem here, but the general outlineof a proof will be given in Section 1.2.1.

In the literature there are many cases where the inverse is instead writtenas a product of several simpler matrices, usually triangular or diagonal [214,225,226,277]. There is also a lot of literature that takes a more algorithmicapproach and tries to find fast ways of computing the elements, classicalexamples include the Parker–Traub algorithm [274] and the Bjorck–Pereyraalgorithm [23], and more recent results can be found in [84].

1.1.4 The alternant matrix

Many generalizations of the Vandermonde matrix have been proposed andstudied in the literature. An early generalization is the alternant matrixwhich is a matrix that exchanges the powers in the Vandermonde matrixwith other functions [219].

Definition 1.3. An alternant matrix is a matrix of the form

Amn(fm; xn) = [fi(xj)]m,ni,j =

f1(x1) f1(x2) · · · f1(xn)f2(x1) f2(x2) · · · f2(xn)

......

. . ....

fm(x1) fm(x2) · · · fm(xn)

(6)

where fi : F → F where F is a field. If the matrix is square, n = m, thenotation An = Anm will be used.

Remark 1.4. Someties the alternant matrix is used as an alternative namefor the Vandermonde matrix or the Vandermonde matrix multiplied by adiagonal matrix [276].

There are several special cases of alternant matrices that are useful orinteresting in various mathematical fields:

Interpolation and curve fitting

Just like the Vandermonde matrix can be used for polynomial interpolationthe alternant matrix can be used to describe interpolation with other setsof function, see Section 1.2.1 and 1.2.2, as well as approximate curve fitting,for example using the least squares method described in Section 1.2.3.

26

28


Alternant codes

As mentioned on page 19 there are several different error-detecting anderror-correcting codes that can be described using the Vandermonde matrix.These and some related codes can also be categorized as alternant codes, aterm introduced in [121]. For a survey on these codes see [295].

Jacobian matrix

One of the most well-known examples of an alternant matrix is the Jaco-bian matrix. Let f : Fn → Fn be a vector-valued function that is n timesdifferentiable with respect to each variable, then the Jacobian matrix is thematrix J, given by

∂y1∂x1

∂y2∂x1

· · · ∂yn∂x1

∂y1∂x2

∂y2∂x2

· · · ∂yn∂x2

......

. . ....

∂y1∂xn

∂y2∂xn

· · · ∂yn∂xn

where y = f(x). The most common application of the Jacobian matrix isto use its determinant to describe how volume elements are deformed whenchanging variables in multivariate calculus [246]. The numerous applicationsand generalizations that follow from this alone are too numerous to list sohere we will only note that it holds a central role in many methods formultivariate optimizations, such as the Marquardt least squares methoddescribed in Section 1.2.6.

Wronskian matrix

If fn = (f1, . . . , fn), fi = di−1

dxi−1 , and gn = (g1, . . . , gn), gi ∈ Cn−1[C], then thealternant matrix An(fn; gn) will be the Wronskian matrix . The Wronskianmatrix has a long history [125] and is commonly used to test if a set offunctions are linearly independent as well as finding solutions to ordinarydifferential equations [101]. If the determinant of the Wronskian matrix isnon-zero then the functions are linearly independent, see [27,32], but provinglinear dependence requires further conditions, see [25,26,230,231,293].

A classical application of the Wronskian is confirming that a set of solu-tions to a linear differential equation are linearly independent, or if n−1 lin-early independent solutions are known, constructing the remaining linearlyindependent solution using Abel’s identity (for n = 2) or a generalisation ofit [34].

If Li is a linear partial differential operator of order i, then the alternantmatrix An(Ln; gn), where Ln = (L1, . . . , Ln), is the generalized Wronskianmatrix [227], has been used in for example diophantine geometry [82, 244]and for solving Korteweg-de Vries equations, see [197] and the references

27


Alternant codes

As mentioned on page 19 there are several different error-detecting anderror-correcting codes that can be described using the Vandermonde matrix.These and some related codes can also be categorized as alternant codes, aterm introduced in [121]. For a survey on these codes see [295].

Jacobian matrix

One of the most well-known examples of an alternant matrix is the Jaco-bian matrix. Let f : Fn → Fn be a vector-valued function that is n timesdifferentiable with respect to each variable, then the Jacobian matrix is thematrix J, given by

∂y1∂x1

∂y2∂x1

· · · ∂yn∂x1

∂y1∂x2

∂y2∂x2

· · · ∂yn∂x2

......

. . ....

∂y1∂xn

∂y2∂xn

· · · ∂yn∂xn

where y = f(x). The most common application of the Jacobian matrix isto use its determinant to describe how volume elements are deformed whenchanging variables in multivariate calculus [246]. The numerous applicationsand generalizations that follow from this alone are too numerous to list sohere we will only note that it holds a central role in many methods formultivariate optimizations, such as the Marquardt least squares methoddescribed in Section 1.2.6.

Wronskian matrix

If fn = (f1, . . . , fn), fi = di−1

dxi−1 , and gn = (g1, . . . , gn), gi ∈ Cn−1[C], then thealternant matrix An(fn; gn) will be the Wronskian matrix . The Wronskianmatrix has a long history [125] and is commonly used to test if a set offunctions are linearly independent as well as finding solutions to ordinarydifferential equations [101]. If the determinant of the Wronskian matrix isnon-zero then the functions are linearly independent, see [27,32], but provinglinear dependence requires further conditions, see [25,26,230,231,293].

A classical application of the Wronskian is confirming that a set of solu-tions to a linear differential equation are linearly independent, or if n−1 lin-early independent solutions are known, constructing the remaining linearlyindependent solution using Abel’s identity (for n = 2) or a generalisation ofit [34].

If Li is a linear partial differential operator of order i, then the alternantmatrix An(Ln; gn), where Ln = (L1, . . . , Ln), is the generalized Wronskianmatrix [227], has been used in for example diophantine geometry [82, 244]and for solving Korteweg-de Vries equations, see [197] and the references

27

29


therein. The generalized Wronskian matrix has similar properties with re-spect to the linear dependence of the functions it is created from as thestandard Wronskian [294]. Both the Wronskian and generalized Wronskianis also useful in algebraic geometry, see [101] for several examples.

Bell matrix

Alternating matrices can be used to convert function composition into ma-trix multiplication. By letting Di = di−1

dxi−1 and gj(x) = (f(x))j , where f isinfinitely differentiable, the alternant matrix B[f ] = An(Dn,gn) is calleda Bell matrix (its transpose is known as the Carleman matrix ). Some au-thors, for instance [159], refer to Bell matrices as Jabotinsky matrices dueto a special case of Bell matrices considered in [137].

That Bell matrices converts function composition into matrix multiplica-tion can be seen by noting that the power series expansion of the jth powerof f can be written as

(f(x))j =∞∑i=1

B[f ]ijxi

and from this equality follows that B[f g] = B[g]B[f ]. This is the basicproperty behind a popular technique called Carleman linearisation or Carle-man embedding that has seen wide use in the theory of non-linear dynamicalsystems. The literature on the subject is vast but a systematic introductionis offered in [165].

Moore matrix

When working in a finite field with prime characteristic p an analogue ofthe Vandermonde and Wronskian matrix can be constructed by taking analternant matrix where the rows are given by power of the Frobenius au-tomorphism, F (ω) = ωp. This matrix is called the Moore matrix and isnamed after its originator E. H. Moore who also calculated its determinant,∣∣∣∣∣∣∣∣∣

ω1 · · · ωnωp1 · · · ωpn...

. . ....

ωpn−1

1 · · · ωpn−1

n

∣∣∣∣∣∣∣∣∣ =

n−1∏i=1

p−1∏ki−1

· · ·p−1∏k1=0

(ωi + ki−1ωi−1 + . . .+ k1ω1)(mod p),

and showed that if this determinant was not equal to zero then ω1, . . . , ωn arelinearly independent [211]. There are several uses for the determinant of theMoore matrix in function field arithmetic, see for instance [113], a classicalexample is finding the modular invariants of the general linear group overa finite field [72, 224]. The determinant also plays an important role in thetheory of Drinfeld modules [221].

28


therein. The generalized Wronskian matrix has similar properties with re-spect to the linear dependence of the functions it is created from as thestandard Wronskian [294]. Both the Wronskian and generalized Wronskianis also useful in algebraic geometry, see [101] for several examples.

Bell matrix

Alternating matrices can be used to convert function composition into ma-trix multiplication. By letting Di = di−1

dxi−1 and gj(x) = (f(x))j , where f isinfinitely differentiable, the alternant matrix B[f ] = An(Dn,gn) is calleda Bell matrix (its transpose is known as the Carleman matrix ). Some au-thors, for instance [159], refer to Bell matrices as Jabotinsky matrices dueto a special case of Bell matrices considered in [137].

That Bell matrices converts function composition into matrix multiplica-tion can be seen by noting that the power series expansion of the jth powerof f can be written as

(f(x))j =∞∑i=1

B[f ]ijxi

and from this equality follows that B[f g] = B[g]B[f ]. This is the basicproperty behind a popular technique called Carleman linearisation or Carle-man embedding that has seen wide use in the theory of non-linear dynamicalsystems. The literature on the subject is vast but a systematic introductionis offered in [165].

Moore matrix

When working in a finite field with prime characteristic p an analogue ofthe Vandermonde and Wronskian matrix can be constructed by taking analternant matrix where the rows are given by power of the Frobenius au-tomorphism, F (ω) = ωp. This matrix is called the Moore matrix and isnamed after its originator E. H. Moore who also calculated its determinant,∣∣∣∣∣∣∣∣∣

ω1 · · · ωnωp1 · · · ωpn...

. . ....

ωpn−1

1 · · · ωpn−1

n

∣∣∣∣∣∣∣∣∣ =

n−1∏i=1

p−1∏ki−1

· · ·p−1∏k1=0

(ωi + ki−1ωi−1 + . . .+ k1ω1)(mod p),

and showed that if this determinant was not equal to zero then ω1, . . . , ωn arelinearly independent [211]. There are several uses for the determinant of theMoore matrix in function field arithmetic, see for instance [113], a classicalexample is finding the modular invariants of the general linear group overa finite field [72, 224]. The determinant also plays an important role in thetheory of Drinfeld modules [221].

28

30


1.1.5 The generalized Vandermonde matrix

There are several types of matrices (or determinants) that have been referredto as generalized Vandermonde matrices, for example the confluent Vander-monde matrix is sometimes referred to as the generalized Vandermonde ma-trix [149,150,175,194,265], this matrix and its role in interpolation problemsis briefly described on page 40. Other examples include modified versionsof confluent Vandermonde matrices [91], as well as matrices with elementsgiven by multivariate monomials of increasing multidegree [39], or similarlyover the algebraic closure of a field [61], matrices with elements given bymultivariate polynomials with univariate terms [283].

In this thesis we call the alternant matrix Amn(xα1 , . . . , xαn ;x1, . . . , xn)the generalized Vandermonde matrix.

Definition 1.4. A generalized Vandermonde matrix is an n×m matrix ofthe form

Gmn(xn) =[xαij

]m,ni,j

=

xα1

1 xα12 · · · xα1

n

xα21 xα2

2 · · · xα2n

......

. . ....

xαm1 xαm2 · · · xαmn

(7)

where xi ∈ C, αi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, thenotation Gn = Gnm will be used.

This name has been used for quite some time, see [120] for instance.The main reason to study this matrix seems to be its connection to Schurpolynomials, see below, and thus the research on the matrix is primarilyfocused on its determinant. Many of the results are algorithmic in nature [47,66–68,157] but there are also more algebraic examinations [85,97,250,296].

There are several of examples where the determinant of generalized Van-dermonde matrices are interesting or useful.

Schur polynomials

Given an integer partition λ = (λ1, . . . , λn), that is 0 < λ1 ≤ λ2 ≤ . . . λnand each λi ∈ N, we can define

a(λ1+n−1,λ2+n−2,...,λn)(x1, . . . , xn)

= det(Gn(λ1 + n− 1, λ2 + n− 2, . . . , λn;x1, . . . , xn)).

Note that a(λ1+n−1,λ2+n−2,...,λn)(x1, . . . , xn) is a polynomial that always havea(n−2,n−1,...,0)(x1, . . . , xn) = vn(x1, . . . , xn) as a factor. The polynomialsgiven by expressions of the form

sλ(x1, . . . , xn) =a(λ1+n−1,λ2+n−2,...,λn)(x1, . . . , xn)

a(n−2,n−1,...,0)(x1, . . . , xn)

29


1.1.5 The generalized Vandermonde matrix

There are several types of matrices (or determinants) that have been referredto as generalized Vandermonde matrices, for example the confluent Vander-monde matrix is sometimes referred to as the generalized Vandermonde ma-trix [149,150,175,194,265], this matrix and its role in interpolation problemsis briefly described on page 40. Other examples include modified versionsof confluent Vandermonde matrices [91], as well as matrices with elementsgiven by multivariate monomials of increasing multidegree [39], or similarlyover the algebraic closure of a field [61], matrices with elements given bymultivariate polynomials with univariate terms [283].

In this thesis we call the alternant matrix Amn(xα1 , . . . , xαn ;x1, . . . , xn)the generalized Vandermonde matrix.

Definition 1.4. A generalized Vandermonde matrix is an n×m matrix ofthe form

Gmn(xn) =[xαij

]m,ni,j

=

xα1

1 xα12 · · · xα1

n

xα21 xα2

2 · · · xα2n

......

. . ....

xαm1 xαm2 · · · xαmn

(7)

where xi ∈ C, αi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, thenotation Gn = Gnm will be used.

This name has been used for quite some time, see [120] for instance.The main reason to study this matrix seems to be its connection to Schurpolynomials, see below, and thus the research on the matrix is primarilyfocused on its determinant. Many of the results are algorithmic in nature [47,66–68,157] but there are also more algebraic examinations [85,97,250,296].

There are several of examples where the determinant of generalized Van-dermonde matrices are interesting or useful.

Schur polynomials

Given an integer partition λ = (λ1, . . . , λn), that is 0 < λ1 ≤ λ2 ≤ . . . λnand each λi ∈ N, we can define

a(λ1+n−1,λ2+n−2,...,λn)(x1, . . . , xn)

= det(Gn(λ1 + n− 1, λ2 + n− 2, . . . , λn;x1, . . . , xn)).

Note that a(λ1+n−1,λ2+n−2,...,λn)(x1, . . . , xn) is a polynomial that always havea(n−2,n−1,...,0)(x1, . . . , xn) = vn(x1, . . . , xn) as a factor. The polynomialsgiven by expressions of the form

sλ(x1, . . . , xn) =a(λ1+n−1,λ2+n−2,...,λn)(x1, . . . , xn)

a(n−2,n−1,...,0)(x1, . . . , xn)

29

31


are called the Schur functions or Schur polynomials and were introducedby Cauchy [42] but named after Issai Schur (1875–1941) that showed thatthey were highly useful in invariant theory and representation theory. Forinstance they can be used to determine the character of conjugacy classesof representations of the symmetric group [98]. They have also been used inother areas, for instance to describe the generating function of many classesof plane partitions, see for instance [36] for several examples. The literatureon Schur polynomials is vast and so are the applications so there will be noattempt to summarise them here.

Integration of an exponential function over a unitary group

If we let U(n) be the n-dimensional unitary group and dU a Haar measurenormalised to 1 then the Harish-Chandra–Itzykson–Zuber integral formula[116, 136], says that if A and B are Hermitian matrices with eigenvaluesλ1(A) ≤ . . . ≤ λn(A) and λ1(B) ≤ . . . ≤ λn(B) then∫

U(n)et tr(AUBU∗) dU =

det([exp(tλj(A)λk(B))]n,nj,k

)tn(n−1)

2 vn(λ(A))vn(λ(B))

n−1∏i=1

i! (8)

where vn is the determinant of the Vandermonde matrix. If t = 1 and Aand B are chosen as diagonal matrices

Aij =

ai if i = j,

0 if i 6= j,Bij =

bi if i = j,

0 if i 6= j,

then formula (8) reduces to an expression involving determinants of a gen-eralized Vandermonde matrix and two Vandermonde matrices,

∫U(n)

etr(AUBU∗) dU =

∣∣∣∣∣∣∣∣∣ea1b1 ea1b2 . . . ea1bn

ea2b1 ea2b2 . . . ea2bn

......

. . ....

eanb1 eanb2 . . . eanbn

∣∣∣∣∣∣∣∣∣vn(a1, . . . , an)vn(b1, . . . , bn)

.

1.1.6 The Vandermonde determinant in systems withCoulombian interactions

Several interesting mathematical problems that feature Vandermonde ma-trices and Vandermonde determinant can be described as questions aboutsystems with Coulombian interactions. The name Coulombian interactioncome from Charles-Augustin Coulomb (1736–1806) who is probably mostwell-known for quantifying the force between two charged particles (whatis today known as Coulomb’s law) in 1785 [59]. Coulombs law states that

30


are called the Schur functions or Schur polynomials and were introducedby Cauchy [42] but named after Issai Schur (1875–1941) that showed thatthey were highly useful in invariant theory and representation theory. Forinstance they can be used to determine the character of conjugacy classesof representations of the symmetric group [98]. They have also been used inother areas, for instance to describe the generating function of many classesof plane partitions, see for instance [36] for several examples. The literatureon Schur polynomials is vast and so are the applications so there will be noattempt to summarise them here.

Integration of an exponential function over a unitary group

If we let U(n) be the n-dimensional unitary group and dU a Haar measurenormalised to 1 then the Harish-Chandra–Itzykson–Zuber integral formula[116, 136], says that if A and B are Hermitian matrices with eigenvaluesλ1(A) ≤ . . . ≤ λn(A) and λ1(B) ≤ . . . ≤ λn(B) then∫

U(n)et tr(AUBU∗) dU =

det([exp(tλj(A)λk(B))]n,nj,k

)tn(n−1)

2 vn(λ(A))vn(λ(B))

n−1∏i=1

i! (8)

where vn is the determinant of the Vandermonde matrix. If t = 1 and Aand B are chosen as diagonal matrices

Aij =

ai if i = j,

0 if i 6= j,Bij =

bi if i = j,

0 if i 6= j,

then formula (8) reduces to an expression involving determinants of a gen-eralized Vandermonde matrix and two Vandermonde matrices,

∫U(n)

etr(AUBU∗) dU =

∣∣∣∣∣∣∣∣∣ea1b1 ea1b2 . . . ea1bn

ea2b1 ea2b2 . . . ea2bn

......

. . ....

eanb1 eanb2 . . . eanbn

∣∣∣∣∣∣∣∣∣vn(a1, . . . , an)vn(b1, . . . , bn)

.

1.1.6 The Vandermonde determinant in systems withCoulombian interactions

Several interesting mathematical problems that feature Vandermonde ma-trices and Vandermonde determinant can be described as questions aboutsystems with Coulombian interactions. The name Coulombian interactioncome from Charles-Augustin Coulomb (1736–1806) who is probably mostwell-known for quantifying the force between two charged particles (whatis today known as Coulomb’s law) in 1785 [59]. Coulombs law states that

30

32


the force between two charged particles is proportional to the product ofthe charges and the inverse of the square of the distance between the twocharges. When talking about Coulombian interactions in mathematics ormathematical physics it usually refers to a system described by an energygiven by

HN (x1, . . . , xN ) =1

2

∑i 6=j

g(xi − xj) +NN∑i=1

V (xi) (9)

where the interaction kernel, g(x), can take a few different forms, more onthis later, and V (x) is an external potential that can behave in many dif-ferent ways. The points xi usually belong to Rd (or some subset thereof)but there is also research that involves more general manifolds. A commongoal is to minimize this energy or find some other extreme points. Thereare many areas where this kind of problems, or closely related problems,appear. See the extended version of [255] for a recent review of the field.In this section we will mention a few examples of interesting systems withCoulombian interactions that are connected to the Vandermonde determi-nant and the properties of the Vandermonde determinant we discuss in thisthesis.

Fekete points

In Section 1.2.1 interpolation of a finite number of points using a polynomialwill be discussed. When a function is approximated by a polynomial usinginterpolation the approximation error depends on the chosen interpolationpoints. The Fekete points is a set of points that provide an almost optimalchoice of interpolation points [248] and they are given by maximizing theVandermonde determinant. Taking the logarithm of the expression for theVandermonde determinant given in Theorem 1.2 gives

log(vn(x1, . . . , xn)) =∑

1≤i<j≤nlog(xj − xi)

and thus −12 log(vn(x1, . . . , xn)) gives the same as setting g(x) = log(x) and

V (x) ≡ 0 in (9). Finding the Fekete points is also of interest in complex-ity theory and would help with finding an appropriate starting polynomialfor a homotopy algorithm for realizing the Fundamental Theorem of Alge-bra [258,262]. In Chapter 2 we will discuss how to find the maximum pointsof the Vandermonde determinant for certain special cases. A common gen-eralisation of the Fekete points is the case where multivariate polynomialsare used, see for example [30, 37, 203]. The case where and points in Cdare interpolated have also been examined, an example of a recent significantresults is [20] and a review can be found in [24].

31


the force between two charged particles is proportional to the product ofthe charges and the inverse of the square of the distance between the twocharges. When talking about Coulombian interactions in mathematics ormathematical physics it usually refers to a system described by an energygiven by

HN (x1, . . . , xN ) =1

2

∑i 6=j

g(xi − xj) +NN∑i=1

V (xi) (9)

where the interaction kernel, g(x), can take a few different forms, more onthis later, and V (x) is an external potential that can behave in many dif-ferent ways. The points xi usually belong to Rd (or some subset thereof)but there is also research that involves more general manifolds. A commongoal is to minimize this energy or find some other extreme points. Thereare many areas where this kind of problems, or closely related problems,appear. See the extended version of [255] for a recent review of the field.In this section we will mention a few examples of interesting systems withCoulombian interactions that are connected to the Vandermonde determi-nant and the properties of the Vandermonde determinant we discuss in thisthesis.

Fekete points

In Section 1.2.1 interpolation of a finite number of points using a polynomialwill be discussed. When a function is approximated by a polynomial usinginterpolation the approximation error depends on the chosen interpolationpoints. The Fekete points is a set of points that provide an almost optimalchoice of interpolation points [248] and they are given by maximizing theVandermonde determinant. Taking the logarithm of the expression for theVandermonde determinant given in Theorem 1.2 gives

log(vn(x1, . . . , xn)) =∑

1≤i<j≤nlog(xj − xi)

and thus −12 log(vn(x1, . . . , xn)) gives the same as setting g(x) = log(x) and

V (x) ≡ 0 in (9). Finding the Fekete points is also of interest in complex-ity theory and would help with finding an appropriate starting polynomialfor a homotopy algorithm for realizing the Fundamental Theorem of Alge-bra [258,262]. In Chapter 2 we will discuss how to find the maximum pointsof the Vandermonde determinant for certain special cases. A common gen-eralisation of the Fekete points is the case where multivariate polynomialsare used, see for example [30, 37, 203]. The case where and points in Cdare interpolated have also been examined, an example of a recent significantresults is [20] and a review can be found in [24].

31

33


Distribution of electrical charges

The most classical example of a system with Coulombian interactions isa system of charged particles confined to some volume, even if it was notstudied (from a mathematical point of view) until almost a hundred yearsafter Coulomb’s law was introduced [119, 267]. The classical mathematicalformulation of this problem considers p+1 charges fixed at points a0,. . .,ap ∈C with weights η0,. . .,ηp and n moveable charges x1,. . .,xn. The questions isthen what x-values give the extreme points of L(x1, . . . , xn) given by

L(x1, . . . , xn) =n∑k=1

p∑j=0

ηj log

(1

|aj − xk|

)+

∑1≤i<k≤n

log

(1

|xk − xi|

).

More background on this type of problem together with a collection of recentresults can be found in [73]. If there are no fixed charges the problembecomes equivalent to maximising the absolute value of the Vandermondedeterminant similar to finding the Fekete points. The problems discussed inChapter 2 belongs to the class of equations that are called Schrodinger-likein [73].

Sphere packing

There are several different interaction kernels apart from the logarithmicinteraction kernel, g(x) = − log(x), that are interesting in mathematicalphysics, especially statistical mechanics and quantum mechanics. One im-portant class of interaction kernels are those given by g(x) = 1

|x|s where sis a positive integer. When this interaction kernel is used value given byformula (9) is called the Riesz s-energy. There is a large body of significantliterature, in [255] over 30 references are listed as introduction to differentrelated problems.

It is worth noting that lims→0

(1− 1

|x|s

)= − log(|x|) which connects min-

imising the Riesz s-energy to the Fekete points.

If we instead s → ∞ the problem of minimising the Riesz s-energyformally corresponds to the optimal sphere-packing problem, that is findingthe arrangement of non-overlapping identical spheres that cover as muchof a space as possible. This is a classical problem where extensive efforthas gone into finding optimal packings but for many years the problemwas only fully solved in one, two and three dimensions, until recently whensurprisingly simple proofs were found for 8 and 24 dimensions (seeminglywithout giving any results for any number of dimensions in-between). Fora thorough collection of classical results see [58] and for the recent resultssee [52–54,284].

32


Distribution of electrical charges

The most classical example of a system with Coulombian interactions isa system of charged particles confined to some volume, even if it was notstudied (from a mathematical point of view) until almost a hundred yearsafter Coulomb’s law was introduced [119, 267]. The classical mathematicalformulation of this problem considers p+1 charges fixed at points a0,. . .,ap ∈C with weights η0,. . .,ηp and n moveable charges x1,. . .,xn. The questions isthen what x-values give the extreme points of L(x1, . . . , xn) given by

L(x1, . . . , xn) =n∑k=1

p∑j=0

ηj log

(1

|aj − xk|

)+

∑1≤i<k≤n

log

(1

|xk − xi|

).

More background on this type of problem together with a collection of recentresults can be found in [73]. If there are no fixed charges the problembecomes equivalent to maximising the absolute value of the Vandermondedeterminant similar to finding the Fekete points. The problems discussed inChapter 2 belongs to the class of equations that are called Schrodinger-likein [73].

Sphere packing

There are several different interaction kernels apart from the logarithmicinteraction kernel, g(x) = − log(x), that are interesting in mathematicalphysics, especially statistical mechanics and quantum mechanics. One im-portant class of interaction kernels are those given by g(x) = 1

|x|s where sis a positive integer. When this interaction kernel is used value given byformula (9) is called the Riesz s-energy. There is a large body of significantliterature, in [255] over 30 references are listed as introduction to differentrelated problems.

It is worth noting that lims→0

(1− 1

|x|s

)= − log(|x|) which connects min-

imising the Riesz s-energy to the Fekete points.

If we instead s → ∞ the problem of minimising the Riesz s-energyformally corresponds to the optimal sphere-packing problem, that is findingthe arrangement of non-overlapping identical spheres that cover as muchof a space as possible. This is a classical problem where extensive efforthas gone into finding optimal packings but for many years the problemwas only fully solved in one, two and three dimensions, until recently whensurprisingly simple proofs were found for 8 and 24 dimensions (seeminglywithout giving any results for any number of dimensions in-between). Fora thorough collection of classical results see [58] and for the recent resultssee [52–54,284].

32

34


Coulomb gas

In mathematical physics a system of particles whose energy can be describedby (9) is often called a Coulomb gas [93, 207, 255]. One of the most wide-reaching results in the analysis of Coulomb gases was that many gas sys-tems can be described using random matrices that belongs to a so-calledβ-ensemble which is defined by matrices with random elements. The foun-dational results were found in the early 1960s and applied to the cases whereβ = 1, β = 2 and β = 4 [78–81]. These cases will be briefly discussed inSection 1.1.7 and describe where the Vandermonde determinant appears theprobability density functions for the eigenvalues of the random matrices. Ifthe same theory is extended to other values of β it can also be connected toequations similar to the Harish-Chandra–Itzykson–Zuber integral formuladescribed on page 30 [93].

1.1.7 The Vandermonde determinant in random matrixtheory

This section is based on Section 1, 3 and 4 in Paper D

Random matrix theory is a large research are with many applications,primarily in quantum mechanics and statistical mechanics [93,207,255] butalso in wireless communication and finance [13] and they appear as an im-portant tool for analysing and evaluating algorithms in numerical linearalgebra [83].

One class of random matrices that have been analysed extensively are theso-called β-ensembles, for a brief motivation see the section on Coulomb gasabove. Here we will define the some well-known β-ensembles and describewhere the Vandermonde determinant appears in their probability densityfunctions.

Definition 1.5. Let X = (X1, · · · , Xn), where Xi ∼ N (µi,ΣΣΣ) and Xi isindependent of Xj , where i 6= j. The matrix W : p×p is said to be Wishartdistributed [292] if and only if W = XX> for some matrix X in a family ofGaussian matrices Gm×n,m ≤ n, that is, X ∼ Nm,n(µµµ,ΣΣΣ, I) where ΣΣΣ ≥ 0.

Next we will look at the expression for the probability density distribu-tion of the eigenvalues of a Wishart distributed matrix taken from [7].

Theorem 1.5. If X is distributed as N (µµµ,ΣΣΣ), then the probability densitydistribution of the eigenvalues of XX>, denoted λλλ = (λ1, . . . , λm), is givenby:

P(λλλ) =π−

12n det(ΣΣΣ)−

12n det(D)

12

(n−p−1)

212npΓp

(12n)Γp(

12p) ∏

i<j

(λi − λj) exp

(−1

2Tr(ΣΣΣ−1D)

)where D = diag(λi) and Γ is the Gamma function.

33


Coulomb gas

In mathematical physics a system of particles whose energy can be describedby (9) is often called a Coulomb gas [93, 207, 255]. One of the most wide-reaching results in the analysis of Coulomb gases was that many gas sys-tems can be described using random matrices that belongs to a so-calledβ-ensemble which is defined by matrices with random elements. The foun-dational results were found in the early 1960s and applied to the cases whereβ = 1, β = 2 and β = 4 [78–81]. These cases will be briefly discussed inSection 1.1.7 and describe where the Vandermonde determinant appears theprobability density functions for the eigenvalues of the random matrices. Ifthe same theory is extended to other values of β it can also be connected toequations similar to the Harish-Chandra–Itzykson–Zuber integral formuladescribed on page 30 [93].

1.1.7 The Vandermonde determinant in random matrixtheory

This section is based on Section 1, 3 and 4 in Paper D

Random matrix theory is a large research are with many applications,primarily in quantum mechanics and statistical mechanics [93,207,255] butalso in wireless communication and finance [13] and they appear as an im-portant tool for analysing and evaluating algorithms in numerical linearalgebra [83].

One class of random matrices that have been analysed extensively are theso-called β-ensembles, for a brief motivation see the section on Coulomb gasabove. Here we will define the some well-known β-ensembles and describewhere the Vandermonde determinant appears in their probability densityfunctions.

Definition 1.5. Let X = (X1, · · · , Xn), where Xi ∼ N (µi,ΣΣΣ) and Xi isindependent of Xj , where i 6= j. The matrix W : p×p is said to be Wishartdistributed [292] if and only if W = XX> for some matrix X in a family ofGaussian matrices Gm×n,m ≤ n, that is, X ∼ Nm,n(µµµ,ΣΣΣ, I) where ΣΣΣ ≥ 0.

Next we will look at the expression for the probability density distribu-tion of the eigenvalues of a Wishart distributed matrix taken from [7].

Theorem 1.5. If X is distributed as N (µµµ,ΣΣΣ), then the probability densitydistribution of the eigenvalues of XX>, denoted λλλ = (λ1, . . . , λm), is givenby:

P(λλλ) =π−

12n det(ΣΣΣ)−

12n det(D)

12

(n−p−1)

212npΓp

(12n)Γp(

12p) ∏

i<j

(λi − λj) exp

(−1

2Tr(ΣΣΣ−1D)

)where D = diag(λi) and Γ is the Gamma function.

33

35


It will prove useful that the formula given in Theorem 1.5 contains the

term∏i<j

(λi − λj) which we recognize from Theorem 1.2 as the determinant

of a Vandermonde matrix.In Section 2.3.7 we will use some results from Section 2.2 to find the

extreme values of the probability density function of a Wishart matrix. Herewe will prove a couple of properties of a Wishart matrix that will be usedin Section 2.3.7.

Lemma 1.1. Let P be a polynomial and A be a symmetric n × n matrix.If the eigenvalues of A, λk, k = 1, . . . , n, are all distinct then

n∑k=1

P (λk) = Tr (P (A)) .

Proof. By definition, for any eigenvalue λ and eigenvector v we must haveAv = λv and thus

P (A)v =

(m∑k=0

ckAk

)v =

m∑k=0

ck(Akv) =

m∑k=0

ckλkv

and thus P (λ) is an eigenvalue of P (A). For any matrix, A, the sum ofeigenvalues is equal to the trace of the matrix

n∑k=1

λk = Tr(A)

when multiplicities are taken into account. For the matrices considered inthe Lemma 1.1 all eigenvalues are distinct. Thus applying this property tothe matrix P (A) gives the desired statement.

Lemma 1.2. A Wishart distributed matrix W as defined in Definition 1.5will be a symmetric n× n matrix.

Proof. From the definition W is a p×p matrix such that W = XX>. Then

W> = (XX>)> = (X>)>X> = XX> = W

and thus W is symmetric.

The Gaussian Orthogonal Ensembles (GOE), the Gaussian Unitary En-sembles (GUE), the Gaussian Symplectic Ensembles (GSE) and the WishartEnsembles (WE) are well-known classical ensembles. More detailed discus-sions on these ensembles can be found in [6,7,75,163,207,220,292], here wewill only give their definitions and look at how the Vandermonde determi-nant appears in the probability density function for their eigenvalues.

34


It will prove useful that the formula given in Theorem 1.5 contains the

term∏i<j

(λi − λj) which we recognize from Theorem 1.2 as the determinant

of a Vandermonde matrix.In Section 2.3.7 we will use some results from Section 2.2 to find the

extreme values of the probability density function of a Wishart matrix. Herewe will prove a couple of properties of a Wishart matrix that will be usedin Section 2.3.7.

Lemma 1.1. Let P be a polynomial and A be a symmetric n × n matrix.If the eigenvalues of A, λk, k = 1, . . . , n, are all distinct then

n∑k=1

P (λk) = Tr (P (A)) .

Proof. By definition, for any eigenvalue λ and eigenvector v we must haveAv = λv and thus

P (A)v =

(m∑k=0

ckAk

)v =

m∑k=0

ck(Akv) =

m∑k=0

ckλkv

and thus P (λ) is an eigenvalue of P (A). For any matrix, A, the sum ofeigenvalues is equal to the trace of the matrix

n∑k=1

λk = Tr(A)

when multiplicities are taken into account. For the matrices considered inthe Lemma 1.1 all eigenvalues are distinct. Thus applying this property tothe matrix P (A) gives the desired statement.

Lemma 1.2. A Wishart distributed matrix W as defined in Definition 1.5will be a symmetric n× n matrix.

Proof. From the definition W is a p×p matrix such that W = XX>. Then

W> = (XX>)> = (X>)>X> = XX> = W

and thus W is symmetric.

The Gaussian Orthogonal Ensembles (GOE), the Gaussian Unitary En-sembles (GUE), the Gaussian Symplectic Ensembles (GSE) and the WishartEnsembles (WE) are well-known classical ensembles. More detailed discus-sions on these ensembles can be found in [6,7,75,163,207,220,292], here wewill only give their definitions and look at how the Vandermonde determi-nant appears in the probability density function for their eigenvalues.

34

36


Definition 1.6. The Gaussian Orthogonal Ensemble (GOE) is characterisedby a symmetric matrix X with real elements. The diagonal entries of X areindependent and identically distributes (i.i.d) with a standard normal distri-bution N (0, 1) while the off-diagonal entries are i.i.d with a standard normaldistribution N1(0, 1/2). That is, a random matrix X gives a GOE, if it issymmetric and real-valued (Xij = Xji) and has

X−ij =

√2ξii ∼ N1(0, 1), if i = j

ξij ∼ N1(0, 1/2), i < j.(10)

Definition 1.7. The Gaussian Unitary Ensemble (GUE) is characterisedby Hermitian (that is H>

∗= H where >

∗denotes the conjugate transpose)

complex-valued matrices H. The diagonal entries of H are independent andidentically distributes (i.i.d) with a standard normal distribution N (0, 1)while the off-diagonal entries are i.i.d with a standard normal distributionN2(0, 1/2). In other words, a random matrix H belongs to the GUE, if it iscomplex-valued, Hermitian, and the entries satisfy

Hij =

√2ξii ∼ N2(0, 1), if i = j

1√2(ξij + iηij) ∼ N2(0, 1/2), i < j,

(11)

where i is the imaginary unit.

Definition 1.8. The Gaussian Symplectic Ensemble (GSE) is characterisedby a matrix, S, with quaternion elements that is self-dual (that is S>

∗= S

where >∗

denotes the conjugate transpose of a quaternion). The diagonalentries H are independent and identically distributes (i.i.d) with a standardnormal distribution N (0, 1) while the off-diagonal entries are i.i.d with astandard normal distribution N4(0, 1/2). In other words, a random matrixS belongs to the GUE, if it is complex-valued, Hermitian, and the entriessatisfy

Hij =

√2ξii ∼ N2(0, 1), if i = j

1√2(ξij + iαij + jβij) + kγij ∼ N4(0, 1/2), i < j,

(12)

where i, j and k are the fundamental quaternion units.

Definition 1.9. The Wishart Ensembles (WE),Wβ(m,n),m ≥ n, are char-acterised by the symmetric, Hermitian or self-dual matrix W = Wβ(N,N)obtained as W = AA>,W = HH>, or W = SS> where > represents theappropriate transpose as given in the definition of the GOE, GUE and GSErespectively.

To obtain the joint eigenvalue densities for random matrices, we applythe the principle of matrix factorization, for instance if the random matrix Xis expressed as X = QΛΛΛQ>, then ΛΛΛ directly gives the eigenvalues of X [138].

35


Definition 1.6. The Gaussian Orthogonal Ensemble (GOE) is characterisedby a symmetric matrix X with real elements. The diagonal entries of X areindependent and identically distributes (i.i.d) with a standard normal distri-bution N (0, 1) while the off-diagonal entries are i.i.d with a standard normaldistribution N1(0, 1/2). That is, a random matrix X gives a GOE, if it issymmetric and real-valued (Xij = Xji) and has

X−ij =

√2ξii ∼ N1(0, 1), if i = j

ξij ∼ N1(0, 1/2), i < j.(10)

Definition 1.7. The Gaussian Unitary Ensemble (GUE) is characterisedby Hermitian (that is H>

∗= H where >

∗denotes the conjugate transpose)

complex-valued matrices H. The diagonal entries of H are independent andidentically distributes (i.i.d) with a standard normal distribution N (0, 1)while the off-diagonal entries are i.i.d with a standard normal distributionN2(0, 1/2). In other words, a random matrix H belongs to the GUE, if it iscomplex-valued, Hermitian, and the entries satisfy

Hij =

√2ξii ∼ N2(0, 1), if i = j

1√2(ξij + iηij) ∼ N2(0, 1/2), i < j,

(11)

where i is the imaginary unit.

Definition 1.8. The Gaussian Symplectic Ensemble (GSE) is characterisedby a matrix, S, with quaternion elements that is self-dual (that is S>

∗= S

where >∗

denotes the conjugate transpose of a quaternion). The diagonalentries H are independent and identically distributes (i.i.d) with a standardnormal distribution N (0, 1) while the off-diagonal entries are i.i.d with astandard normal distribution N4(0, 1/2). In other words, a random matrixS belongs to the GUE, if it is complex-valued, Hermitian, and the entriessatisfy

Hij =

√2ξii ∼ N2(0, 1), if i = j

1√2(ξij + iαij + jβij) + kγij ∼ N4(0, 1/2), i < j,

(12)

where i, j and k are the fundamental quaternion units.

Definition 1.9. The Wishart Ensembles (WE),Wβ(m,n),m ≥ n, are char-acterised by the symmetric, Hermitian or self-dual matrix W = Wβ(N,N)obtained as W = AA>,W = HH>, or W = SS> where > represents theappropriate transpose as given in the definition of the GOE, GUE and GSErespectively.

To obtain the joint eigenvalue densities for random matrices, we applythe the principle of matrix factorization, for instance if the random matrix Xis expressed as X = QΛΛΛQ>, then ΛΛΛ directly gives the eigenvalues of X [138].

35

37


Applying the Jacobian technique for joint density transformation, see forexample [7], this yields the joint densities of eigenvalues and eigenvectors.

Lemma 1.3. The three Gaussian ensembles have joint eigenvalues proba-bility density function given by

Gaussian: Pβ(λλλ) = CβN

∏i<j

|λ1 − λ2|β exp

(−1

2

N∑i=1

λ2i

)(13)

where β = 1 representing reals, β = 2 representing the complexes, and β = 4representing the quaternion, and

CβN = (2π)−N/2N∏j=1

Γ (1 + β/2)

Γ (1 + jβ/2).

Lemma 1.4. The Wishart ensembles have a joint eigenvalue probabilitydensity distribution given by

Wishart: Pβ(λ) = Cβ,αN

∏i<j

|λ1 − λ2|β∏i

λα−pi exp

(−1

2

N∑i=1

λ2i

)(14)

where α = β2m and p = 1 + β

2 (N − 1). The β parameter is decided by whattype of elements are in the Wishart matrix, real-valued elements correspondsto β = 1, complex-valued elements correspond to β = 2 and quaternionelements correspond to β = 4, and the normalizing constant Cβ,αN is givenby

Cβ,αN = 2−NαN∏j=1

Γ (1 + β/2)

Γ (1 + jβ/2) Γ(α− β

2 (n− j)) . (15)

More information on Lemma 1.3 and 1.4 can be found in standard texton random matrix theory, see for example [138,207,220].

Thus the joint eigenvalue probability density distribution for all the en-sembles can be summarized in the following theorem (for more detail see forexample [83,163,207]).

Theorem 1.6. Suppose that X belongs to one of the ensembles discussedgiven by Definitions 1.6–1.9. Then the distribution of eigenvalues of XN isgiven by

PX(x1, · · · , xN ) = CβN

∏i<j

|xi − xj |β exp

(−β

4

∑i

x2i

)(16)

where C(β)N are normalized constants and can be computed explicitly and β

is determined by the elements of X as in Lemma 1.4.

36


Applying the Jacobian technique for joint density transformation, see forexample [7], this yields the joint densities of eigenvalues and eigenvectors.

Lemma 1.3. The three Gaussian ensembles have joint eigenvalues proba-bility density function given by

Gaussian: Pβ(λλλ) = CβN

∏i<j

|λ1 − λ2|β exp

(−1

2

N∑i=1

λ2i

)(13)

where β = 1 representing reals, β = 2 representing the complexes, and β = 4representing the quaternion, and

CβN = (2π)−N/2N∏j=1

Γ (1 + β/2)

Γ (1 + jβ/2).

Lemma 1.4. The Wishart ensembles have a joint eigenvalue probabilitydensity distribution given by

Wishart: Pβ(λ) = Cβ,αN

∏i<j

|λ1 − λ2|β∏i

λα−pi exp

(−1

2

N∑i=1

λ2i

)(14)

where α = β2m and p = 1 + β

2 (N − 1). The β parameter is decided by whattype of elements are in the Wishart matrix, real-valued elements correspondsto β = 1, complex-valued elements correspond to β = 2 and quaternionelements correspond to β = 4, and the normalizing constant Cβ,αN is givenby

Cβ,αN = 2−NαN∏j=1

Γ (1 + β/2)

Γ (1 + jβ/2) Γ(α− β

2 (n− j)) . (15)

More information on Lemma 1.3 and 1.4 can be found in standard texton random matrix theory, see for example [138,207,220].

Thus the joint eigenvalue probability density distribution for all the en-sembles can be summarized in the following theorem (for more detail see forexample [83,163,207]).

Theorem 1.6. Suppose that X belongs to one of the ensembles discussedgiven by Definitions 1.6–1.9. Then the distribution of eigenvalues of XN isgiven by

PX(x1, · · · , xN ) = CβN

∏i<j

|xi − xj |β exp

(−β

4

∑i

x2i

)(16)

where C(β)N are normalized constants and can be computed explicitly and β

is determined by the elements of X as in Lemma 1.4.

36

38

1.2. CURVE FITTING

From (16) it should be noted that the properties of a probability densityfunction, that is,

0 ≤ P(x) ≤ 1 and

∫RN

P(x)N∏i=1

dxi = 1

do hold as verified in [207]. We also notice that the term∏i<j

|xi − xj |β in

expression (16) is the determinant of the Vandermonde matrix raised to thepower β = 1, 2, 4, c.f. Theorem 1.2.

In Section 2.2 we will derive some results about the extreme points ofthe Vandermonde determinant and in Section 2.3.7 we will show how thisalso gives the extreme points for the probability density functions discussedhere.

1.2 Curve fitting

The process of constructing a mathematical curve so that it has the bestpossible fit to some series of data pints is usually referred to as curve fitting .Exactly what fit means and what constraints are put on the constructedcurve varies depending on context. In this section we will discuss a fewdifferent scenarios and methods that are related to the Vandermonde ma-trix and the methods used in later chapters to construct phenomenologicalmathematical models.

We will give an introduction to a few different interpolation methods inSections 1.2.1–1.2.2, that gives a curve that passes exactly through a finiteset of points. If we cannot make a curve that passes through the pointsexactly we will need to choose how to measure the distance between thecurve and the points in order to determine what curve fits the data pointsbest. In Sections 1.2.3–1.2.6 the so called least squares approach to this kindof problem is presented.

1.2.1 Linear interpolation

The problem of finding a function that generates a given set of points isusually referred to as an interpolation problem and the function generatingthe points is called an interpolating function. A common type of inter-polation problem is to find a continuous function, f , such that the givenset of points (x1, y1), (x2, y2), . . . can be generated by calculating the set(x1, f(x1)), (x2, f(x2)), . . .. Often the interpolating function is also a lin-ear combination of elementary functions, but interpolation can also be donein other ways, for instance with fractals (the classical texts on this is [15,16])or parametrised curves. For some examples, see Figure 1.2.

37

1.2. CURVE FITTING

From (16) it should be noted that the properties of a probability densityfunction, that is,

0 ≤ P(x) ≤ 1 and

∫RN

P(x)N∏i=1

dxi = 1

do hold as verified in [207]. We also notice that the term∏i<j

|xi − xj |β in

expression (16) is the determinant of the Vandermonde matrix raised to thepower β = 1, 2, 4, c.f. Theorem 1.2.

In Section 2.2 we will derive some results about the extreme points ofthe Vandermonde determinant and in Section 2.3.7 we will show how thisalso gives the extreme points for the probability density functions discussedhere.

1.2 Curve fitting

The process of constructing a mathematical curve so that it has the bestpossible fit to some series of data pints is usually referred to as curve fitting .Exactly what fit means and what constraints are put on the constructedcurve varies depending on context. In this section we will discuss a fewdifferent scenarios and methods that are related to the Vandermonde ma-trix and the methods used in later chapters to construct phenomenologicalmathematical models.

We will give an introduction to a few different interpolation methods inSections 1.2.1–1.2.2, that gives a curve that passes exactly through a finiteset of points. If we cannot make a curve that passes through the pointsexactly we will need to choose how to measure the distance between thecurve and the points in order to determine what curve fits the data pointsbest. In Sections 1.2.3–1.2.6 the so called least squares approach to this kindof problem is presented.

1.2.1 Linear interpolation

The problem of finding a function that generates a given set of points isusually referred to as an interpolation problem and the function generatingthe points is called an interpolating function. A common type of inter-polation problem is to find a continuous function, f , such that the givenset of points (x1, y1), (x2, y2), . . . can be generated by calculating the set(x1, f(x1)), (x2, f(x2)), . . .. Often the interpolating function is also a lin-ear combination of elementary functions, but interpolation can also be donein other ways, for instance with fractals (the classical texts on this is [15,16])or parametrised curves. For some examples, see Figure 1.2.

37

39


Figure 1.2: Some examples of different interpolating curves. The set of redpoints are interpolated by a polynomial (left), a self-affine fractal(middle) and a Lissajous curve (right).

In the case of the interpolating function being a linear combination ofother functions and the interpolation is achieved by changing the coefficientsof the linear combination this is said to be a linear model (not to be confusedwith linear interpolation that is interpolation with piecewise straight lines).

For linear models the interpolation problem can be described using al-ternant matrices. Suppose we want to find a function

f(x) =m∑i=1

aigi(x) (17)

that fits as well as possible to the data points (xi, yi), i = 1, . . . , n. Wethen get an interpolation problem described by the linear equation systemAa = y where a are the coefficients of f , y are the data values and X is theappropriate alternant matrix,

X =

g1(x1) g2(x1) . . . gm(x1)g1(x2) g2(x2) . . . gm(x2)

......

. . ....

g1(xn) g2(xn) . . . gm(xn)

, a =

a1

a2...an

, y =

y1

y2...yn

.Polynomial interpolation

A classical form of interpolation is polynomial interpolation where n datapoints are interpolated by a polynomial of at most degree n− 1.

The Vandermonde matrix can be used to describe this type of interpola-tion problem simply by rewriting the equation system given by p(xk) = yk,k = 1, . . . , n as a matrix equation

1 x1 · · · xn−11

1 x2 · · · xn−12

......

. . ....

1 xn · · · xn−1n

a1

a2...an

=

y1

y2...yn

.

38


Figure 1.2: Some examples of different interpolating curves. The set of redpoints are interpolated by a polynomial (left), a self-affine fractal(middle) and a Lissajous curve (right).

In the case of the interpolating function being a linear combination ofother functions and the interpolation is achieved by changing the coefficientsof the linear combination this is said to be a linear model (not to be confusedwith linear interpolation that is interpolation with piecewise straight lines).

For linear models the interpolation problem can be described using al-ternant matrices. Suppose we want to find a function

f(x) =m∑i=1

aigi(x) (17)

that fits as well as possible to the data points (xi, yi), i = 1, . . . , n. Wethen get an interpolation problem described by the linear equation systemAa = y where a are the coefficients of f , y are the data values and X is theappropriate alternant matrix,

X =

g1(x1) g2(x1) . . . gm(x1)g1(x2) g2(x2) . . . gm(x2)

......

. . ....

g1(xn) g2(xn) . . . gm(xn)

, a =

a1

a2...an

, y =

y1

y2...yn

.Polynomial interpolation

A classical form of interpolation is polynomial interpolation where n datapoints are interpolated by a polynomial of at most degree n− 1.

The Vandermonde matrix can be used to describe this type of interpola-tion problem simply by rewriting the equation system given by p(xk) = yk,k = 1, . . . , n as a matrix equation

1 x1 · · · xn−11

1 x2 · · · xn−12

......

. . ....

1 xn · · · xn−1n

a1

a2...an

=

y1

y2...yn

.

38

40

1.2. CURVE FITTING

That the polynomial is unique (if it exists) is easy to see when consideringthe determinant of the Vandermonde matrix

det(Vn(x1, . . . , xn)) =∏


Clearly this determinant is non-zero whenever all xi are distinct which meansthat the matrix is invertible whenever all xi are distinct. If not all xi aredistinct there is no function of the x coordinate that can interpolate all thepoints.

There are several ways to construct the interpolating polynomial withoutexplicitly inverting the Vandermonde matrix. The most straight-forward isprobably Lagrange interpolation, named after Joseph-Louis Lagrange (1736–1813) [167] who independently discovered it a few years after Edward Waring(1736–1798) [288].

The idea behind Lagrange interpolation is simple, construct a set of npolynomials p1, p2, . . . , pn such that

pi(xj) =

0, i 6= j

1, i = j

and then construct the final interpolating polynomial by the sum of thesepi weighted by the corresponding yi.

The pi polynomials are called Lagrange basis polynomials and can easilybe constructed by placing the roots appropriately and then normalizing theresult such that pi(xi) = 1, which gives the expression

pi(x) =(x− x1) · · · (x− xi−1)(x− xi+1) · · · (x− xn)

(xi − x1) · · · (xi − xi−1)(xi − xi+1) · · · (xi − xn).

The explicit formula for the full interpolating polynomial is

p(x) =

n∑k=1

yk(x− x1) · · · (x− xk−1)(x− xk+1) · · · (x− xn)

(xk − x1) · · · (xk − xk−1)(xk − xk+1) · · · (xk − xn)

and from this formula the expression for the inverse of the Vandermondematrix can be found by noting that the jth row of the inverse will consistof the coefficients of pj , the resulting expression for the elements is given inTheorem 1.4.

Polynomial interpolation is mostly used when the data set we wish tointerpolate is small. The main reason for this is the instability of the inter-polation method. One example of this is Runge’s phenomenon that showsthat when certain functions are approximated by polynomial interpolationfitted to equidistantly sampled points will sometimes lose precision when thenumber of interpolating points is increased, see Figure 1.4 for an example.

39

1.2. CURVE FITTING

That the polynomial is unique (if it exists) is easy to see when consideringthe determinant of the Vandermonde matrix

det(Vn(x1, . . . , xn)) =∏


Clearly this determinant is non-zero whenever all xi are distinct which meansthat the matrix is invertible whenever all xi are distinct. If not all xi aredistinct there is no function of the x coordinate that can interpolate all thepoints.

There are several ways to construct the interpolating polynomial withoutexplicitly inverting the Vandermonde matrix. The most straight-forward isprobably Lagrange interpolation, named after Joseph-Louis Lagrange (1736–1813) [167] who independently discovered it a few years after Edward Waring(1736–1798) [288].

The idea behind Lagrange interpolation is simple, construct a set of npolynomials p1, p2, . . . , pn such that

pi(xj) =

0, i 6= j

1, i = j

and then construct the final interpolating polynomial by the sum of thesepi weighted by the corresponding yi.

The pi polynomials are called Lagrange basis polynomials and can easilybe constructed by placing the roots appropriately and then normalizing theresult such that pi(xi) = 1, which gives the expression

pi(x) =(x− x1) · · · (x− xi−1)(x− xi+1) · · · (x− xn)

(xi − x1) · · · (xi − xi−1)(xi − xi+1) · · · (xi − xn).

The explicit formula for the full interpolating polynomial is

p(x) =

n∑k=1

yk(x− x1) · · · (x− xk−1)(x− xk+1) · · · (x− xn)

(xk − x1) · · · (xk − xk−1)(xk − xk+1) · · · (xk − xn)

and from this formula the expression for the inverse of the Vandermondematrix can be found by noting that the jth row of the inverse will consistof the coefficients of pj , the resulting expression for the elements is given inTheorem 1.4.

Polynomial interpolation is mostly used when the data set we wish tointerpolate is small. The main reason for this is the instability of the inter-polation method. One example of this is Runge’s phenomenon that showsthat when certain functions are approximated by polynomial interpolationfitted to equidistantly sampled points will sometimes lose precision when thenumber of interpolating points is increased, see Figure 1.4 for an example.

39

41


0 1 2 3 4 5 6 7 8

−2

0

2

p1(x) p2(x) p3(x) p4(x) p(x) (x, y)

Figure 1.3: Illustration of Lagrange interpolation of 4 data points. The red

dots are the data set and p(x) =4∑

k=1

ykp(xk) is the interpolating

polynomial.

One way to predict this instability of polynomial interpolation is thatthe conditional number of the Vandermonde matrix can be very large forequidistant points [108].

There are different ways to mitigate the issue of stability, for examplechoosing data points that minimize the conditional number of the relevantmatrix [106, 108] or by choosing a polynomial basis that is more stable forthe given set of data points such as Bernstein polynomials in the case ofequidistant points [222]. Other polynomial schemes can also be considered,for instance by interpolating with different basis functions in different inter-vals, for example using polynomial splines.

Naturally another choice is to instead of polynomials choose basis func-tions that are more suitable to the problem at hand. For an example of thissee Section 3.3.

While the instability of polynomial interpolation does not prevent it frombeing useful for analytical examinations it is generally considered imprac-tical when there is noise present or when calculations are performed withlimited precision. Often interpolating polynomials are not constructed byinverting the Vandermonde matrix or calculating the Lagrange basis poly-nomials, instead a more computationally efficient method such as Newtoninterpolation or Neville’s algorithm are used [235]. There are some variantsof Lagrange interpolation, such as barycentric Lagrange interpolation, thathave good computational performance [21].

In applications where the data is noisy it is often suitable to use leastsquares fitting, which is discussed in Section 1.2.3, instead of interpolation.

40


0 1 2 3 4 5 6 7 8

−2

0

2

p1(x) p2(x) p3(x) p4(x) p(x) (x, y)

Figure 1.3: Illustration of Lagrange interpolation of 4 data points. The red

dots are the data set and p(x) =4∑

k=1

ykp(xk) is the interpolating

polynomial.

One way to predict this instability of polynomial interpolation is thatthe conditional number of the Vandermonde matrix can be very large forequidistant points [108].

There are different ways to mitigate the issue of stability, for examplechoosing data points that minimize the conditional number of the relevantmatrix [106, 108] or by choosing a polynomial basis that is more stable forthe given set of data points such as Bernstein polynomials in the case ofequidistant points [222]. Other polynomial schemes can also be considered,for instance by interpolating with different basis functions in different inter-vals, for example using polynomial splines.

Naturally another choice is to instead of polynomials choose basis func-tions that are more suitable to the problem at hand. For an example of thissee Section 3.3.

While the instability of polynomial interpolation does not prevent it frombeing useful for analytical examinations it is generally considered imprac-tical when there is noise present or when calculations are performed withlimited precision. Often interpolating polynomials are not constructed byinverting the Vandermonde matrix or calculating the Lagrange basis poly-nomials, instead a more computationally efficient method such as Newtoninterpolation or Neville’s algorithm are used [235]. There are some variantsof Lagrange interpolation, such as barycentric Lagrange interpolation, thathave good computational performance [21].

In applications where the data is noisy it is often suitable to use leastsquares fitting, which is discussed in Section 1.2.3, instead of interpolation.

40

42

1.2. CURVE FITTING

−40−20 0 20 400

0.5

1

1.5

−40−20 0 20 400

0.5

1

1.5

−40−20 0 20 400

0.5

1

1.5

Figure 1.4: Illustration of Runge’s phenomenon. Here we attempt to approx-imate a function (dashed line) by polynomial interpolation (solidline). With 7 equidistant sample points (left figure) the approx-imation is poor near the edges of the interval and increasing thenumber of sample points to 14 (center) and 19 (right) clearly re-duces accuracy at the edges further.

Finally we will discuss an interesting and important (but for the restof the thesis irrelevant) form of polynomial interpolation called Hermiteinterpolation where it is not only required that p(xk) = yk but also thatthe derivatives up to a certain order (sometimes allowed vary per point) arealso given. This requires a higher degree polynomial that can be found bysolving the equation system

p(xk) = yk0

p′(xk) = yk1...

p(i)(xk) = yki

for all k = 1, 2, . . . , n where ki are integers that defines the order of thederivative that needs to match at the point given by xk.

When this equation system is written as a matrix equation the resulting

matrix, C, will have dimension m×m with m =

n∑i=1

ki with rows given by

Ca,b =

0, b ≤ kj

(b−1)!(b−c−1)!x

b−c−1k , b > kj

with c = a−j∑i=1

ki and c < a ≤ c+ kj+1.

The matrix C is called a confluent Vandermonde matrix and has beenstudied extensively since Hermite interpolation is important both for nu-merical and analytical purposes. For example the confluent Vandermonde

41

1.2. CURVE FITTING

−40−20 0 20 400

0.5

1

1.5

−40−20 0 20 400

0.5

1

1.5

−40−20 0 20 400

0.5

1

1.5

Figure 1.4: Illustration of Runge’s phenomenon. Here we attempt to approx-imate a function (dashed line) by polynomial interpolation (solidline). With 7 equidistant sample points (left figure) the approx-imation is poor near the edges of the interval and increasing thenumber of sample points to 14 (center) and 19 (right) clearly re-duces accuracy at the edges further.

Finally we will discuss an interesting and important (but for the restof the thesis irrelevant) form of polynomial interpolation called Hermiteinterpolation where it is not only required that p(xk) = yk but also thatthe derivatives up to a certain order (sometimes allowed vary per point) arealso given. This requires a higher degree polynomial that can be found bysolving the equation system

p(xk) = yk0

p′(xk) = yk1...

p(i)(xk) = yki

for all k = 1, 2, . . . , n where ki are integers that defines the order of thederivative that needs to match at the point given by xk.

When this equation system is written as a matrix equation the resulting

matrix, C, will have dimension m×m with m =

n∑i=1

ki with rows given by

Ca,b =

0, b ≤ kj

(b−1)!(b−c−1)!x

b−c−1k , b > kj

with c = a−j∑i=1

ki and c < a ≤ c+ kj+1.

The matrix C is called a confluent Vandermonde matrix and has beenstudied extensively since Hermite interpolation is important both for nu-merical and analytical purposes. For example the confluent Vandermonde

41

43


matrix also has a very elegant formula for the determinant [3]

det(C) =∏

1≤i<j≤n(xj − xi)(ki+1)(kj+1).

There are also many results related to its inverse and numerical proper-ties, classical examples are [104, 105, 107], some further examples are men-tioned on page 29 but this is a vanishingly small part of the total literatureon the subject.

1.2.2 Generalized divided differences and interpolation

In Section 1.2.1 we saw how the coefficients of an interpolating polyno-mial could be computed by inverting the Vandermonde matrix or using theLagrange basis polynomials. Another method for the coefficients of thepolynomials is based on a computation called divided differences.

Definition 1.10. Let x0,. . . , xn then the divided differences operator thatacts on a function f(x) is defined as

[x0, . . . , xn]f(x) =

f(x0), n = 0,[x1, . . . , xn]f(x)− [x0, . . . , xn−1]f(x)

xn − x1, n > 0.

The reason that the divided difference operator is interesting in polyno-mial interpolation is that if we apply it to two distinct points, x0 and x1,and a function f(x) then the result is the slope of a line that passes throughthe two points (x0, f(x0)) and (x1, f(x1)),

[x0, x1]f(x) =[x1]f(x)− [x0]f(x)

x1 − x0=f(x1)− f(x0)

x1 − x0.

A line that passes through the two points can then be constructed like this

p(x) = f(x0) + (x− x0)[x0, x1]f(x).

It can similarly be shown that a polynomial that interpolates a set of points

(x0, f(x0)), . . . , (xn, f(xn))

can be written

p(x) =f(x0) + (x− x0)[x0, x1]f(x) + (x− x0)(x− x1)[x0, x1, x2]f(x) + . . .

+ (x− x0) · · · (x− xn−1)[x0, . . . , xn]f(x).

This method for interpolation is usually referred to as Newton interpolationand is probably the most well-known application of divided differences. In

42


matrix also has a very elegant formula for the determinant [3]

det(C) =∏

1≤i<j≤n(xj − xi)(ki+1)(kj+1).

There are also many results related to its inverse and numerical proper-ties, classical examples are [104, 105, 107], some further examples are men-tioned on page 29 but this is a vanishingly small part of the total literatureon the subject.

1.2.2 Generalized divided differences and interpolation

In Section 1.2.1 we saw how the coefficients of an interpolating polyno-mial could be computed by inverting the Vandermonde matrix or using theLagrange basis polynomials. Another method for the coefficients of thepolynomials is based on a computation called divided differences.

Definition 1.10. Let x0,. . . , xn then the divided differences operator thatacts on a function f(x) is defined as

[x0, . . . , xn]f(x) =

f(x0), n = 0,[x1, . . . , xn]f(x)− [x0, . . . , xn−1]f(x)

xn − x1, n > 0.

The reason that the divided difference operator is interesting in polyno-mial interpolation is that if we apply it to two distinct points, x0 and x1,and a function f(x) then the result is the slope of a line that passes throughthe two points (x0, f(x0)) and (x1, f(x1)),

[x0, x1]f(x) =[x1]f(x)− [x0]f(x)

x1 − x0=f(x1)− f(x0)

x1 − x0.

A line that passes through the two points can then be constructed like this

p(x) = f(x0) + (x− x0)[x0, x1]f(x).

It can similarly be shown that a polynomial that interpolates a set of points

(x0, f(x0)), . . . , (xn, f(xn))

can be written

p(x) =f(x0) + (x− x0)[x0, x1]f(x) + (x− x0)(x− x1)[x0, x1, x2]f(x) + . . .

+ (x− x0) · · · (x− xn−1)[x0, . . . , xn]f(x).

This method for interpolation is usually referred to as Newton interpolationand is probably the most well-known application of divided differences. In

42

44

1.2. CURVE FITTING

some literature, e.g. [65], this property is even used as a definition for divideddifferences.

Since we expect to find the same polynomial whether we use the Lan-grange interpolation method described in Section 1.2.1 or the Newton inter-polation method described above we also expect there to be some relationbetween the divided difference operator and the Vandermonde determinant.Turns out there is a fairly simple relation, see [253] for details.

Lemma 1.5. The divided difference operator defined in Definition 1.10 canalso be written as

[x0, . . . , xn]f(x) =

∣∣∣∣∣∣∣∣∣1 x0 · · · xn−1

0 f(x0)

1 x1 · · · xn−11 f(x1)

......

. . ....

...1 xn · · · xn−1

n f(xn)

∣∣∣∣∣∣∣∣∣vn(x0, . . . , xn)

, (18)

where vn(x0, . . . , xn) denotes the Vandermonde determinant.

Remark 1.5. Sometimes, see for example [253], the relation in Lemma 1.5is used as the definition of the divided difference operator.

The divided differences operator can also be used to describe the er-ror that one gets when a function is approximated by interpolating with apolynomial, the following lemma is from [156].

Lemma 1.6. Let p(x) be a polynomial of degree smaller than or equal ton that interpolates the points (xi, f(xi)), i = 0, . . . , n. For any x 6= xi,i = 0, . . . , n the error f(x)− p(x) is given by

f(x)− p(x) = [x0, . . . , xn, x]f(x)n∏i=0

(x− xi).

Combining Lemma 1.5 and Lemma 1.6 gives

f(x)− p(x) =

∣∣∣∣∣∣∣∣∣∣∣

1 x0 · · · xn−10 f(x0)

1 x1 · · · xn−11 f(x1)

......

. . ....

...1 xn · · · xn−1

n f(xn)1 x · · · xn−1 f(x)

∣∣∣∣∣∣∣∣∣∣∣vn(x0, . . . , xn, x)

n∏i=0

(x− xi).

which gives some insight to why the value of the Vandermonde determinantis important when choosing interpolation points.

Another popular application of the divided differences operator is theconstruction of so calledB-splines, piecewise polynomial functions that allow

43

1.2. CURVE FITTING

some literature, e.g. [65], this property is even used as a definition for divideddifferences.

Since we expect to find the same polynomial whether we use the Lan-grange interpolation method described in Section 1.2.1 or the Newton inter-polation method described above we also expect there to be some relationbetween the divided difference operator and the Vandermonde determinant.Turns out there is a fairly simple relation, see [253] for details.

Lemma 1.5. The divided difference operator defined in Definition 1.10 canalso be written as

[x0, . . . , xn]f(x) =

∣∣∣∣∣∣∣∣∣1 x0 · · · xn−1

0 f(x0)

1 x1 · · · xn−11 f(x1)

......

. . ....

...1 xn · · · xn−1

n f(xn)

∣∣∣∣∣∣∣∣∣vn(x0, . . . , xn)

, (18)

where vn(x0, . . . , xn) denotes the Vandermonde determinant.

Remark 1.5. Sometimes, see for example [253], the relation in Lemma 1.5is used as the definition of the divided difference operator.

The divided differences operator can also be used to describe the er-ror that one gets when a function is approximated by interpolating with apolynomial, the following lemma is from [156].

Lemma 1.6. Let p(x) be a polynomial of degree smaller than or equal ton that interpolates the points (xi, f(xi)), i = 0, . . . , n. For any x 6= xi,i = 0, . . . , n the error f(x)− p(x) is given by

f(x)− p(x) = [x0, . . . , xn, x]f(x)n∏i=0

(x− xi).

Combining Lemma 1.5 and Lemma 1.6 gives

f(x)− p(x) =

∣∣∣∣∣∣∣∣∣∣∣

1 x0 · · · xn−10 f(x0)

1 x1 · · · xn−11 f(x1)

......

. . ....

...1 xn · · · xn−1

n f(xn)1 x · · · xn−1 f(x)

∣∣∣∣∣∣∣∣∣∣∣vn(x0, . . . , xn, x)

n∏i=0

(x− xi).

which gives some insight to why the value of the Vandermonde determinantis important when choosing interpolation points.

Another popular application of the divided differences operator is theconstruction of so calledB-splines, piecewise polynomial functions that allow

43

45


for very efficient storage and computation of a variety of shapes. The conceptof (mathematical) splines first appeared in the 1940s [251,252] and B-splineswere developed in the 1960s and 1970s [22,63,64]. We can define a B-splineusing the divided differences as follows.

Definition 1.11. Given a sequence, · · · ≤ t−1 ≤ t0 ≤ t1 ≤ t2 ≤ · · · we candefine the kth B-spline of order m as

Bk,m(x) =

(−1)m[tk, . . . , tk+m]gk(x, t), tk ≤ x < tk+1,

0, otherwise,

where

gk(x, t) =

(x− t)k−1, x ≥ t,0, otherwise.

and the divided difference operator acts with respect to t.

Remark 1.6. There are several different ways to define B-splines, abovewe followed the definition in [253]. In modern literature it is more commonthat B-splines and their computation are described from the perspective ofso-called blossoms [99,239,240] rather that the divided difference description.

B-splines can be used for many things, for example approximation theory[204], geometric modelling [99] and wavelets construction [49]. We will notdiscuss their use further in this thesis.

If we want to do linear interpolation and use some other set of basisfunctions other than the monomials, as in (17), then we need to define ageneralized version of the divided difference operator.

Definition 1.12. Given a set of m linearly independent functions, G = gi,and n values, x0,. . .,xn, then the generalized divided differences operator thatacts on a function f(x) is defined as

[x1, . . . , xn]Gf(x) =

∣∣∣∣∣∣∣∣∣g1(x1) g2(x1) · · · gn−1(x1) f(x1)g1(x2) g2(x2) · · · gn−1(x2) f(x2)

......

. . ....

...g1(xn) g2(xn) · · · gn−1(xn) f(xn)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣g1(x1) g2(x1) · · · gn(x1)g1(x2) g2(x2) · · · gn(x2)

......

. . ....

g1(xn) g2(xn) · · · gn(xn)

∣∣∣∣∣∣∣∣∣

.

Remark 1.7. We mentioned previously that the divided difference operatorcan be used to construct B-splines and using the generalized divided differ-ence operator similar tools can be constructed using other sets functionsthan polynomials as a basis, see for example [196].

44


for very efficient storage and computation of a variety of shapes. The conceptof (mathematical) splines first appeared in the 1940s [251,252] and B-splineswere developed in the 1960s and 1970s [22,63,64]. We can define a B-splineusing the divided differences as follows.

Definition 1.11. Given a sequence, · · · ≤ t−1 ≤ t0 ≤ t1 ≤ t2 ≤ · · · we candefine the kth B-spline of order m as

Bk,m(x) =

(−1)m[tk, . . . , tk+m]gk(x, t), tk ≤ x < tk+1,

0, otherwise,

where

gk(x, t) =

(x− t)k−1, x ≥ t,0, otherwise.

and the divided difference operator acts with respect to t.

Remark 1.6. There are several different ways to define B-splines, abovewe followed the definition in [253]. In modern literature it is more commonthat B-splines and their computation are described from the perspective ofso-called blossoms [99,239,240] rather that the divided difference description.

B-splines can be used for many things, for example approximation theory[204], geometric modelling [99] and wavelets construction [49]. We will notdiscuss their use further in this thesis.

If we want to do linear interpolation and use some other set of basisfunctions other than the monomials, as in (17), then we need to define ageneralized version of the divided difference operator.

Definition 1.12. Given a set of m linearly independent functions, G = gi,and n values, x0,. . .,xn, then the generalized divided differences operator thatacts on a function f(x) is defined as

[x1, . . . , xn]Gf(x) =

∣∣∣∣∣∣∣∣∣g1(x1) g2(x1) · · · gn−1(x1) f(x1)g1(x2) g2(x2) · · · gn−1(x2) f(x2)

......

. . ....

...g1(xn) g2(xn) · · · gn−1(xn) f(xn)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣g1(x1) g2(x1) · · · gn(x1)g1(x2) g2(x2) · · · gn(x2)

......

. . ....

g1(xn) g2(xn) · · · gn(xn)

∣∣∣∣∣∣∣∣∣

.

Remark 1.7. We mentioned previously that the divided difference operatorcan be used to construct B-splines and using the generalized divided differ-ence operator similar tools can be constructed using other sets functionsthan polynomials as a basis, see for example [196].

44

46

1.2. CURVE FITTING

1.2.3 Least squares fitting

If it is not necessary to exactly reproduce the series of data points a com-monly applied alternative to interpolation is least squares fitting.

A least squares fitting of a mathematical model to a set of data points(xi, yi), i = 1, . . . , n is the choice of parameters of the model, here denotedβ, chosen such that the sum of the squares of the residuals

S(β) =n∑i=1

(yi − f(β;xi))2

is minimized. This choice is appropriate if data series is affected by inde-pendent and normally distributed noise, see Section 1.3.1.

The most wide-spread form of least squares fitting is linear least squaresfitting where, analogously to linear interpolation, the function f(β;x) de-pends linearly on β. This case has a unique solution that is simple to find.It is commonly known as the least squares method and we describe it indetail in the next section. With a non-linear f(β;x) it is usually much moredifficult to find the least squares fitting and often numerical methods areused, e.g. the Marquardt least squares method described in Section 1.2.6.

In Section 3.2 we present a scheme for approximating electrostaticaldischarges to ensure electromagnetic compatibility (see Section 1.5) thatuses both the least squares method and the Marquard least squares method.

In Chapter 4 we fit several models to estimated mortality rates using non-linear least squares fitting and compare the result in various way describedin Sections 1.3.1–1.3.3.

1.2.4 Linear least squares fitting

Suppose we want to find a function

f(x) =

m∑i=1

βigi(x) (19)

that fits as well as possible in the least squares sense to the data points(xi, yi), i = 1, . . . , n, n > m. We then get a curve fitting problem describedby the linear equation system Aβ = y where β are the coefficients of f , yis the vector of data values and A is the appropriate alternant matrix,

A =

g1(x1) g2(x1) . . . gm(x1)g1(x2) g2(x2) . . . gm(x2)

......

. . ....

g1(xn) g2(xn) . . . gm(xn)

, β =

β1

β2...βn

, y =

y1

y2...yn

.This is an overdetermined version of the linear interpolation problem

described in Section 1.2.1.

45

1.2. CURVE FITTING

1.2.3 Least squares fitting

If it is not necessary to exactly reproduce the series of data points a com-monly applied alternative to interpolation is least squares fitting.

A least squares fitting of a mathematical model to a set of data points(xi, yi), i = 1, . . . , n is the choice of parameters of the model, here denotedβ, chosen such that the sum of the squares of the residuals

S(β) =n∑i=1

(yi − f(β;xi))2

is minimized. This choice is appropriate if data series is affected by inde-pendent and normally distributed noise, see Section 1.3.1.

The most wide-spread form of least squares fitting is linear least squaresfitting where, analogously to linear interpolation, the function f(β;x) de-pends linearly on β. This case has a unique solution that is simple to find.It is commonly known as the least squares method and we describe it indetail in the next section. With a non-linear f(β;x) it is usually much moredifficult to find the least squares fitting and often numerical methods areused, e.g. the Marquardt least squares method described in Section 1.2.6.

In Section 3.2 we present a scheme for approximating electrostaticaldischarges to ensure electromagnetic compatibility (see Section 1.5) thatuses both the least squares method and the Marquard least squares method.

In Chapter 4 we fit several models to estimated mortality rates using non-linear least squares fitting and compare the result in various way describedin Sections 1.3.1–1.3.3.

1.2.4 Linear least squares fitting

Suppose we want to find a function

f(x) =

m∑i=1

βigi(x) (19)

that fits as well as possible in the least squares sense to the data points(xi, yi), i = 1, . . . , n, n > m. We then get a curve fitting problem describedby the linear equation system Aβ = y where β are the coefficients of f , yis the vector of data values and A is the appropriate alternant matrix,

A =

g1(x1) g2(x1) . . . gm(x1)g1(x2) g2(x2) . . . gm(x2)

......

. . ....

g1(xn) g2(xn) . . . gm(xn)

, β =

β1

β2...βn

, y =

y1

y2...yn

.This is an overdetermined version of the linear interpolation problem

described in Section 1.2.1.

45

47


How can we actually find the coefficients that minimize the sum of thesquares of the residuals? First we can define the square of the length of theresidual vector, e = Aβ − y, as a function

S(e) = e>e =n∑i=1

|ei|2 = (Aβ − y)>(Aβ − y).

This kind of function is a positive second degree polynomial with no mixedterms and thus has a global minima where ∂s

∂ei= 0 for all 1 ≤ i ≤ n. We

can find the global minima by looking at the derivative of the function, eiis determined by βi and

∂ei∂βj

= Ai,j

thus

∂S

∂βi=

n∑i=1

2ei∂ei∂βj

=

n∑i=1

2(Ai,·β − yi)Ai,j = 0⇔ A>Aβ = A>y

This givesA>Aβ = A>y⇔ β = (A>A)−1A>y

and by the Gauss–Markov theorem ([102,103,201], see for instance [208] fora more modern description), if (A>A)−1 exists then (19) gives the linear,unbiased estimator that gives the lowest variance possible for any linear,unbiased estimator. The matrix given by (A>A)−1A> is sometimes referredto as the Moore–Penrose pseudoinverse of A.

Clearly a linear curve fitting model with gi(x) = xi−1 gives an equationsystem described by a rectangular Vandermonde matrix.

1.2.5 Non-linear least squares fitting

So far we have only considered models that are linear with respect to theparameters that specify them. If we relax the linearity condition and simplyconsider fitting a function with m parameters, f(β1, . . . , βm;x), to n datapoints in the least squares sense it is usually referred to as a non-linear leastsquares fitting.

There is no general analogue to the Gauss–Markov theorem for non-linear least squares fitting and therefore finding the appropriate estimatorrequires more knowledge about the specifics of the model. In practice non-linear least squares fittings are often found using some numerical methodfor non-linear optimization of which there are many (see for instance [247]for an overview).

In the next section we will give an overview of a standard method calledthe Marquardt least squares method. In Section 3.2.2 we will use a combi-nation of the Marquardt least squares method and methods for linear least

46


How can we actually find the coefficients that minimize the sum of thesquares of the residuals? First we can define the square of the length of theresidual vector, e = Aβ − y, as a function

S(e) = e>e =n∑i=1

|ei|2 = (Aβ − y)>(Aβ − y).

This kind of function is a positive second degree polynomial with no mixedterms and thus has a global minima where ∂s

∂ei= 0 for all 1 ≤ i ≤ n. We

can find the global minima by looking at the derivative of the function, eiis determined by βi and

∂ei∂βj

= Ai,j

thus

∂S

∂βi=

n∑i=1

2ei∂ei∂βj

=

n∑i=1

2(Ai,·β − yi)Ai,j = 0⇔ A>Aβ = A>y

This givesA>Aβ = A>y⇔ β = (A>A)−1A>y

and by the Gauss–Markov theorem ([102,103,201], see for instance [208] fora more modern description), if (A>A)−1 exists then (19) gives the linear,unbiased estimator that gives the lowest variance possible for any linear,unbiased estimator. The matrix given by (A>A)−1A> is sometimes referredto as the Moore–Penrose pseudoinverse of A.

Clearly a linear curve fitting model with gi(x) = xi−1 gives an equationsystem described by a rectangular Vandermonde matrix.

1.2.5 Non-linear least squares fitting

So far we have only considered models that are linear with respect to theparameters that specify them. If we relax the linearity condition and simplyconsider fitting a function with m parameters, f(β1, . . . , βm;x), to n datapoints in the least squares sense it is usually referred to as a non-linear leastsquares fitting.

There is no general analogue to the Gauss–Markov theorem for non-linear least squares fitting and therefore finding the appropriate estimatorrequires more knowledge about the specifics of the model. In practice non-linear least squares fittings are often found using some numerical methodfor non-linear optimization of which there are many (see for instance [247]for an overview).

In the next section we will give an overview of a standard method calledthe Marquardt least squares method. In Section 3.2.2 we will use a combi-nation of the Marquardt least squares method and methods for linear least

46

48

1.2. CURVE FITTING

squares fitting to fit a non-linear model described by

G (β; t)η = y

where β, η are vectors of parameters to be fitted, y is the data we wish tofit the model to and G (β; t) is the generalized Vandermonde matrix

G (β; t) =

(t1e

1−t1)β1 (t1e1−t1)β2 · · · (t1e

1−t1)βm

(t2e1−t2)β1 (t2e

1−t2)β2 · · · (t2e1−t2)βm

......

. . ....

(tne1−tn)β1 (tne

1−tn)β2 · · · (tne1−tn)βm

.

1.2.6 The Marquardt least squares method

This section is based on Section 3.1 of Paper E

The Marquardt least squares method , also known as the Levenberg-Marquardtalgorithm or damped least squares, is an efficient method for least squaresestimation for functions with non-linear parameters that was developed inthe middle of the 20th century (see [174], [202]).

The least squares estimation problem for functions with non-linear pa-rameters arises when a function of m independent variables and describedby k unknown parameters needs to be fitted to a set of n data points suchthat the sum of squares of residuals is minimized.

The vector containing the independent variables is x = (x1, · · · , xn), thevector containing the parameters β = (β1, · · · , βk) and the data points

(Yi, X1i, X2i, · · · , Xmi) = (Yi,Xi) , i = 1, 2, · · · , n.

Let the residuals be denoted by Ei = f(Xi;β) − Yi and the sum ofsquares of Ei is then written as

S =n∑i=1

(f(Xi;β)− Yi)2 ,

which is the function to be minimized with respect to β.The Marquardt least squares method is an iterative method that gives

approximate values of β by combining the Gauss-Newton method (alsoknown as the inverse Hessian method) and the steepest descent (also knownas the gradient) method to minimize S. The method is based around solvingthe linear equation system(

A∗(r) + λ(r)I)δ∗(r) = g∗(r), (20)

where A∗(r) is a modified Hessian matrix of E(b) (or f(Xi; b)), g∗(r) is arescaled version of the gradient of S, r is the number of the current iteration

47

1.2. CURVE FITTING

squares fitting to fit a non-linear model described by

G (β; t)η = y

where β, η are vectors of parameters to be fitted, y is the data we wish tofit the model to and G (β; t) is the generalized Vandermonde matrix

G (β; t) =

(t1e

1−t1)β1 (t1e1−t1)β2 · · · (t1e

1−t1)βm

(t2e1−t2)β1 (t2e

1−t2)β2 · · · (t2e1−t2)βm

......

. . ....

(tne1−tn)β1 (tne

1−tn)β2 · · · (tne1−tn)βm

.

1.2.6 The Marquardt least squares method


The Marquardt least squares method , also known as the Levenberg-Marquardtalgorithm or damped least squares, is an efficient method for least squaresestimation for functions with non-linear parameters that was developed inthe middle of the 20th century (see [174], [202]).

The least squares estimation problem for functions with non-linear pa-rameters arises when a function of m independent variables and describedby k unknown parameters needs to be fitted to a set of n data points suchthat the sum of squares of residuals is minimized.

The vector containing the independent variables is x = (x1, · · · , xn), thevector containing the parameters β = (β1, · · · , βk) and the data points

(Yi, X1i, X2i, · · · , Xmi) = (Yi,Xi) , i = 1, 2, · · · , n.

Let the residuals be denoted by Ei = f(Xi;β) − Yi and the sum ofsquares of Ei is then written as

S =n∑i=1

(f(Xi;β)− Yi)2 ,

which is the function to be minimized with respect to β.The Marquardt least squares method is an iterative method that gives

approximate values of β by combining the Gauss-Newton method (alsoknown as the inverse Hessian method) and the steepest descent (also knownas the gradient) method to minimize S. The method is based around solvingthe linear equation system(

A∗(r) + λ(r)I)δ∗(r) = g∗(r), (20)

where A∗(r) is a modified Hessian matrix of E(b) (or f(Xi; b)), g∗(r) is arescaled version of the gradient of S, r is the number of the current iteration

47

49


of the method, and λ is a real positive number sometimes referred to as thefudge factor [235]. The Hessian, the gradient and their modifications aredefined as follows:

A = J>J,

Jij =∂fi∂bj

=∂Ei∂bj

, i = 1, 2, · · · ,m; j = 1, 2, · · · , k,

and

(A∗)ij =aij√aii√ajj

,

while

g = J>(Y − f0), f0i = f(Xi,b, c), g∗i =giaii.

Solving (20) gives a vector which, after some scaling, describes how theparameters b should be changed in order to get a new approximation of β,

b(r+1) = b(r) + δ(r), δ(r) =δ∗(r)i√aii. (21)

It is obvious from (20) that δ(r) depends on the value of the fudge factorλ. Note that if λ = 0, then (20) reduces to the regular Gauss-Newtonmethod [202], and if λ→∞ the method will converge towards the steepestdescent method [202]. The reason that the two methods are combined is thatthe Gauss-Newton method often has faster convergence than the steepestdescent method, but is also an unstable method [202]. Therefore, λ must bechosen appropriately in each step. In the Marquardt least squares methodthis amounts to increasing λ with a chosen factor v whenever an iterationincreases S, and if an iteration reduces S then λ is reduced by a factor v asmany times as possible. Below follows a detailed description of the methodusing the following notation:

S(r) =n∑i=1

(Yi − f(Xi,b

(r), c))2, (22)

S(λ(r)

)=

n∑i=1

(Yi − f(Xi,b

(r) + δ(r), c))2. (23)

The iteration step of the Marquardt least squares method can be describedas follows:

• Input: v > 1 and b(r), λ(r).

/ Compute S(λ(r)

).

• If λ(r) 1 then compute S(λ(r)

v

), else go to ..

48


of the method, and λ is a real positive number sometimes referred to as thefudge factor [235]. The Hessian, the gradient and their modifications aredefined as follows:

A = J>J,

Jij =∂fi∂bj

=∂Ei∂bj

, i = 1, 2, · · · ,m; j = 1, 2, · · · , k,

and

(A∗)ij =aij√aii√ajj

,

while

g = J>(Y − f0), f0i = f(Xi,b, c), g∗i =giaii.

Solving (20) gives a vector which, after some scaling, describes how theparameters b should be changed in order to get a new approximation of β,

b(r+1) = b(r) + δ(r), δ(r) =δ∗(r)i√aii. (21)

It is obvious from (20) that δ(r) depends on the value of the fudge factorλ. Note that if λ = 0, then (20) reduces to the regular Gauss-Newtonmethod [202], and if λ→∞ the method will converge towards the steepestdescent method [202]. The reason that the two methods are combined is thatthe Gauss-Newton method often has faster convergence than the steepestdescent method, but is also an unstable method [202]. Therefore, λ must bechosen appropriately in each step. In the Marquardt least squares methodthis amounts to increasing λ with a chosen factor v whenever an iterationincreases S, and if an iteration reduces S then λ is reduced by a factor v asmany times as possible. Below follows a detailed description of the methodusing the following notation:

S(r) =n∑i=1

(Yi − f(Xi,b

(r), c))2, (22)

S(λ(r)

)=

n∑i=1

(Yi − f(Xi,b

(r) + δ(r), c))2. (23)

The iteration step of the Marquardt least squares method can be describedas follows:

• Input: v > 1 and b(r), λ(r).

/ Compute S(λ(r)

).

• If λ(r) 1 then compute S(λ(r)

v

), else go to ..

48

50

1.2. CURVE FITTING

• If S(λ(r)

v

)≤ S(r) let λ(r+1) = λ(r)

v .

. If S(λ(r)

)≤ S(r) let λ(r+1) = λ(r).

• If S(λ(r)

)> S(r) then find the smallest integer ω > 0 such that

S(λ(r)vω

)≤ S(r), and set λ(r+1) = λ(r)vω.

• Output: b(r+1) = b(r) + δ(r), δ(r).

This iteration procedure is also described in Figure 1.5. Naturally, somecondition for what constitutes an acceptable fit for the function must alsobe chosen. If this condition is not satisfied the new values for b(r+1) andλ(r+1) will be used as input for the next iteration and if the condition issatisfied the algorithm terminates. The quality of the fitting, in other wordsthe value of S, is determined by the stopping condition and the initial valuesfor b(0). The initial value of λ(0) affects the performance of the algorithmto some extent since after the first iteration λ(r) will be self-regulating.Suitable values for b(0) are challenging to find for many functions f andthey are often, together with λ(0), found using heuristic methods.

Input:b(r), λ(r) and v > 1

Compute S(λ(r)

)

ω = ω + 1 λ(r) 1 Compute S(λ(r)

v

)

S(λ(r)vω

)≤ S(r)

ω = 1

S(λ(r)

)≤ S(r) S

(λ(r)

v

)≤ S(r)

λ(r+1) = λ(r)vω λ(r+1) = λ(r) λ(r+1) = λ(r)

v

Output:b(r+1) = b(r) + δ(r), δ(r)

YES

YES

NO

NO

YES

NONO

YES

Figure 1.5: The basic iteration step of the Marquardt least squares method,definitions of computed quantities are given in (21), (22) and (23).

In Section 3.2 the Marquardt least squares method will be used for leastsquares fitting with power-exponential functions.

49

1.2. CURVE FITTING

• If S(λ(r)

v

)≤ S(r) let λ(r+1) = λ(r)

v .

. If S(λ(r)

)≤ S(r) let λ(r+1) = λ(r).

• If S(λ(r)

)> S(r) then find the smallest integer ω > 0 such that

S(λ(r)vω

)≤ S(r), and set λ(r+1) = λ(r)vω.

• Output: b(r+1) = b(r) + δ(r), δ(r).

This iteration procedure is also described in Figure 1.5. Naturally, somecondition for what constitutes an acceptable fit for the function must alsobe chosen. If this condition is not satisfied the new values for b(r+1) andλ(r+1) will be used as input for the next iteration and if the condition issatisfied the algorithm terminates. The quality of the fitting, in other wordsthe value of S, is determined by the stopping condition and the initial valuesfor b(0). The initial value of λ(0) affects the performance of the algorithmto some extent since after the first iteration λ(r) will be self-regulating.Suitable values for b(0) are challenging to find for many functions f andthey are often, together with λ(0), found using heuristic methods.

Input:b(r), λ(r) and v > 1

Compute S(λ(r)

)

ω = ω + 1 λ(r) 1 Compute S(λ(r)

v

)

S(λ(r)vω

)≤ S(r)

ω = 1

S(λ(r)

)≤ S(r) S

(λ(r)

v

)≤ S(r)

λ(r+1) = λ(r)vω λ(r+1) = λ(r) λ(r+1) = λ(r)

v

Output:b(r+1) = b(r) + δ(r), δ(r)

YES

YES

NO

NO

YES

NONO

YES

Figure 1.5: The basic iteration step of the Marquardt least squares method,definitions of computed quantities are given in (21), (22) and (23).

In Section 3.2 the Marquardt least squares method will be used for leastsquares fitting with power-exponential functions.

49

51


1.3 Analysing how well a curve fits

In this thesis we will discuss several ways to construct mathematical mod-els. With several mathematical models available it is needed to have somemethod for comparing the methods and choose the most suitable one. Whenthe model in constructed with a certain application in mind there is often aset of required or desired properties given by the application and choosingthe best model is a matter of seeing which model matches the requirementsthe best. In many cases this process is not straightforward and often there isnot one model that is better than the other candidate models in all aspects,a common example is the trade-off between accuracy and complexity of themodel. It is often easy to improve the model by increasing its complex-ity (either by introducing more general and flexible mathematical conceptsthat are more difficult to analyse or less well understood or by extendingthe model in a way that increases the cost of computations and simulationsusing the model), but finding the best compromise between accuracy andcomplexity can be difficult. In this section we will discuss how to com-pare models primarily with respect to accuracy and the number of requiredparameters.

1.3.1 Regression

Regression is similar to interpolation except that the presence of noise in thedata is taken into consideration. The typical regression problem assumesthat the data points (xi, yi), i = 1, . . . , n are sample from a stochasticvariable of the form

Yi = f(β;xi) + εi

where f(β;x) is a given function with a fixed number of undetermined pa-rameters β ∈ B and εi for i = 1, . . . , n are samples of a random variablewith expected value zero, called the errors or the noise for the data set.

There are many different classes of regression problems defined by thetype of function f(β1, . . . , βm;x) and the distribution of errors.

Here we will only consider the situation when the εi variables are in-dependent and normally distributed with identical variance and that theparameter space B is a compact subset of Rk and that for all xi the functionf(β;xi) is a continuous function of β ∈ B.

Suppose we want to choose the appropriate set of parameters for f basedon some set of observed data points. A common approach to this is so calledmaximum likelihood estimation.

Definition 1.13. The likelihood function, L is the function that gives usthe probability that a certain observation, x, of a stochastic variable X ismade given a certain set of parameters, β,

Lx(β) = Pr(X = x|β).

50


1.3 Analysing how well a curve fits

In this thesis we will discuss several ways to construct mathematical mod-els. With several mathematical models available it is needed to have somemethod for comparing the methods and choose the most suitable one. Whenthe model in constructed with a certain application in mind there is often aset of required or desired properties given by the application and choosingthe best model is a matter of seeing which model matches the requirementsthe best. In many cases this process is not straightforward and often there isnot one model that is better than the other candidate models in all aspects,a common example is the trade-off between accuracy and complexity of themodel. It is often easy to improve the model by increasing its complex-ity (either by introducing more general and flexible mathematical conceptsthat are more difficult to analyse or less well understood or by extendingthe model in a way that increases the cost of computations and simulationsusing the model), but finding the best compromise between accuracy andcomplexity can be difficult. In this section we will discuss how to com-pare models primarily with respect to accuracy and the number of requiredparameters.

1.3.1 Regression

Regression is similar to interpolation except that the presence of noise in thedata is taken into consideration. The typical regression problem assumesthat the data points (xi, yi), i = 1, . . . , n are sample from a stochasticvariable of the form

Yi = f(β;xi) + εi

where f(β;x) is a given function with a fixed number of undetermined pa-rameters β ∈ B and εi for i = 1, . . . , n are samples of a random variablewith expected value zero, called the errors or the noise for the data set.

There are many different classes of regression problems defined by thetype of function f(β1, . . . , βm;x) and the distribution of errors.

Here we will only consider the situation when the εi variables are in-dependent and normally distributed with identical variance and that theparameter space B is a compact subset of Rk and that for all xi the functionf(β;xi) is a continuous function of β ∈ B.

Suppose we want to choose the appropriate set of parameters for f basedon some set of observed data points. A common approach to this is so calledmaximum likelihood estimation.

Definition 1.13. The likelihood function, L is the function that gives usthe probability that a certain observation, x, of a stochastic variable X ismade given a certain set of parameters, β,

Lx(β) = Pr(X = x|β).

50

52

1.3. ANALYSING HOW WELL A CURVE FITS

Thus choosing parameters that maximize the likelihood function givesthe set of parameters that seem to be most likely based on available infor-mation. Typically these parameters cannot be calculated exactly and mustbe estimated, this estimation is called the Maximum Likelihood Estimation(MLE).

To find the MLE we need to find the maximum of the likelihood function.Note that here we will only consider the case where the noise variables, εi,are independent and normally distributed with mean zero.

Lemma 1.7. For the stochastic variables Yi = f(β;xi) + εi where f(β;x)is a given function with a fixed number of undetermined parameters β ∈ Band εi for i = 1, . . . , n are independent random variables with expected valuezero and standard deviation σ the likelihood function is given by the jointprobability density function for the noise,

Ly(β) = (2π)n2 σn

n∏i=1

exp

(−(yi − f(β;xi))

2

σ2

).

Proof. Since each εi is normally distributed with mean zero and standard de-viation σ the difference between the observed value and the given function,yi − f(β;xi) is normally distributed with mean zero and standard devia-tion σ. Since all the errors are independent the joint probability densityfunction is just the product of n probability density functions of the form

pi(β; (xi, yi)) = 1√2πσ

exp(− (yi−f(β;xi))

2

σ2

)for i = 1, . . . , n.

For the MLE we only care about what parameters give the maximum ofthe likelihood function, not the actual value of the likelihood function so wecan ignore the constant factor and in practice it also often simple to considerthe maximum of the logarithm of the likelihood function. This leads to thefollowing lemma.

Lemma 1.8. Consider a regression problem described by a set of data points(xi, yi), i = 1, . . . , n and the stochastic variables Yi = f(β;xi) + εi wheref(β;x) is a given function with a fixed number of undetermined parametersβ ∈ B and εi for i = 1, . . . , n are independent normally distributed randomvariables with expected value zero and standard deviation σ. The MLE forthe parameters β will minimize the sum of the squares of the residuals,

S(β) =n∑i=1

(yi − f(β;xi))2.

Proof. Since the natural logarithm is a monotonically increasing function− ln(Ly(β)) will have a minimum point where Ly(β) has a maximum point.

51


Thus choosing parameters that maximize the likelihood function givesthe set of parameters that seem to be most likely based on available infor-mation. Typically these parameters cannot be calculated exactly and mustbe estimated, this estimation is called the Maximum Likelihood Estimation(MLE).

To find the MLE we need to find the maximum of the likelihood function.Note that here we will only consider the case where the noise variables, εi,are independent and normally distributed with mean zero.

Lemma 1.7. For the stochastic variables Yi = f(β;xi) + εi where f(β;x)is a given function with a fixed number of undetermined parameters β ∈ Band εi for i = 1, . . . , n are independent random variables with expected valuezero and standard deviation σ the likelihood function is given by the jointprobability density function for the noise,

Ly(β) = (2π)n2 σn

n∏i=1

exp

(−(yi − f(β;xi))

2

σ2

).

Proof. Since each εi is normally distributed with mean zero and standard de-viation σ the difference between the observed value and the given function,yi − f(β;xi) is normally distributed with mean zero and standard devia-tion σ. Since all the errors are independent the joint probability densityfunction is just the product of n probability density functions of the form

pi(β; (xi, yi)) = 1√2πσ

exp(− (yi−f(β;xi))

2

σ2

)for i = 1, . . . , n.

For the MLE we only care about what parameters give the maximum ofthe likelihood function, not the actual value of the likelihood function so wecan ignore the constant factor and in practice it also often simple to considerthe maximum of the logarithm of the likelihood function. This leads to thefollowing lemma.

Lemma 1.8. Consider a regression problem described by a set of data points(xi, yi), i = 1, . . . , n and the stochastic variables Yi = f(β;xi) + εi wheref(β;x) is a given function with a fixed number of undetermined parametersβ ∈ B and εi for i = 1, . . . , n are independent normally distributed randomvariables with expected value zero and standard deviation σ. The MLE forthe parameters β will minimize the sum of the squares of the residuals,

S(β) =n∑i=1

(yi − f(β;xi))2.

Proof. Since the natural logarithm is a monotonically increasing function− ln(Ly(β)) will have a minimum point where Ly(β) has a maximum point.

51

53


By Lemma 1.7

− ln(Ly(β)) =− ln

((2π)

n2 σn

n∏i=1

exp

(−yi − f(β;xi)

σ2

))

=− ln(

(2π)n2 σn

)+

1

σ2

n∑i=1

(yi − f(β;xi))2

=− ln(

(2π)n2 σn

)+

1

σ2S(β).

Since the first term and the factor in front of S(β) does not depend on β theminimum point of S(β) will coincide with the maximum of the likelihoodfunction.

Here we can see that finding the MLE is equivalent to using the curvefitting technique describes in Section 1.2.3. In Section 1.2.4 we saw thatsolving this problem in the case when the model was linear with respectto its parameters was relatively straightforward and in Section 1.2.5 wesaw that when the model was nonlinear with respect to its parameters theproblem was considerably harder.

1.3.2 Quantile-Quantile plots

In the regression problem discussed in Section 1.3.1 is assumed that thenoise of the model follows some distribution, for the purpose of this thesiswe only considered the case of normally distributed noise but the essentialproblem formulation is the same regardless of noise distribution. Testingthis assumption can be done in different ways and in Section 4.4.1 we willdemonstrate some cases where the assumption is not entirely true using aquantile-quantile plot (Q-Q plot). Q-Q plots are a common tool for graphi-cally analysing how close sampled data is to a given distribution [273].

Suppose we have n samples of a stochastic variable X from an unknowndistribution. Suppose we want to make a Q-Q plot that tests if X belongto a distribution with cumulative distribution function F . First we sort thesamples in ascending order, x1 ≤ x2 ≤ . . . ≤ xn and then choose what weexpect to be the corresponding probability for each sample size. A common

approach is computing thek − 0.5

n-th quantile for sample xk, k = 1, . . . , n,

i.e. finding F−1

(k − 0.5

n

). It is then expected that the points given by(

F−1

(k − 0.5

n

), xk

)should mostly follow a straight line in the Q-Q plot

apart from some random noise. If the residuals show some other pattern orsome points lie very far from the line this indicates that the residuals wouldbe better described by some other distribution or that there is a significantnumber of outliers.

52


By Lemma 1.7

− ln(Ly(β)) =− ln

((2π)

n2 σn

n∏i=1

exp

(−yi − f(β;xi)

σ2

))

=− ln(

(2π)n2 σn

)+

1

σ2

n∑i=1

(yi − f(β;xi))2

=− ln(

(2π)n2 σn

)+

1

σ2S(β).

Since the first term and the factor in front of S(β) does not depend on β theminimum point of S(β) will coincide with the maximum of the likelihoodfunction.

Here we can see that finding the MLE is equivalent to using the curvefitting technique describes in Section 1.2.3. In Section 1.2.4 we saw thatsolving this problem in the case when the model was linear with respectto its parameters was relatively straightforward and in Section 1.2.5 wesaw that when the model was nonlinear with respect to its parameters theproblem was considerably harder.

1.3.2 Quantile-Quantile plots

In the regression problem discussed in Section 1.3.1 is assumed that thenoise of the model follows some distribution, for the purpose of this thesiswe only considered the case of normally distributed noise but the essentialproblem formulation is the same regardless of noise distribution. Testingthis assumption can be done in different ways and in Section 4.4.1 we willdemonstrate some cases where the assumption is not entirely true using aquantile-quantile plot (Q-Q plot). Q-Q plots are a common tool for graphi-cally analysing how close sampled data is to a given distribution [273].

Suppose we have n samples of a stochastic variable X from an unknowndistribution. Suppose we want to make a Q-Q plot that tests if X belongto a distribution with cumulative distribution function F . First we sort thesamples in ascending order, x1 ≤ x2 ≤ . . . ≤ xn and then choose what weexpect to be the corresponding probability for each sample size. A common

approach is computing thek − 0.5

n-th quantile for sample xk, k = 1, . . . , n,

i.e. finding F−1

(k − 0.5

n

). It is then expected that the points given by(

F−1

(k − 0.5

n

), xk

)should mostly follow a straight line in the Q-Q plot

apart from some random noise. If the residuals show some other pattern orsome points lie very far from the line this indicates that the residuals wouldbe better described by some other distribution or that there is a significantnumber of outliers.

52

54


There are many versions of this kind of tool and many alternatives forwhich quantiles to choose, see for examples Table 2.1 in [273], but here wewill only use quantiles given above.

This kind of tool does not provide a rigorous test, rather it is up towhoever analyses the sample to determine if the sample points are closeenough to linear or not. These types of plot can also help identifying anotherdistribution that might be a more reasonable assumption, for example weuse a Q-Q plot to compare to a normal distribution and the lower end of thecurve turns downwards and the higher tail turns upwards this indicates thatthe samples come a relatively long-tailed distribution, while if the ends of thecurve turn in the opposite directions this indicates a relatively short-taileddistribution [273].

1.3.3 The Akaike information criterion

When constructing a mathematical model of an observable process withoutdescribing the underlying causes of the process, i.e. a phenomenologicalmodel, there are many tools available for creating a model that can recreatea finite set of points with arbitrarily high precision, for example the inter-polation methods described in Sections 1.2.1 – 1.2.2 or the least squaresmethods in Sections 1.2.3 – 1.2.6. Regardless of what method is chosenthe accuracy (unless already exact) can be improved by adding more (free)parameters to the model (exactly how this is accomplished depends on themodel). There is a well known anecdote, see [77], where Freeman Dysondescribes how Enrico Fermi dissuaded him from pursuing a research projectin particle physics where the model had many free parameters by saying

I remember my friend Johnny von Neumann used to say, ’withfour parameters I can fit an elephant, and with five I can makehim wiggle his trunk’.

Sometimes, especially when working with data that has significant noise,you can have a model that is ’too accurate’ in the sense that the model de-scribes some of the noise as well as the underlying process in a curve fittingor regression problem. For an example see Figure 4.4 where some modelsdesigned to reproduce a certain pattern in data gives unreasonable resultswhen noise makes the pattern indistinct. This phenomena is called overfit-ting and can cause problems with extrapolation based on and interpretationof the model. A common sign of overfitting is that the model gives unrea-sonable results for points not in the original data set, similar to Runge’sphenomenon, illustrated in Figure 1.4.

One way to detect possible overfitting is to compare the model to a sim-ilar model with fewer parameters and see if the improvement in accuracyis sufficiently large to warrant the use of the more complex model. A for-

53


There are many versions of this kind of tool and many alternatives forwhich quantiles to choose, see for examples Table 2.1 in [273], but here wewill only use quantiles given above.

This kind of tool does not provide a rigorous test, rather it is up towhoever analyses the sample to determine if the sample points are closeenough to linear or not. These types of plot can also help identifying anotherdistribution that might be a more reasonable assumption, for example weuse a Q-Q plot to compare to a normal distribution and the lower end of thecurve turns downwards and the higher tail turns upwards this indicates thatthe samples come a relatively long-tailed distribution, while if the ends of thecurve turn in the opposite directions this indicates a relatively short-taileddistribution [273].

1.3.3 The Akaike information criterion

When constructing a mathematical model of an observable process withoutdescribing the underlying causes of the process, i.e. a phenomenologicalmodel, there are many tools available for creating a model that can recreatea finite set of points with arbitrarily high precision, for example the inter-polation methods described in Sections 1.2.1 – 1.2.2 or the least squaresmethods in Sections 1.2.3 – 1.2.6. Regardless of what method is chosenthe accuracy (unless already exact) can be improved by adding more (free)parameters to the model (exactly how this is accomplished depends on themodel). There is a well known anecdote, see [77], where Freeman Dysondescribes how Enrico Fermi dissuaded him from pursuing a research projectin particle physics where the model had many free parameters by saying

I remember my friend Johnny von Neumann used to say, ’withfour parameters I can fit an elephant, and with five I can makehim wiggle his trunk’.

Sometimes, especially when working with data that has significant noise,you can have a model that is ’too accurate’ in the sense that the model de-scribes some of the noise as well as the underlying process in a curve fittingor regression problem. For an example see Figure 4.4 where some modelsdesigned to reproduce a certain pattern in data gives unreasonable resultswhen noise makes the pattern indistinct. This phenomena is called overfit-ting and can cause problems with extrapolation based on and interpretationof the model. A common sign of overfitting is that the model gives unrea-sonable results for points not in the original data set, similar to Runge’sphenomenon, illustrated in Figure 1.4.

One way to detect possible overfitting is to compare the model to a sim-ilar model with fewer parameters and see if the improvement in accuracyis sufficiently large to warrant the use of the more complex model. A for-

53

55


malised way of doing this is using the Akaike Information Criterion (AIC)originally developed in the 1970s [4, 5].

Definition 1.14. Let g be a model of some data, y, with k estimatedparameters and let L(g|y) be the maximum value of the likelihood functionfor the model. Then the Akaike Information Criterion is given by

AIC(g|y) = 2(k + 1)− 2 log(L(g|y)

).

The AIC is used for comparing models to each other and a lower AICindicates a model that describes the data better but without overfitting orneedless complexity, essentially a model that contains more information.

The key concept for explaining why the AIC works is Kullback–Leiblerdivergence, also known as Kullback–Leibler information or relative entropy.

Definition 1.15. The Kullback–Leibler divergence of two probability dis-tributions over the real numbers with probability density functions f and gis defined as

I(f, g) =

∫ ∞−∞

f(x) ln

(f(x)

g(x)

)dx.

Remark 1.8. There are definitions of the Kullback–Leibler divergence fordiscrete and multivariate distributions as well, but for the purposes of thisthesis this is the only definition we will need.

The Kullback–Leibler divergence has two properties that makes it usefulfor evaluating mathematical models.

Lemma 1.9. Let f and g be the probability density functions of two prob-ability distributions. Then I(f, g) ≥ 0 and if and only if I(f, g) = 0 thenf = g.

Proof. Since f(x) and g(x) are probability density functions they must be

non-negative and thus g(x)f(x) must also be non-negative. Then the following

inequality holds

ln

(g(x)

f(x)

)≤ g(x)

f(x)− 1.

If we multiply by f(x) and integrate on both sides of the inequality and note

that ln(g(x)f(x)

)= − ln

(f(x)g(x)

)then we get

−I(f, g) = −∫ ∞−∞

f(x) ln

(f(x)

g(x)

)dx ≤

∫ ∞−∞

f(x)

(g(x)

f(x)− 1

)dx

Since f and g are probability density distributions we can conclude that∫ ∞−∞

f(x)

(g(x)

f(x)− 1

)dx =

∫ ∞−∞

g(x)− f(x) dx

=

∫ ∞−∞

g(x) dx−∫ ∞−∞

f(x) dx = 1− 1 = 0

54


malised way of doing this is using the Akaike Information Criterion (AIC)originally developed in the 1970s [4, 5].

Definition 1.14. Let g be a model of some data, y, with k estimatedparameters and let L(g|y) be the maximum value of the likelihood functionfor the model. Then the Akaike Information Criterion is given by

AIC(g|y) = 2(k + 1)− 2 log(L(g|y)

).

The AIC is used for comparing models to each other and a lower AICindicates a model that describes the data better but without overfitting orneedless complexity, essentially a model that contains more information.

The key concept for explaining why the AIC works is Kullback–Leiblerdivergence, also known as Kullback–Leibler information or relative entropy.

Definition 1.15. The Kullback–Leibler divergence of two probability dis-tributions over the real numbers with probability density functions f and gis defined as

I(f, g) =

∫ ∞−∞

f(x) ln

(f(x)

g(x)

)dx.

Remark 1.8. There are definitions of the Kullback–Leibler divergence fordiscrete and multivariate distributions as well, but for the purposes of thisthesis this is the only definition we will need.

The Kullback–Leibler divergence has two properties that makes it usefulfor evaluating mathematical models.

Lemma 1.9. Let f and g be the probability density functions of two prob-ability distributions. Then I(f, g) ≥ 0 and if and only if I(f, g) = 0 thenf = g.

Proof. Since f(x) and g(x) are probability density functions they must be

non-negative and thus g(x)f(x) must also be non-negative. Then the following

inequality holds

ln

(g(x)

f(x)

)≤ g(x)

f(x)− 1.

If we multiply by f(x) and integrate on both sides of the inequality and note

that ln(g(x)f(x)

)= − ln

(f(x)g(x)

)then we get

−I(f, g) = −∫ ∞−∞

f(x) ln

(f(x)

g(x)

)dx ≤

∫ ∞−∞

f(x)

(g(x)

f(x)− 1

)dx

Since f and g are probability density distributions we can conclude that∫ ∞−∞

f(x)

(g(x)

f(x)− 1

)dx =

∫ ∞−∞

g(x)− f(x) dx

=

∫ ∞−∞

g(x) dx−∫ ∞−∞

f(x) dx = 1− 1 = 0

54

56


and thus

−I(f, g) = −∫ ∞−∞

f(x) ln

(f(x)

g(x)

)dx ≤ 0⇔ I(f, g)) ≥ 0.

To prove that I(f, g) = 0 if and only if f = g we can argue similarly asbefore and get

I(f, g) ≥∫ ∞−∞

f(x)− g(x) dx

and since f and g are probability density function the right hand side willonly be zero if f = g.

A common interpretation of I(f, g) is that the larger the Kullback–Leibler divergence is, the more information is lost by using g as an ap-proximation of f [40,164]. If we have a distribution with probability densityfunction f and a number of candidates for approximating this distributionthat have probability density functions g1, g2, . . ., gn, then the best can-didate for approximation would be the candidate with I(f, gk) closest tozero. Often this is not useful when trying to model a process based on ob-servations since the true distribution is unknown. One solution to this is toestimate Kullback–Leibler divergence from the true model. This is the mainidea behind the AIC. Fully deriving the AIC is somewhat complicated, seefor example [35], and here we will only give a short motivation (based onSection 7.2 in [40]).

First we must consider the situation that we can apply the AIC. Wewill have taken some set of observations from a stochastic variable Y withan unknown probability density function f and based on those constructedour model (using for example a curve fitting technique). Let us call theobservations y and the model we construct based on the data g(·|y). Thiswill not necessarily give the best possible version of the model so simplylooking at I(f, g(·|y)) can be misleading, it is better to consider the expectedvalue of I(f, g(·|y)) with respect to y that has the property

EY [I(f, g(·|Y ))] > I(f, g(·|y∗))

where y∗ is the set of observations that gives the best possible version ofthe candidate model. Since f is unknown we cannot estimate this expectedvalue directly but it can be rewritten as follows

EY [I(f, g(·|Y ))] =

∫ ∞−∞

f(y) ln(f(y)) dy

− EY

[∫ ∞−∞

f(x) ln(g(x|Y )) ln(f(x)) dx

]= c− EY EX [ln(g(X|Y )]

55


and thus

−I(f, g) = −∫ ∞−∞

f(x) ln

(f(x)

g(x)

)dx ≤ 0⇔ I(f, g)) ≥ 0.

To prove that I(f, g) = 0 if and only if f = g we can argue similarly asbefore and get

I(f, g) ≥∫ ∞−∞

f(x)− g(x) dx

and since f and g are probability density function the right hand side willonly be zero if f = g.

A common interpretation of I(f, g) is that the larger the Kullback–Leibler divergence is, the more information is lost by using g as an ap-proximation of f [40,164]. If we have a distribution with probability densityfunction f and a number of candidates for approximating this distributionthat have probability density functions g1, g2, . . ., gn, then the best can-didate for approximation would be the candidate with I(f, gk) closest tozero. Often this is not useful when trying to model a process based on ob-servations since the true distribution is unknown. One solution to this is toestimate Kullback–Leibler divergence from the true model. This is the mainidea behind the AIC. Fully deriving the AIC is somewhat complicated, seefor example [35], and here we will only give a short motivation (based onSection 7.2 in [40]).

First we must consider the situation that we can apply the AIC. Wewill have taken some set of observations from a stochastic variable Y withan unknown probability density function f and based on those constructedour model (using for example a curve fitting technique). Let us call theobservations y and the model we construct based on the data g(·|y). Thiswill not necessarily give the best possible version of the model so simplylooking at I(f, g(·|y)) can be misleading, it is better to consider the expectedvalue of I(f, g(·|y)) with respect to y that has the property

EY [I(f, g(·|Y ))] > I(f, g(·|y∗))

where y∗ is the set of observations that gives the best possible version ofthe candidate model. Since f is unknown we cannot estimate this expectedvalue directly but it can be rewritten as follows

EY [I(f, g(·|Y ))] =

∫ ∞−∞

f(y) ln(f(y)) dy

− EY

[∫ ∞−∞

f(x) ln(g(x|Y )) ln(f(x)) dx

]= c− EY EX [ln(g(X|Y )]

55

57


where X is an independent stochastic variables with the same distributionas Y . Since c is a constant value that is independent of our candidate modelwe can ignore it and focus on EY EX [ln(g(X|Y )].

We saw in Section 1.3.1 that when we construct the model from the datawe use the MLE for the parameters. If we denote the parameters given bythe MLE by β then instead of g(x|y) we can denote the chosen model withg(β;x). Thus

EY EX [ln(g(X|Y )] = EβEX

[ln(g(β;X)

].

Using a Taylor expansion we can see that

ln(g(β;X)) ≈ ln(g(β;X)) +

∂

∂β1ln(g(β;X))

...∂

∂βnln(g(β;X))

> β1 − β1

...

βn − βn

+1

2

β1 − β1...

βn − βn

> [

∂2

∂βi∂βjln(g(β;X))

]n,n1,1

β1 − β1...

βn − βn

. (24)

Since β were given by the MLE the first order derivatives will disappear,∂

∂β1ln(g(β;X)) = 0. Taking the expectation of the remaining part of the

expression with respect to X gives

EX [ln(g(β;X))] ≈ EX [ln(g(β;X))]

+1

2

β1 − β1...

βn − βn

> [

EX

[∂2


] ]n,n1,1

β1 − β1...

βn − βn

= EX [ln(g(β;X))] + T (β).

Next we take the expectation with respect to β and get

EβEX

[ln(g(β;X)

]≈Eβ[EX [ln(g(β;X))] + T (β)]

=EX [ln(g(β;X))] + Eβ[T (β)].

It can be shown that the first term is given by the maximum of the likelihoodfunction, EX [ln(g(β;X))] = L, and that the second therm is approximatelyequal to the number of free parameters which is the number of parametersof the model plus the standard deviation of the noise, Eβ[T (β)] ≈ k + 1.Combining this gives the expression for the AIC given in Definition 1.14apart from the factor −2 which is used as a matter of convention [40].

56


where X is an independent stochastic variables with the same distributionas Y . Since c is a constant value that is independent of our candidate modelwe can ignore it and focus on EY EX [ln(g(X|Y )].

We saw in Section 1.3.1 that when we construct the model from the datawe use the MLE for the parameters. If we denote the parameters given bythe MLE by β then instead of g(x|y) we can denote the chosen model withg(β;x). Thus

EY EX [ln(g(X|Y )] = EβEX

[ln(g(β;X)

].

Using a Taylor expansion we can see that

ln(g(β;X)) ≈ ln(g(β;X)) +

∂

∂β1ln(g(β;X))

...∂

∂βnln(g(β;X))

> β1 − β1

...

βn − βn

+1

2

β1 − β1...

βn − βn

> [

∂2


]n,n1,1

β1 − β1...

βn − βn

. (24)

Since β were given by the MLE the first order derivatives will disappear,∂

∂β1ln(g(β;X)) = 0. Taking the expectation of the remaining part of the

expression with respect to X gives

EX [ln(g(β;X))] ≈ EX [ln(g(β;X))]

+1

2

β1 − β1...

βn − βn

> [

EX

[∂2


] ]n,n1,1

β1 − β1...

βn − βn

= EX [ln(g(β;X))] + T (β).

Next we take the expectation with respect to β and get

EβEX

[ln(g(β;X)

]≈Eβ[EX [ln(g(β;X))] + T (β)]

=EX [ln(g(β;X))] + Eβ[T (β)].

It can be shown that the first term is given by the maximum of the likelihoodfunction, EX [ln(g(β;X))] = L, and that the second therm is approximatelyequal to the number of free parameters which is the number of parametersof the model plus the standard deviation of the noise, Eβ[T (β)] ≈ k + 1.Combining this gives the expression for the AIC given in Definition 1.14apart from the factor −2 which is used as a matter of convention [40].

56

58

1.4. D-OPTIMAL EXPERIMENT DESIGN

Remark 1.9. For models with many parameters and small sample sizesit is recommended to add a second order correction to the AIC called theAICC [40,129] and in the case of least squares fitting is given by the followingexpression

AICC = AIC +2(k + 1)(k + 2)

n− k − 2.

There are several other information criterion that could be used in asimilar way to the AIC, for example the Takeuchi information criterion [270]or the Bayesian information criterion [254]. Here we will use the AIC sinceit is considered a reliable criterion that is simple to calculate [40,164] and isasymptotically optimal for selecting the model with the least mean squareerrors [297].

1.4 D-optimal experiment design

For the class of linear non-weighted regression problems described in Sec-tion 1.3.1 minimizing the square of the sum of residuals gives the maximum-likelihood estimation of the parameters that specify the fitted function. Thisestimation naturally has a variance as well and minimizing this variance canbe interpreted as improving the reliability of the fitted function by minimiz-ing its sensitivity to noise in measurements. This minimization is usuallydone by choosing where to sample the data carefully, in other words, giventhe regression problem defined by

yi = f(β;xi) + εi

for i = 1, . . . , n with the same conditions on f(β;x) and εi as in Section 1.3.1we want to choose a design xi, i = 1, . . . , n that minimizes the variance ofthe values predicted by the regression model. This is usually referred to asG-optimality.

To give a proper definition of G-optimality we will need the concept ofthe Fisher information matrix. When motivating the expression for the AICin Section 1.3.3 the matrix[

∂2


]n,n1,1

appeared, see expression (24). In that context we were interested how muchinformation was lost when the model g was used instead of the data. Ifthe model g is the true distribution, twice differentiable, and has only oneparameter, β, it is possible to describe how information about the modelthat is contained in the parameter using the Fisher information

I(β) = −EX

[∂2 log(g(β;X))

∂β2

].

57


Remark 1.9. For models with many parameters and small sample sizesit is recommended to add a second order correction to the AIC called theAICC [40,129] and in the case of least squares fitting is given by the followingexpression

AICC = AIC +2(k + 1)(k + 2)

n− k − 2.

There are several other information criterion that could be used in asimilar way to the AIC, for example the Takeuchi information criterion [270]or the Bayesian information criterion [254]. Here we will use the AIC sinceit is considered a reliable criterion that is simple to calculate [40,164] and isasymptotically optimal for selecting the model with the least mean squareerrors [297].

1.4 D-optimal experiment design

For the class of linear non-weighted regression problems described in Sec-tion 1.3.1 minimizing the square of the sum of residuals gives the maximum-likelihood estimation of the parameters that specify the fitted function. Thisestimation naturally has a variance as well and minimizing this variance canbe interpreted as improving the reliability of the fitted function by minimiz-ing its sensitivity to noise in measurements. This minimization is usuallydone by choosing where to sample the data carefully, in other words, giventhe regression problem defined by

yi = f(β;xi) + εi

for i = 1, . . . , n with the same conditions on f(β;x) and εi as in Section 1.3.1we want to choose a design xi, i = 1, . . . , n that minimizes the variance ofthe values predicted by the regression model. This is usually referred to asG-optimality.

To give a proper definition of G-optimality we will need the concept ofthe Fisher information matrix. When motivating the expression for the AICin Section 1.3.3 the matrix[

∂2


]n,n1,1

appeared, see expression (24). In that context we were interested how muchinformation was lost when the model g was used instead of the data. Ifthe model g is the true distribution, twice differentiable, and has only oneparameter, β, it is possible to describe how information about the modelthat is contained in the parameter using the Fisher information

I(β) = −EX

[∂2 log(g(β;X))

∂β2

].

57

59


Essentially this expression measures the probability of a particular outcomebeing observed for a known value of β, so if the Fischer information is onlylarge in near a certain points it is easy to tell which parameter value is thetrue parameter value and if the Fisher information does not have a clear peait is difficult to estimate the correct value of β. When the model has severalparameters the Fisher information is replaced by the Fischer informationmatrix.

Definition 1.16. For a finite design x ∈ X ⊆ Rn the Fisher informationmatrix , M, is the matrix defined by

M(β) = −EX

[∂2


]n,n1,1

Remark 1.10. The concept of information in the AIC and the concept ofinformation here are two different but related concepts, for a discussion ofthis relation see Section 7.7.8 in [40].

There is a lot of literature on the Fisher information matrix and but inthe context of the least squares problems discussed here we have a fairlysimple expression for its elements, see [208] for details.

Lemma 1.10. For a finite design x ∈ X ⊆ Rn the Fisher informationmatrix for the type of least squares fitting problem considered in this sectioncan be computed by

M(x) =n∑i=1

f(xi)f(xi)>

where f(x) =[f1(x) f2(x) · · · fn(x)

]>.

Definition 1.17 (The G-optimality criterion). A design ξ is said to beG-optimal if it minimizes the maximum variance of any predicted value

Var(y(ξ)) = minxi, i=1,2,...,n

maxx∈X

Var(y(x)) = minz∈X

maxx∈X

f(x)>M(z)f(x).

The G-optimality condition was first introduced in [264] (the name G-optimality comes from later work by Kiefer and Wolfowitz where they de-scribe several different types of optimal design using alphabetical letters[153], [154]) and is an example of a minimax criterion, since it minimizesthe maximum variance of the values given by the regression model [208].

There are many kinds of optimality conditions related to G-optimality.One which is suitable for us to consider is D-optimality. This type of opti-mality was first introduced in [285] and instead of focusing on the varianceof the predicted values of the model it instead minimizes the volume of theconfidence ellipsoid for the parameters (for a given confidence level).

58


Essentially this expression measures the probability of a particular outcomebeing observed for a known value of β, so if the Fischer information is onlylarge in near a certain points it is easy to tell which parameter value is thetrue parameter value and if the Fisher information does not have a clear peait is difficult to estimate the correct value of β. When the model has severalparameters the Fisher information is replaced by the Fischer informationmatrix.

Definition 1.16. For a finite design x ∈ X ⊆ Rn the Fisher informationmatrix , M, is the matrix defined by

M(β) = −EX

[∂2


]n,n1,1

Remark 1.10. The concept of information in the AIC and the concept ofinformation here are two different but related concepts, for a discussion ofthis relation see Section 7.7.8 in [40].

There is a lot of literature on the Fisher information matrix and but inthe context of the least squares problems discussed here we have a fairlysimple expression for its elements, see [208] for details.

Lemma 1.10. For a finite design x ∈ X ⊆ Rn the Fisher informationmatrix for the type of least squares fitting problem considered in this sectioncan be computed by

M(x) =n∑i=1

f(xi)f(xi)>

where f(x) =[f1(x) f2(x) · · · fn(x)

]>.

Definition 1.17 (The G-optimality criterion). A design ξ is said to beG-optimal if it minimizes the maximum variance of any predicted value

Var(y(ξ)) = minxi, i=1,2,...,n

maxx∈X

Var(y(x)) = minz∈X

maxx∈X

f(x)>M(z)f(x).

The G-optimality condition was first introduced in [264] (the name G-optimality comes from later work by Kiefer and Wolfowitz where they de-scribe several different types of optimal design using alphabetical letters[153], [154]) and is an example of a minimax criterion, since it minimizesthe maximum variance of the values given by the regression model [208].

There are many kinds of optimality conditions related to G-optimality.One which is suitable for us to consider is D-optimality. This type of opti-mality was first introduced in [285] and instead of focusing on the varianceof the predicted values of the model it instead minimizes the volume of theconfidence ellipsoid for the parameters (for a given confidence level).

58

60


Definition 1.18 (The D-optimality criterion). A design ξ is said to beD-optimal if it maximizes the determinant of the Fisher information matrix

det(M(ξ)) = maxx∈X

det(M(x)).

The D-optimal designs are often good design with respect to other typesof criterion (see for example [112] for a brief discussion on this) and isoften practical to consider due to being invariant with respect to lineartransformations of the design matrix. A well-known theorem called theKiefer–Wolfowitz equivalence theorem shows that under certain conditionsG-optimality is equivalent to D-optimality.

Theorem 1.7 (Kiefer–Wolfowitz equivalence theorem). For any linear re-gression model with independent, uncorrelated errors and continuous andlinearly independent basis functions fi(x) defined on a fixed compact topo-logical space X there exists a D-optimal design and any D-optimal design isalso G-optimal.

This equivalence theorem was originally proven in [155] but the for-mulation above is taken from [208]. Thus maximizing the determinantof the Fisher information matrix corresponds to minimizing the varianceof the estimated β. Interpolation can be considered a special case of re-gression when the sum of the square of the residuals can be reduced tozero. Thus we can speak of D-optimal design for interpolation as well, infact optimal experiment design is often used to find the minimum numberof points needed for a certain model. For a linear interpolation problemdefined by the alternant matrix A(f ; x) the Fisher information matrix isM(x) = A(f ; x)>A(f ; x) and since A(f ; x) is an n× n matrix det(M(x)) =det(A(f ; x)>) det(A(f ; x)) = det(f ; x))2. Thus the maximization of the de-terminant of the Fisher information matrix is equivalent to finding the ex-treme points of the determinant of an alternant matrix in some volume givenby the set of possible designs.

A standard case of this is polynomial interpolation where the x-valuesare in a limited interval, for instance −1 ≤ xi ≤ 1 for i = 1, 2, . . . , n. In thiscase the regression problem can be written as Vn(x)>β = y where Vn(x)is a Vandermonde matrix as defined in equation (1) and the constraints onthe elements of β means that the volume we want to optimize over is a cubein n dimensions. There is a number of classical results that describe how tofind the D-optimal designs for weighted univariate polynomials with variousefficiency functions, e.g. [87], and in Section 2.3.3 we will demonstrate oneway to optimize the Vandermonde determinant over a cube.

The shape of the volume to optimize the determinant in is given by con-straints on the data points. For example, if there is a cost associated witheach data point that increases quadratically with x and there is a total bud-get, C, for the experiment that cannot be exceeded the constraint on the

59


Definition 1.18 (The D-optimality criterion). A design ξ is said to beD-optimal if it maximizes the determinant of the Fisher information matrix

det(M(ξ)) = maxx∈X

det(M(x)).

The D-optimal designs are often good design with respect to other typesof criterion (see for example [112] for a brief discussion on this) and isoften practical to consider due to being invariant with respect to lineartransformations of the design matrix. A well-known theorem called theKiefer–Wolfowitz equivalence theorem shows that under certain conditionsG-optimality is equivalent to D-optimality.

Theorem 1.7 (Kiefer–Wolfowitz equivalence theorem). For any linear re-gression model with independent, uncorrelated errors and continuous andlinearly independent basis functions fi(x) defined on a fixed compact topo-logical space X there exists a D-optimal design and any D-optimal design isalso G-optimal.

This equivalence theorem was originally proven in [155] but the for-mulation above is taken from [208]. Thus maximizing the determinantof the Fisher information matrix corresponds to minimizing the varianceof the estimated β. Interpolation can be considered a special case of re-gression when the sum of the square of the residuals can be reduced tozero. Thus we can speak of D-optimal design for interpolation as well, infact optimal experiment design is often used to find the minimum numberof points needed for a certain model. For a linear interpolation problemdefined by the alternant matrix A(f ; x) the Fisher information matrix isM(x) = A(f ; x)>A(f ; x) and since A(f ; x) is an n× n matrix det(M(x)) =det(A(f ; x)>) det(A(f ; x)) = det(f ; x))2. Thus the maximization of the de-terminant of the Fisher information matrix is equivalent to finding the ex-treme points of the determinant of an alternant matrix in some volume givenby the set of possible designs.

A standard case of this is polynomial interpolation where the x-valuesare in a limited interval, for instance −1 ≤ xi ≤ 1 for i = 1, 2, . . . , n. In thiscase the regression problem can be written as Vn(x)>β = y where Vn(x)is a Vandermonde matrix as defined in equation (1) and the constraints onthe elements of β means that the volume we want to optimize over is a cubein n dimensions. There is a number of classical results that describe how tofind the D-optimal designs for weighted univariate polynomials with variousefficiency functions, e.g. [87], and in Section 2.3.3 we will demonstrate oneway to optimize the Vandermonde determinant over a cube.

The shape of the volume to optimize the determinant in is given by con-straints on the data points. For example, if there is a cost associated witheach data point that increases quadratically with x and there is a total bud-get, C, for the experiment that cannot be exceeded the constraint on the

59

61


x-values becomes x21 + x2

2 + . . . + x2n ≤ C and the determinant needs to be

optimized in a ball. In Chapter 2 we examine the optimization of the Van-dermonde determinant over several different surfaces in several dimensions.

In Section 3.3 we use a D-optimal design to improve the stability of aninterpolation problem as an alternative to the non-linear fitting from Section3.2. Note that while choosing a D-optimal design can give an approximationmethod that is more stable since it minimizes the variance of the parameters,the approximating function can still be highly sensitive to changes in param-eters (the variance of the predicted values can be minimized but still high)so it does necessarily maximize stability or stop instability phenomenonssimilar to Runge’s phenomenon for polynomial interpolation.

1.5 Electromagnetic compatibility andelectrostatic discharge currents

There are many examples of electromagnetic phenomena that involve twoobjects influencing each other without touching. Almost everyone is familiarwith magnets that attract or repel other object, sparks that bridge physicalgaps and radio waves that send messages across the globe. While this action-at-a-distance can be very useful it can also cause unintended interactionsbetween different systems. This is usually referred to as electromagneticdisturbance or electromagnetic interference and the field of electromagneticcompatibility (EMC) is the study and design of systems that are not sus-ceptible to disturbances from other systems and does not cause interferencewith other systems or themselves [228,290].

There are many possible causes of electromagnetic disturbance includinga multitude of sources. Some examples are man-made sources such as broad-casting and receiving devices, power generators and converters, power con-version and ignition systems for combustion engines, manufacturing equip-ment like ovens, saws, mills, welders, blenders and mixers, other equipmentsuch as fans, heaters, coolers, lights, computers, instruments for measure-ments and control, examples of natural sources are atmospheric-, solar- andcosmic noise, static discharges and lightning [212].

Mathematical modelling is an important tool for EMC [212]. Using com-puters for electromagnetic analysis have been done since the 1950s [115] andit rapidly became more and more useful and important over time [234]. Inpractice many different types of models and methods are used, all with theirown advantages and disadvantages, and the design process often involves acombination of analytical and numerical techniques [94]. The sources of elec-tromagnetic disturbances are not always well understood or cannot be welldescribed and deriving all parts of the model from first principles requiresa combination of many different techniques, both numerical, stochastic andanalytical, see [76,229] for examples. In practice it is often reasonable to use

60


x-values becomes x21 + x2

2 + . . . + x2n ≤ C and the determinant needs to be

optimized in a ball. In Chapter 2 we examine the optimization of the Van-dermonde determinant over several different surfaces in several dimensions.

In Section 3.3 we use a D-optimal design to improve the stability of aninterpolation problem as an alternative to the non-linear fitting from Section3.2. Note that while choosing a D-optimal design can give an approximationmethod that is more stable since it minimizes the variance of the parameters,the approximating function can still be highly sensitive to changes in param-eters (the variance of the predicted values can be minimized but still high)so it does necessarily maximize stability or stop instability phenomenonssimilar to Runge’s phenomenon for polynomial interpolation.

1.5 Electromagnetic compatibility andelectrostatic discharge currents

There are many examples of electromagnetic phenomena that involve twoobjects influencing each other without touching. Almost everyone is familiarwith magnets that attract or repel other object, sparks that bridge physicalgaps and radio waves that send messages across the globe. While this action-at-a-distance can be very useful it can also cause unintended interactionsbetween different systems. This is usually referred to as electromagneticdisturbance or electromagnetic interference and the field of electromagneticcompatibility (EMC) is the study and design of systems that are not sus-ceptible to disturbances from other systems and does not cause interferencewith other systems or themselves [228,290].

There are many possible causes of electromagnetic disturbance includinga multitude of sources. Some examples are man-made sources such as broad-casting and receiving devices, power generators and converters, power con-version and ignition systems for combustion engines, manufacturing equip-ment like ovens, saws, mills, welders, blenders and mixers, other equipmentsuch as fans, heaters, coolers, lights, computers, instruments for measure-ments and control, examples of natural sources are atmospheric-, solar- andcosmic noise, static discharges and lightning [212].

Mathematical modelling is an important tool for EMC [212]. Using com-puters for electromagnetic analysis have been done since the 1950s [115] andit rapidly became more and more useful and important over time [234]. Inpractice many different types of models and methods are used, all with theirown advantages and disadvantages, and the design process often involves acombination of analytical and numerical techniques [94]. The sources of elec-tromagnetic disturbances are not always well understood or cannot be welldescribed and deriving all parts of the model from first principles requiresa combination of many different techniques, both numerical, stochastic andanalytical, see [76,229] for examples. In practice it is often reasonable to use

60

62

1.5. ELECTROMAGNETIC COMPATIBILITY ANDELECTROSTATIC DISCHARGE CURRENTS

phenomenological models reproducing typical patterns based on statisticaldata [45,148].

Requirements for a product or system to be considered electromagneti-cally compatible can be found in standards such as the IEC 61000-4-2 [132]and IEC 62305-1 [133]. In several of these standard approximations of typ-ical currents for various phenomena are given and electromagnetic compat-ibility requirements are based on the effects of the system being exposedto these currents, such as the radiated electromagnetic fields. Ideally thedescriptions of these currents should give an accurate description of the ob-served behaviour that the standard is based on as well being computationallyefficient (since computer simulations replacing construction of prototypescan save both time and resources) and be compatible with the mathematicaltools that are commonly used in electromagnetic calculations, for instanceLaplace and Fourier transforms.

In this thesis we will discuss approximations of electrostatic dischargecurrents, either from a standard or based on experimental data. In Section1.5.1 a review of models in the literature can be found and in Chapter3 we propose a new function, the analytically extended function (AEF),for modelling these currents that has some advantages compared to thecommonly used models and can be applied to many different cases, typicallyat the cost of some extra manual work in fitting the model.

Electrostatic discharge (ESD) is a common phenomenon where a sud-den flow of electricity between two charged object occurs, examples includesparks and lightning strikes. The main mechanism behind is usually saidto be contact electrification, this phenomena is due to all materials occa-sionally emitting electrons, usually at a higher rate when they are heated.Typically the emission and absorption balances out but since the rate ofemission varies between different materials an imbalance can occur whentwo materials come sufficiently close to each other. When the materials areseparated this charge imbalance might remain for some time, it can be re-stored by the charged objects slowly emitting electrons to the surroundingobjects but in the right conditions, for example if the charged object comesnear a conductive material with an opposite charge, the restoration of thecharge balance can be very rapid resulting in an electrostatic discharge. Thereader is likely to be familiar with the case of two materials rubbing againsteach other building up a charge imbalance and one of the objects generatinga spark when moved close to a metal object. This case is common since fric-tion between objects typically means a larger contact area where charges cantransfer and movement is necessary for charge separation. For this reasonthis mechanism is often referred to as friction charging or the triboelectriceffect. Contact charging can happen between any material, including liq-uids and gases, and can also be affected by many other types of phenomena,such as ion transfer or energetic charged particles colliding with other ob-jects [100]. Therefore the exact mechanisms behind electrostatic discharges

61


phenomenological models reproducing typical patterns based on statisticaldata [45,148].

Requirements for a product or system to be considered electromagneti-cally compatible can be found in standards such as the IEC 61000-4-2 [132]and IEC 62305-1 [133]. In several of these standard approximations of typ-ical currents for various phenomena are given and electromagnetic compat-ibility requirements are based on the effects of the system being exposedto these currents, such as the radiated electromagnetic fields. Ideally thedescriptions of these currents should give an accurate description of the ob-served behaviour that the standard is based on as well being computationallyefficient (since computer simulations replacing construction of prototypescan save both time and resources) and be compatible with the mathematicaltools that are commonly used in electromagnetic calculations, for instanceLaplace and Fourier transforms.

In this thesis we will discuss approximations of electrostatic dischargecurrents, either from a standard or based on experimental data. In Section1.5.1 a review of models in the literature can be found and in Chapter3 we propose a new function, the analytically extended function (AEF),for modelling these currents that has some advantages compared to thecommonly used models and can be applied to many different cases, typicallyat the cost of some extra manual work in fitting the model.

Electrostatic discharge (ESD) is a common phenomenon where a sud-den flow of electricity between two charged object occurs, examples includesparks and lightning strikes. The main mechanism behind is usually saidto be contact electrification, this phenomena is due to all materials occa-sionally emitting electrons, usually at a higher rate when they are heated.Typically the emission and absorption balances out but since the rate ofemission varies between different materials an imbalance can occur whentwo materials come sufficiently close to each other. When the materials areseparated this charge imbalance might remain for some time, it can be re-stored by the charged objects slowly emitting electrons to the surroundingobjects but in the right conditions, for example if the charged object comesnear a conductive material with an opposite charge, the restoration of thecharge balance can be very rapid resulting in an electrostatic discharge. Thereader is likely to be familiar with the case of two materials rubbing againsteach other building up a charge imbalance and one of the objects generatinga spark when moved close to a metal object. This case is common since fric-tion between objects typically means a larger contact area where charges cantransfer and movement is necessary for charge separation. For this reasonthis mechanism is often referred to as friction charging or the triboelectriceffect. Contact charging can happen between any material, including liq-uids and gases, and can also be affected by many other types of phenomena,such as ion transfer or energetic charged particles colliding with other ob-jects [100]. Therefore the exact mechanisms behind electrostatic discharges

61

63


can be difficult to understand and describe, even when the circumstanceswhere the electrostatic discharge are likely are well known [195].

In this thesis we focus on two types of electrostatic discharge, lightningdischarge and human-to-object (human-to-metal or human-to-human).

Lightning discharges can cause electromagnetic disturbances in threeways, by passing through a system directly, by passing through a nearbyobject which then radiates electrical fields that disturbs the system, or byindirectly inducing transient currents in systems when the electrical fieldassociated with a thundercloud disappears when the lightning discharge re-moves the charge imbalance between cloud and ground [45]. We discussmodelling of some lightning discharges from standards and experimentaldata in Section 3.2.

Electrostatical discharges from humans are very common and are typ-ically just a nuisance, but they can damage sensitive electronics and cancause severe accidents, either by the shock from the discharge causing ahuman error or by directly causing gas or dust explosions [148, 195]. Wediscuss modelling of a simulated human-to-object electrostatical dischargein Section 3.3.

1.5.1 Electrostatic discharge modelling

Well-defined representation of real electrostatic discharge currents is neededin order to establish realistic requirements for ESD generators used in testingthe equipment and devices, as well as to provide and improve the repeata-bility of tests. It should be able to approximate the current for varioustest levels, test set-ups and procedures, and also for various ESD conditionssuch as approach speeds, types of electrodes, relative arc length, humidity,etc. A mathematical function is necessary for computer simulation of suchphenomena, for verification of test generators and for improving standardwaveshape definitions.

A number of current functions, mostly based on exponential functions,have been proposed in the literature to model the ESD currents, [44,95,96,142,144,152,266,278,286,287,301,302]. Here we will give a brief presentationof some of them and in Section 3.1 we will propose an alternative functionand a scheme for fitting it to a waveshape.

A number of mathematical expressions have been introduced in the liter-ature for the purpose of representation of the ESD currents, either the IEC61000-4-2 Standard one [132], or experimentally measured ones, e.g. [95].In this section we give an overview of most commonly applied ESD currentapproximations.

A double-exponential function has been proposed by Cerri et al. [44] forrepresentation of ESD currents for commercial simulators in the form

i(t) = I1e− tτ1 − I2e

− tτ2 ,

62


can be difficult to understand and describe, even when the circumstanceswhere the electrostatic discharge are likely are well known [195].

In this thesis we focus on two types of electrostatic discharge, lightningdischarge and human-to-object (human-to-metal or human-to-human).

Lightning discharges can cause electromagnetic disturbances in threeways, by passing through a system directly, by passing through a nearbyobject which then radiates electrical fields that disturbs the system, or byindirectly inducing transient currents in systems when the electrical fieldassociated with a thundercloud disappears when the lightning discharge re-moves the charge imbalance between cloud and ground [45]. We discussmodelling of some lightning discharges from standards and experimentaldata in Section 3.2.

Electrostatical discharges from humans are very common and are typ-ically just a nuisance, but they can damage sensitive electronics and cancause severe accidents, either by the shock from the discharge causing ahuman error or by directly causing gas or dust explosions [148, 195]. Wediscuss modelling of a simulated human-to-object electrostatical dischargein Section 3.3.

1.5.1 Electrostatic discharge modelling

Well-defined representation of real electrostatic discharge currents is neededin order to establish realistic requirements for ESD generators used in testingthe equipment and devices, as well as to provide and improve the repeata-bility of tests. It should be able to approximate the current for varioustest levels, test set-ups and procedures, and also for various ESD conditionssuch as approach speeds, types of electrodes, relative arc length, humidity,etc. A mathematical function is necessary for computer simulation of suchphenomena, for verification of test generators and for improving standardwaveshape definitions.

A number of current functions, mostly based on exponential functions,have been proposed in the literature to model the ESD currents, [44,95,96,142,144,152,266,278,286,287,301,302]. Here we will give a brief presentationof some of them and in Section 3.1 we will propose an alternative functionand a scheme for fitting it to a waveshape.

A number of mathematical expressions have been introduced in the liter-ature for the purpose of representation of the ESD currents, either the IEC61000-4-2 Standard one [132], or experimentally measured ones, e.g. [95].In this section we give an overview of most commonly applied ESD currentapproximations.

A double-exponential function has been proposed by Cerri et al. [44] forrepresentation of ESD currents for commercial simulators in the form

i(t) = I1e− tτ1 − I2e

− tτ2 ,

62

64


this type of function is also applied in other types of engineering, see Sec-tion 3.1 for some examples.

This model was also extended with a four-exponential version by Keenanand Rossi [152]:

i(t) = I1

(e− tτ1 − e−

tτ2

)− I2

(e− tτ3 − e−

tτ4

). (25)

The Pulse function was proposed in [89],

i(t) = I0

(1− e−

tτ1

)pe− tτ2 ,

and has been used for representation of lightning discharge currents both inits single term form [181] as well as linear combinations of two [266], threeor four Pulse functions [301].

The Heidler function [117] is one of the most commonly used functionsfor lightning discharge modelling

i(t) =I0

η

(tτ1

)n1 +

(tτ1

)n e− tτ2 ,

Wang et al. [286] proposed an ESD model in the form of a sum of two Heidlerfunctions:

i(t) =I1

η1

(tτ1

)n1 +

(tτ1

)n e− tτ2 +

I2

η2

(tτ3

)n1 +

(tτ3

)n e− tτ4 , (26)

with η1 = exp

(− τ1τ2

(nτ2τ1

)1/n)

and η2 = exp

(− τ3τ4

(nτ4τ3

)1/n)

being the

peak correction factors. The function has been used to fit different electro-static discharge currents using different methods [95,286,302].

Berghe and Zutter [278] proposed an ESD current model constructed asa sum of two Gaussian functions in the form:

i(t) = A exp

(−(t− τ1

σ1

)2)

+Bt exp

(−(t− τ2

σ2

)2). (27)

The following approximation using exponential polynomials is presentedin [287],

i(t) = Ate−Ct +Bte−Dt, (28)

and has been used for design of simple electric circuits which can be usedto simulate ESD currents.

One of the most commonly used ESD standard currents is the IEC 61000-4-2 current that represents a typical electrostatic discharge generated bythe human body [132]. In the IEC 61000-4-2 standard [132] this current

63


this type of function is also applied in other types of engineering, see Sec-tion 3.1 for some examples.

This model was also extended with a four-exponential version by Keenanand Rossi [152]:

i(t) = I1

(e− tτ1 − e−

tτ2

)− I2

(e− tτ3 − e−

tτ4

). (25)

The Pulse function was proposed in [89],

i(t) = I0

(1− e−

tτ1

)pe− tτ2 ,

and has been used for representation of lightning discharge currents both inits single term form [181] as well as linear combinations of two [266], threeor four Pulse functions [301].

The Heidler function [117] is one of the most commonly used functionsfor lightning discharge modelling

i(t) =I0

η

(tτ1

)n1 +

(tτ1

)n e− tτ2 ,

Wang et al. [286] proposed an ESD model in the form of a sum of two Heidlerfunctions:

i(t) =I1

η1

(tτ1

)n1 +

(tτ1

)n e− tτ2 +

I2

η2

(tτ3

)n1 +

(tτ3

)n e− tτ4 , (26)

with η1 = exp

(− τ1τ2

(nτ2τ1

)1/n)

and η2 = exp

(− τ3τ4

(nτ4τ3

)1/n)

being the

peak correction factors. The function has been used to fit different electro-static discharge currents using different methods [95,286,302].

Berghe and Zutter [278] proposed an ESD current model constructed asa sum of two Gaussian functions in the form:

i(t) = A exp

(−(t− τ1

σ1

)2)

+Bt exp

(−(t− τ2

σ2

)2). (27)

The following approximation using exponential polynomials is presentedin [287],

i(t) = Ate−Ct +Bte−Dt, (28)

and has been used for design of simple electric circuits which can be usedto simulate ESD currents.

One of the most commonly used ESD standard currents is the IEC 61000-4-2 current that represents a typical electrostatic discharge generated bythe human body [132]. In the IEC 61000-4-2 standard [132] this current

63

65


is given by a graphical representation, see Figure 3.10, together with someconstraints, see page 150. In Figure 1.6 the models discussed in this sectionhave been fitted to the graph given in the standard. The data from thestandard is not included in this figure since some features, notably the initialdelay visible in the standard is not reproduced in either model. The differentmodels give quite different quantitative behaviour in the region 2.5− 25 ns.In Section 3.1 we propose a new scheme for modelling this type of functionsand in Section 3.3 we fit this model to the IEC 61000-4-2 standard currentand some experimental data.

0 5 10 15 20 25 30 35 40 45 50 550

2

4

6

8

10

12

14

t [ns]

i(t)

[A]

Two Heidler, [95] Two Heidler, [302]

Pulse binomial, [266] Exponential polynomial, [287]

Two Gaussians, [278] Four exponential, [152]

Figure 1.6: Comparison of different functions representing the Standard ESDcurrent waveshape for 4kV.

The model given in Section 3.1 is also fitted to both lightning dischargecurrent from the standard and from measured data in Section 3.2.

64


is given by a graphical representation, see Figure 3.10, together with someconstraints, see page 150. In Figure 1.6 the models discussed in this sectionhave been fitted to the graph given in the standard. The data from thestandard is not included in this figure since some features, notably the initialdelay visible in the standard is not reproduced in either model. The differentmodels give quite different quantitative behaviour in the region 2.5− 25 ns.In Section 3.1 we propose a new scheme for modelling this type of functionsand in Section 3.3 we fit this model to the IEC 61000-4-2 standard currentand some experimental data.

0 5 10 15 20 25 30 35 40 45 50 550

2

4

6

8

10

12

14

t [ns]

i(t)

[A]

Two Heidler, [95] Two Heidler, [302]

Pulse binomial, [266] Exponential polynomial, [287]

Two Gaussians, [278] Four exponential, [152]

Figure 1.6: Comparison of different functions representing the Standard ESDcurrent waveshape for 4kV.

The model given in Section 3.1 is also fitted to both lightning dischargecurrent from the standard and from measured data in Section 3.2.

64

66

1.6. MODELLING MORTALITY RATES

1.6 Modelling mortality rates

This section is based on Paper H

Understanding how the probability of surviving to or beyond a certainage is, can be an important question for insurers, actuaries, demographersand policies makers. For some purposes a simple mathematical model canbe desirable. Here we will discuss some basic mathematical concepts usefulfor this type of understanding.

Definition 1.19. Consider an individual whose current age is x and whoseremaining lifetime is denoted Tx > 0 then the survival function, Sx(∆x), isdefined as

Sx(∆x) = Pr[Tx > ∆x].

It is typically assumed [71] that the remaining lifetime Tx obeys therelation

Pr[Tx > ∆x] = Pr[T0 > x+ ∆x|T0 > x] =Pr[T0 > x+ ∆x]

Pr[T0 > x](29)

or phrased in terms of the survival function Sx(∆x) =S0(x+ ∆x)

S0(x).

There are three conditions a survival function must satisfy [71] in orderto have a reasonable interpretation in terms of lifespan

• Only individuals with positive remaining lifetime are considered thusan individual must survive at least 0 units of time, Sx(0) = 1.

• There are no immortal individuals, lim∆x→∞

Sx(∆x) = 0.

• Since the definition only contains an upper bound on remaining life-time Sx(∆x) must be non-increasing.

Here we will not work with the survival function directly, instead we willmodel the mortality rate.

Definition 1.20. The mortality rate, µ, (also known as force of mortality,death rate or hazard rate) for an individual of age x is defined as

µ(x) = limdx→0+

Pr[T0 ≤ x+ dx|T0 > x]. (30)

We can express the mortality rate using the survival function and viceversa using the following lemma.

Lemma 1.11. If Sx(∆x) is a survival function whose derivative exists whenx and ∆x are both non-negative and µ(x) is the corresponding mortality ratethen

µx = −dS0dx

S0(x)

65


1.6 Modelling mortality rates


Understanding how the probability of surviving to or beyond a certainage is, can be an important question for insurers, actuaries, demographersand policies makers. For some purposes a simple mathematical model canbe desirable. Here we will discuss some basic mathematical concepts usefulfor this type of understanding.

Definition 1.19. Consider an individual whose current age is x and whoseremaining lifetime is denoted Tx > 0 then the survival function, Sx(∆x), isdefined as

Sx(∆x) = Pr[Tx > ∆x].

It is typically assumed [71] that the remaining lifetime Tx obeys therelation

Pr[Tx > ∆x] = Pr[T0 > x+ ∆x|T0 > x] =Pr[T0 > x+ ∆x]

Pr[T0 > x](29)

or phrased in terms of the survival function Sx(∆x) =S0(x+ ∆x)

S0(x).

There are three conditions a survival function must satisfy [71] in orderto have a reasonable interpretation in terms of lifespan

• Only individuals with positive remaining lifetime are considered thusan individual must survive at least 0 units of time, Sx(0) = 1.

• There are no immortal individuals, lim∆x→∞

Sx(∆x) = 0.

• Since the definition only contains an upper bound on remaining life-time Sx(∆x) must be non-increasing.

Here we will not work with the survival function directly, instead we willmodel the mortality rate.

Definition 1.20. The mortality rate, µ, (also known as force of mortality,death rate or hazard rate) for an individual of age x is defined as

µ(x) = limdx→0+

Pr[T0 ≤ x+ dx|T0 > x]. (30)

We can express the mortality rate using the survival function and viceversa using the following lemma.

Lemma 1.11. If Sx(∆x) is a survival function whose derivative exists whenx and ∆x are both non-negative and µ(x) is the corresponding mortality ratethen

µx = −dS0dx

S0(x)

65

67


and

Sx(∆x) = exp

(−∫ x+∆x

xµ(t) dt

).

Proof. Using (29) and that the derivative of S0(x) exists we can rewrite thedefinition of the mortality rate as

µ(x) = limdx→0+

1− Pr[Tx > dx]

dx= −

dS0dx

S0(x)= − d

dxln(S0(x)).

Thus we have expressed the mortality rate in terms of the survival functionand using some calculus we can express the survival function in terms of themortality rate. First note that if the derivative of S0(x) exists then

µ(x) = −dS0dx

S0(x)= − d

dxln(S0(x))

and integrating on both sides gives∫ x+∆x

xµ(t) dt = ln(Sx(∆x))− ln(Sx(0))

and since Sx(0) = 1 then ln(Sx(0)) = 0

Sx(∆x) = exp

(−∫ x+∆x

xµ(t) dt

).

In Chapter 4 we will apply these concepts to models of the human lifes-pans in different countries and different years. Mortality rates are typicallyestimated from demographic data for a country using the central mortalityrate which is defined differently than the mortality rate given by (30). Thecentral mortality rate of a group of individuals of age x at time t is denotedmx,t and defined as mx,t = dx

Lxwhere dx is the number of deaths at age x

during some time period and Lx is the average number of living individu-als of age x during that same interval. In this thesis we will only considertime intervals of one year and thus the estimates or the central mortalityrate mx,t is estimated the same way as µ(x) so for any given year t we canassume that mx,t ≈ µ(x).

When examining the mortality rate for developed countries there arethree patterns that are recurring all over the world, an increased mortalityrate for infants that decreases rapidly, in other words µ(x) ∼ 1

x for small x,exponential growth of mortality rate for higher ages, µ(x) ∼ ecx for largex, and a ’hump’ for young adults where the mortality rate first increasesquickly and then remains constant or slowly decreases for some years. Someexamples of mortality rates that demonstrate these patterns can be seen

66


and

Sx(∆x) = exp

(−∫ x+∆x

xµ(t) dt

).

Proof. Using (29) and that the derivative of S0(x) exists we can rewrite thedefinition of the mortality rate as

µ(x) = limdx→0+

1− Pr[Tx > dx]

dx= −

dS0dx

S0(x)= − d

dxln(S0(x)).

Thus we have expressed the mortality rate in terms of the survival functionand using some calculus we can express the survival function in terms of themortality rate. First note that if the derivative of S0(x) exists then

µ(x) = −dS0dx

S0(x)= − d

dxln(S0(x))

and integrating on both sides gives∫ x+∆x

xµ(t) dt = ln(Sx(∆x))− ln(Sx(0))

and since Sx(0) = 1 then ln(Sx(0)) = 0

Sx(∆x) = exp

(−∫ x+∆x

xµ(t) dt

).

In Chapter 4 we will apply these concepts to models of the human lifes-pans in different countries and different years. Mortality rates are typicallyestimated from demographic data for a country using the central mortalityrate which is defined differently than the mortality rate given by (30). Thecentral mortality rate of a group of individuals of age x at time t is denotedmx,t and defined as mx,t = dx

Lxwhere dx is the number of deaths at age x

during some time period and Lx is the average number of living individu-als of age x during that same interval. In this thesis we will only considertime intervals of one year and thus the estimates or the central mortalityrate mx,t is estimated the same way as µ(x) so for any given year t we canassume that mx,t ≈ µ(x).

When examining the mortality rate for developed countries there arethree patterns that are recurring all over the world, an increased mortalityrate for infants that decreases rapidly, in other words µ(x) ∼ 1

x for small x,exponential growth of mortality rate for higher ages, µ(x) ∼ ecx for largex, and a ’hump’ for young adults where the mortality rate first increasesquickly and then remains constant or slowly decreases for some years. Someexamples of mortality rates that demonstrate these patterns can be seen

66

68


in Figure 1.7. The typical explanations for the rapid decrease in mortalityrate is that small children are sensitive to disease, disorders and accidentsbut becomes more resilient as they mature. The ’hump’ for young adults isusually attributed to a lifestyle change, starting in their early to mid teensindividuals tend to become more independent and take more risks, especiallyyoung men. Sometimes this phenomena is known as the accident hump sinceaccidents (often vehicular accidents) are believed to explain a large part ofthe shape of the hump, e.g. in the USA in 2017 approximately 40% of thedeaths in the age range 15-35 [161]. The increase in mortality rate for higherages is explained by the increased risk of health issues that follows naturallyfrom aging.

In some countries there is also a visible trend that the growth of themortality rate starts to slow down for very high ages, whether this trend isgenerally present and at which age it should be taken into consideration isstill being debated, see [14,17,88,109,177] for examples of varying views.

age, x, years

0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

USA 1992

age, x, years

0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Sweden 1992

age, x, years

0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Switzerland 1992

age, x, years

0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Ukraine 1992

age, x, years0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Japan 1992

age, x, years0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Taiwan 1992

age, x, years0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Australia 1992

age, x, years0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Chile 1992

Figure 1.7: Examples of central mortality rate curves for men demonstratingthe typical patterns of rapidly decreasing mortality rate for veryyoung ages followed by a ’hump’ for young adult and a rapid in-crease for high ages.

In Section 4.2 an overview of models in literature will be given and threenew models introduced. The different models will then be compared to eachother by fitting the models to the central mortality rate for men in variousdifferent countries and computing the corresponding AIC values.

It is not only a models ability to reproduce observed patterns in datathat determines its usefulness. Choosing the appropriate model dependson the intended application. For many applications it is not just desirableto understand what the mortality rate is now and how it has changed his-

67


in Figure 1.7. The typical explanations for the rapid decrease in mortalityrate is that small children are sensitive to disease, disorders and accidentsbut becomes more resilient as they mature. The ’hump’ for young adults isusually attributed to a lifestyle change, starting in their early to mid teensindividuals tend to become more independent and take more risks, especiallyyoung men. Sometimes this phenomena is known as the accident hump sinceaccidents (often vehicular accidents) are believed to explain a large part ofthe shape of the hump, e.g. in the USA in 2017 approximately 40% of thedeaths in the age range 15-35 [161]. The increase in mortality rate for higherages is explained by the increased risk of health issues that follows naturallyfrom aging.

In some countries there is also a visible trend that the growth of themortality rate starts to slow down for very high ages, whether this trend isgenerally present and at which age it should be taken into consideration isstill being debated, see [14,17,88,109,177] for examples of varying views.

age, x, years

0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

USA 1992

age, x, years

0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Sweden 1992

age, x, years

0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Switzerland 1992

age, x, years

0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Ukraine 1992

age, x, years0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Japan 1992

age, x, years0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Taiwan 1992

age, x, years0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Australia 1992

age, x, years0 20 40 60

ln(m

x,t)

-10

-8

-6

-4

Chile 1992

Figure 1.7: Examples of central mortality rate curves for men demonstratingthe typical patterns of rapidly decreasing mortality rate for veryyoung ages followed by a ’hump’ for young adult and a rapid in-crease for high ages.

In Section 4.2 an overview of models in literature will be given and threenew models introduced. The different models will then be compared to eachother by fitting the models to the central mortality rate for men in variousdifferent countries and computing the corresponding AIC values.

It is not only a models ability to reproduce observed patterns in datathat determines its usefulness. Choosing the appropriate model dependson the intended application. For many applications it is not just desirableto understand what the mortality rate is now and how it has changed his-

67

69


torically, but also how it can be predicted. In Section 4.5 we will see whateffects replacing historical data on mortality rates with values given by a fewdifferent fitted models will have when using a method of forecasting calledthe Lee–Carter method. The Lee–Carter forecasting method is described inthe next section.

1.6.1 Lee–Carter method for forecasting

This section is based on Paper I

For many applications it is also important to be able to forecast howthe mortality rate will change in the future. There are several methodsof producing the forecasts but the method proposed by Lee and Carter in1992 [170], seems to be generally accepted, because it produces satisfactoryfits and forecasts of mortality rates for various countries. Secondly, thestructure of the Lee–Carter (L–C) method allows for easy computation ofconfidence intervals related to mortality projections.

Lee and Carter developed their approach specifically for U.S. mortalitydata, 1900-1989 and forecasted (over a 50 year forecast horizon), 1990-2065.However, the method has now been applied to mortality data from manyother countries and time periods, e.g. Chile [172], China [147], Japan [291],the seven most economically developed nations (G7) [275], India [46], theNordic countries [162], Sri Lanka [1] and Thailand [298].

Lee and Carter assumed [170] that the central mortality rate for a givenage changes as a log-normal random walk with drift

ln(mx,t) = ax + bxkt + εx,t, (31)

where ln(mx,t) is the central mortality rate at age x in year t, ax is theaverage pattern of mortality at age x, bx represents how mortality at eachage varies when the general level of mortality changes, kt is mortality indexthat captures the evolution of rates over time and εx,t an error term whichcauses the deviation of the model from the observed mortality rates, assumedto be normally distributed N(0, σ2

t ).The parametrization given in (31) is not unique. For example, if we

have a solution ax, bx and kt, then there might exist any non-zero constantc ∈ R which gives another solution of ax − cbx, cbx and kt = c, for whichthese transformations might produce identical forecasts. In order to get aunique solution when fitting a L–C model, constraints must be imposed.The constraints can be chosen in different ways but here we will use the

following: bx is constrained to sum to 1,∑x

bx = 1, and kt to sum to 0,∑t

kt = 0, which gives ax to be as the average over time of the ln(mx,t),

ax = 1T

∑t

ln(mx,t).

68


torically, but also how it can be predicted. In Section 4.5 we will see whateffects replacing historical data on mortality rates with values given by a fewdifferent fitted models will have when using a method of forecasting calledthe Lee–Carter method. The Lee–Carter forecasting method is described inthe next section.

1.6.1 Lee–Carter method for forecasting


For many applications it is also important to be able to forecast howthe mortality rate will change in the future. There are several methodsof producing the forecasts but the method proposed by Lee and Carter in1992 [170], seems to be generally accepted, because it produces satisfactoryfits and forecasts of mortality rates for various countries. Secondly, thestructure of the Lee–Carter (L–C) method allows for easy computation ofconfidence intervals related to mortality projections.

Lee and Carter developed their approach specifically for U.S. mortalitydata, 1900-1989 and forecasted (over a 50 year forecast horizon), 1990-2065.However, the method has now been applied to mortality data from manyother countries and time periods, e.g. Chile [172], China [147], Japan [291],the seven most economically developed nations (G7) [275], India [46], theNordic countries [162], Sri Lanka [1] and Thailand [298].

Lee and Carter assumed [170] that the central mortality rate for a givenage changes as a log-normal random walk with drift

ln(mx,t) = ax + bxkt + εx,t, (31)

where ln(mx,t) is the central mortality rate at age x in year t, ax is theaverage pattern of mortality at age x, bx represents how mortality at eachage varies when the general level of mortality changes, kt is mortality indexthat captures the evolution of rates over time and εx,t an error term whichcauses the deviation of the model from the observed mortality rates, assumedto be normally distributed N(0, σ2

t ).The parametrization given in (31) is not unique. For example, if we

have a solution ax, bx and kt, then there might exist any non-zero constantc ∈ R which gives another solution of ax − cbx, cbx and kt = c, for whichthese transformations might produce identical forecasts. In order to get aunique solution when fitting a L–C model, constraints must be imposed.The constraints can be chosen in different ways but here we will use the

following: bx is constrained to sum to 1,∑x

bx = 1, and kt to sum to 0,∑t

kt = 0, which gives ax to be as the average over time of the ln(mx,t),

ax = 1T

∑t

ln(mx,t).

68

70


The parameters ax, bx and the mortality indices kt are found as follows:Given a set of ages (or age ranges), xi, i = 1, . . . , n, and a set of years,

tj , j = 1, . . . , T, first estimate ax = 1T

∑t

ln(mx,t). Then construct the

matrix given by Zij = ln(mxi,tj )− axi . From the conditions imposed on bxand kt we now know that Zij = bxiktj and thereby the values of bx and ktcan be found using the singular value decomposition (SVD) of Z. Findingthe standard SVD Z = USV > gives bx as the first column of U and kt isgiven by the largest singular value multiplied by the first column of V >.

Forecasting future mortality indices can be done in different ways, but inpractice the random walk with drift model (RWD) for kt is common becauseof its simplicity and straight forward interpretation, so we will also use theRWD model to estimate kt by kt = kt−1 + θ + εt. In this specification, θis the drift term, and kt is forecast to decline linearly with increments ofθ, while deviations from this path, εt, are permanently incorporated in thetrajectory. The drift term θ is estimated as below, which shows that θ onlydepends on the first and last values of kt estimates,

θ =kT − k1

T − 1.

We can now forecast the mortality index with the formula kt+∆t =kt + θ∆t+ εt and then predict the logarithm of the central mortality rate asln(mx,t+∆t) = ax + bxkt. To accompany this prediction we also want a con-fidence interval for the forecast at time ∆t. This can be done by computingthe confidence interval for the mortality index by computing the standarddeviation of the mortality indices compared with the RWD model and thenmultiplying the result with square root of ∆t. Thus if we used the centralmortality rate for T different years we can with with confidence level α saythat

kt+∆t − λα2· σ ·√

∆t ≤ kt+∆t ≤ kt+∆t + λα2· σ ·√

∆t,

where λβ is the inverse of the cumulative normal distribution function for

β and σ =

(T−1∑t=1

(kt+1 − kt − θ)2

). In Section 4.5 examples of forecasts

are illustrated, see Figure 4.7 for central mortality rates and Figure 4.8 formortality indices.

The L–C model is not without flaws, a common remark is that the as-sumptions on how mortality rate changes are quite restrictive. It cannot cap-ture age specific changes of pattern, for example medical breakthroughs inreducing a specific cause of death that is common in a certain age range [169].It also often fails when applied to specific causes of mortality, for examplemotor vehicle accidents showed a rising trend initially as the availabilityof motor vehicles increased but over time it has decreased due to improved

69


The parameters ax, bx and the mortality indices kt are found as follows:Given a set of ages (or age ranges), xi, i = 1, . . . , n, and a set of years,

tj , j = 1, . . . , T, first estimate ax = 1T

∑t

ln(mx,t). Then construct the

matrix given by Zij = ln(mxi,tj )− axi . From the conditions imposed on bxand kt we now know that Zij = bxiktj and thereby the values of bx and ktcan be found using the singular value decomposition (SVD) of Z. Findingthe standard SVD Z = USV > gives bx as the first column of U and kt isgiven by the largest singular value multiplied by the first column of V >.

Forecasting future mortality indices can be done in different ways, but inpractice the random walk with drift model (RWD) for kt is common becauseof its simplicity and straight forward interpretation, so we will also use theRWD model to estimate kt by kt = kt−1 + θ + εt. In this specification, θis the drift term, and kt is forecast to decline linearly with increments ofθ, while deviations from this path, εt, are permanently incorporated in thetrajectory. The drift term θ is estimated as below, which shows that θ onlydepends on the first and last values of kt estimates,

θ =kT − k1

T − 1.

We can now forecast the mortality index with the formula kt+∆t =kt + θ∆t+ εt and then predict the logarithm of the central mortality rate asln(mx,t+∆t) = ax + bxkt. To accompany this prediction we also want a con-fidence interval for the forecast at time ∆t. This can be done by computingthe confidence interval for the mortality index by computing the standarddeviation of the mortality indices compared with the RWD model and thenmultiplying the result with square root of ∆t. Thus if we used the centralmortality rate for T different years we can with with confidence level α saythat

kt+∆t − λα2· σ ·√

∆t ≤ kt+∆t ≤ kt+∆t + λα2· σ ·√

∆t,

where λβ is the inverse of the cumulative normal distribution function for

β and σ =

(T−1∑t=1

(kt+1 − kt − θ)2

). In Section 4.5 examples of forecasts

are illustrated, see Figure 4.7 for central mortality rates and Figure 4.8 formortality indices.

The L–C model is not without flaws, a common remark is that the as-sumptions on how mortality rate changes are quite restrictive. It cannot cap-ture age specific changes of pattern, for example medical breakthroughs inreducing a specific cause of death that is common in a certain age range [169].It also often fails when applied to specific causes of mortality, for examplemotor vehicle accidents showed a rising trend initially as the availabilityof motor vehicles increased but over time it has decreased due to improved

69

71


safety of vehicles and roads as well as increased urbanisation [111]. The Lee-Carter model has also been criticized for giving age-profiles that evolve inimplausible ways for long-run forecasts [111] as well as extrapolation back-wards in time [169]. It can also misrepresent the temporal dependence be-tween age groups [210]. Several variants and extensions of the L–C approachthat improves performance have been suggested, see [29,57,162,169,171,176]for examples. These models extended the L–C approach by including addi-tional period effects and in some cases cohort effects.

In Section 4.5 we will fit a few models to central mortality rate data forseveral countries and then use the fitted model to produce values for theL–C method and examine the differences in the predictions based on thedifferent sets of data.

70


safety of vehicles and roads as well as increased urbanisation [111]. The Lee-Carter model has also been criticized for giving age-profiles that evolve inimplausible ways for long-run forecasts [111] as well as extrapolation back-wards in time [169]. It can also misrepresent the temporal dependence be-tween age groups [210]. Several variants and extensions of the L–C approachthat improves performance have been suggested, see [29,57,162,169,171,176]for examples. These models extended the L–C approach by including addi-tional period effects and in some cases cohort effects.

In Section 4.5 we will fit a few models to central mortality rate data forseveral countries and then use the fitted model to produce values for theL–C method and examine the differences in the predictions based on thedifferent sets of data.

70

72

1.7. SUMMARIES OF PAPERS

1.7 Summaries of papers

Paper A [187]

This paper examines the extreme points of the Vandermonde determinant onthe sphere in three or more dimensions. A few different ways to analyse thethree-dimensional case are shown in Section 2.1.2, and a detailed descriptionof the method used to solve the n-dimensional problem from [269] can befound in Section 2.2.1. The extreme points are given in terms of roots ofrescaled Hermite polynomials. For dimensions three to seven explicit expres-sions are given the results are visualized dimensions by using symmetries ofthe answers to project all the extreme points onto a two-dimensional surface,see Section 2.2.2. The thesis author contributed primarily to the derivationof some of the recursive properties of the Vandermonde determinant and itsderivatives and to a lesser extent to the visualisation aspects of the problem.

Paper B [186]

The Vandermonde determinant is optimized over the ellipsoid and cylinderin three dimensions, see Section 2.1.4 and 2.1.5. Lagrange multipliers areused to find a system of polynomial equations which give the local extremepoints. Using Grobner basis and other techniques the extreme points aregiven either explicitly or as roots of univariate polynomials. The results alsopresented visually for some special cases. The method is also extended tosurfaces defined by homogeneous polynomials, see Section 2.1.6. The ex-treme points on sphere defined by the p - norm (primarily p = 4) are alsodiscussed. The thesis author primarily contributed to the examination ofthe ellipsoid, cylinder and surfaces defined by homogenous polynomials.

Paper C [216]

The sphere in n dimensions with respect to a p - norm can be thought of as asurface defined implicitly by a univariate polynomial. Here it is shown thatthe extreme points of the Vandermonde determinant on a bounded surfacedefined by a univariate polynomial are given the zeroes of the polynomialsolution of a differential equation with polynomial coefficients. Expressionsfor polynomials whose roots give the coordinates of the extreme points aregiven for the cases of a surface given by a general first or second-degree poly-nomial, some higher degree monomials and cubes (Sections 2.3.1–2.3.3 and2.3.6). Some results that can be used to reduce the dimension of the prob-lem, but not solve it entirely, for even n and p are also discussed. The thesisauthor contributed by extending previous results to cubes and the generalpolynomials of low degree and, based on contributions from the other au-thors, he found how the Newton–Girard formulae can be used to compactlyexpress and simplify the equation system corresponding to the case wherethe surface is a sphere defined by a p -norm for even n and p.

71


1.7 Summaries of papers

Paper A [187]

This paper examines the extreme points of the Vandermonde determinant onthe sphere in three or more dimensions. A few different ways to analyse thethree-dimensional case are shown in Section 2.1.2, and a detailed descriptionof the method used to solve the n-dimensional problem from [269] can befound in Section 2.2.1. The extreme points are given in terms of roots ofrescaled Hermite polynomials. For dimensions three to seven explicit expres-sions are given the results are visualized dimensions by using symmetries ofthe answers to project all the extreme points onto a two-dimensional surface,see Section 2.2.2. The thesis author contributed primarily to the derivationof some of the recursive properties of the Vandermonde determinant and itsderivatives and to a lesser extent to the visualisation aspects of the problem.

Paper B [186]

The Vandermonde determinant is optimized over the ellipsoid and cylinderin three dimensions, see Section 2.1.4 and 2.1.5. Lagrange multipliers areused to find a system of polynomial equations which give the local extremepoints. Using Grobner basis and other techniques the extreme points aregiven either explicitly or as roots of univariate polynomials. The results alsopresented visually for some special cases. The method is also extended tosurfaces defined by homogeneous polynomials, see Section 2.1.6. The ex-treme points on sphere defined by the p - norm (primarily p = 4) are alsodiscussed. The thesis author primarily contributed to the examination ofthe ellipsoid, cylinder and surfaces defined by homogenous polynomials.

Paper C [216]

The sphere in n dimensions with respect to a p - norm can be thought of as asurface defined implicitly by a univariate polynomial. Here it is shown thatthe extreme points of the Vandermonde determinant on a bounded surfacedefined by a univariate polynomial are given the zeroes of the polynomialsolution of a differential equation with polynomial coefficients. Expressionsfor polynomials whose roots give the coordinates of the extreme points aregiven for the cases of a surface given by a general first or second-degree poly-nomial, some higher degree monomials and cubes (Sections 2.3.1–2.3.3 and2.3.6). Some results that can be used to reduce the dimension of the prob-lem, but not solve it entirely, for even n and p are also discussed. The thesisauthor contributed by extending previous results to cubes and the generalpolynomials of low degree and, based on contributions from the other au-thors, he found how the Newton–Girard formulae can be used to compactlyexpress and simplify the equation system corresponding to the case wherethe surface is a sphere defined by a p -norm for even n and p.

71

73


Paper D [217]

This paper reviews the role of the Vandermonde matrix in random matrixtheory and shows how the problem of finding the extreme points of the prob-ability distribution of the eigenvalues of a Wishart matrix can be rewrittenas a problem of finding the extreme points of the Vandermonde determinanton a sphere with a radius given by the trace of the square of the Wishart ma-trix. The thesis authors contribution was showing that the extreme pointsof the probability distribution of the eigenvalues must lie on a sphere with aparticular radius and how to use the properties of the Vandermonde deter-minant to find a polynomial whose roots give the coordinates of the extremepoints of the probability distribution of the eigenvalues, see Section 2.3.7.

Paper E∗ [185]

This paper is a detailed description and derivation of some properties ofthe analytically extended function (AEF) and a scheme for how it can beused in approximation of lightning discharge currents, see sections 3.1.1 and3.2.2. Lightning discharge currents are classified in the IEC 62305-1 Stan-dard into waveshapes representing important observed phenomena. Thesewaveshapes are approximated with mathematical functions in order to beused in lightning discharge models for ensuring electromagnetic compatibil-ity. A general framework for estimating the parameters of the AEF using theMarquardt least squares method (MLSM) for a waveform with an arbitrary(finite) number of peaks as well as for the given charge transfer and specificenergy is described, see sections 1.2.6, 3.2 and 3.2.3. This framework is usedto find parameters for some single-peak waveshapes and advantages anddisadvantages of the approach are discussed, see Section 3.2.6. The thesisauthor contributed with the p -peak formulation of the AEF, modificationto the MLSM and basic software for fitting the AEF to data.

Paper F∗[184]

In this paper it is examined how the analytically extended function (AEF)can be used to approximate multi-peaked lightning current waveforms. Ageneral framework for estimating the parameters of the AEF using the Mar-quardt least squares method (MLSM) for a waveform with an arbitrary(finite) number of peaks is presented, see Section 3.2. This framework isused to find parameters for some waveforms, such as lightning currents fromthe IEC 62305-1 Standard and recorded lightning current data, see Section3.2.6. The thesis author contributed with improved software for fitting theAEF to the more complicated waveforms (compared to Paper E).

∗The model and techniques in Paper E and F are applied to various waveforms in[144,145,188–190].

72


Paper D [217]

This paper reviews the role of the Vandermonde matrix in random matrixtheory and shows how the problem of finding the extreme points of the prob-ability distribution of the eigenvalues of a Wishart matrix can be rewrittenas a problem of finding the extreme points of the Vandermonde determinanton a sphere with a radius given by the trace of the square of the Wishart ma-trix. The thesis authors contribution was showing that the extreme pointsof the probability distribution of the eigenvalues must lie on a sphere with aparticular radius and how to use the properties of the Vandermonde deter-minant to find a polynomial whose roots give the coordinates of the extremepoints of the probability distribution of the eigenvalues, see Section 2.3.7.

Paper E∗ [185]

This paper is a detailed description and derivation of some properties ofthe analytically extended function (AEF) and a scheme for how it can beused in approximation of lightning discharge currents, see sections 3.1.1 and3.2.2. Lightning discharge currents are classified in the IEC 62305-1 Stan-dard into waveshapes representing important observed phenomena. Thesewaveshapes are approximated with mathematical functions in order to beused in lightning discharge models for ensuring electromagnetic compatibil-ity. A general framework for estimating the parameters of the AEF using theMarquardt least squares method (MLSM) for a waveform with an arbitrary(finite) number of peaks as well as for the given charge transfer and specificenergy is described, see sections 1.2.6, 3.2 and 3.2.3. This framework is usedto find parameters for some single-peak waveshapes and advantages anddisadvantages of the approach are discussed, see Section 3.2.6. The thesisauthor contributed with the p -peak formulation of the AEF, modificationto the MLSM and basic software for fitting the AEF to data.

Paper F∗[184]

In this paper it is examined how the analytically extended function (AEF)can be used to approximate multi-peaked lightning current waveforms. Ageneral framework for estimating the parameters of the AEF using the Mar-quardt least squares method (MLSM) for a waveform with an arbitrary(finite) number of peaks is presented, see Section 3.2. This framework isused to find parameters for some waveforms, such as lightning currents fromthe IEC 62305-1 Standard and recorded lightning current data, see Section3.2.6. The thesis author contributed with improved software for fitting theAEF to the more complicated waveforms (compared to Paper E).

∗The model and techniques in Paper E and F are applied to various waveforms in[144,145,188–190].

72

74


Paper G [191]

The multi-peaked analytically extended function (AEF) is used in this pa-per for representation of electrostatic discharge (ESD) currents. In order tominimize unstable behaviour and the number of free parameters the expo-nents of the AEF are chosen from an arithmetic sequence. The function isfitted by interpolating data chosen according to a D-optimal design. ESDcurrent modelling is illustrated through two examples: an approximation ofthe IEC Standard 61000-4-2 waveshape, and a representation of some mea-sured ESD current. The contents of this paper is in Section 3.3. The thesisauthor contributed with the derivation of the D-optimal design, motivatingits use as well as software for fitting the AEF to the example currents.

Paper H [192]

There are many models for the mortality rates for various years and coun-tries. A phenomenon that complicates the modelling of human mortalityrates is a rapid increase in mortality rate for young adults (in many de-veloped countries this is especially pronounced at the age of 25). In thispaper a model for mortality rates based on power-exponential functions isintroduced and compared to empirical data for mortality rates from sev-eral countries and other mathematical models for mortality rate. The thesisauthors contribution is the formulation of the model and writing softwarefor fitting the various models to empirical data and computing the AkaikeInformation Criterion to facilitate comparison between the models.

Paper I [33]

Mortality rate forecasting is important in actuarial science and demogra-phy. There are many models for mortality rates with different propertiesand varying complexity. In this paper several models are used to mortalityrates listings by fitting the models to empirical data using non-linear leastsquare fitting. These listings are then used to forecast the mortality rateusing the Lee–Carter method and the results for the different models arecompared. The thesis authors contribution was assisting with writing soft-ware that computed the mortality rate listings as well as devise the methodfor comparing the reliability of the forecast in a simple manner.

73


Paper G [191]

The multi-peaked analytically extended function (AEF) is used in this pa-per for representation of electrostatic discharge (ESD) currents. In order tominimize unstable behaviour and the number of free parameters the expo-nents of the AEF are chosen from an arithmetic sequence. The function isfitted by interpolating data chosen according to a D-optimal design. ESDcurrent modelling is illustrated through two examples: an approximation ofthe IEC Standard 61000-4-2 waveshape, and a representation of some mea-sured ESD current. The contents of this paper is in Section 3.3. The thesisauthor contributed with the derivation of the D-optimal design, motivatingits use as well as software for fitting the AEF to the example currents.

Paper H [192]

There are many models for the mortality rates for various years and coun-tries. A phenomenon that complicates the modelling of human mortalityrates is a rapid increase in mortality rate for young adults (in many de-veloped countries this is especially pronounced at the age of 25). In thispaper a model for mortality rates based on power-exponential functions isintroduced and compared to empirical data for mortality rates from sev-eral countries and other mathematical models for mortality rate. The thesisauthors contribution is the formulation of the model and writing softwarefor fitting the various models to empirical data and computing the AkaikeInformation Criterion to facilitate comparison between the models.

Paper I [33]

Mortality rate forecasting is important in actuarial science and demogra-phy. There are many models for mortality rates with different propertiesand varying complexity. In this paper several models are used to mortalityrates listings by fitting the models to empirical data using non-linear leastsquare fitting. These listings are then used to forecast the mortality rateusing the Lee–Carter method and the results for the different models arecompared. The thesis authors contribution was assisting with writing soft-ware that computed the mortality rate listings as well as devise the methodfor comparing the reliability of the forecast in a simple manner.

73

75

76

Chapter 2

Extreme points of theVandermonde determinant

This chapter is based on Papers A, B, C, and D




Paper D Asaph Keikara Muhumuza, Karl Lundengard, Jonas Osterberg,

Sergei Silvestrov, John Magero Mango and Godwin Kakuba.

Optimization of the Wishart joint eigenvalue probability density

distribution based on the Vandermonde determinant.

Accepted for publication in Algebraic structures and Applications.

SPAS2017, Vasteras and Stockholm, Sweden, October 4 – 6, 2017,

Sergei Silvestrov, Anatoliy Malyarenko, Milica Rancic (Eds),

Springer International Publishing, 2019.

Chapter 2

Extreme points of theVandermonde determinant

This chapter is based on Papers A, B, C, and D




Paper D Asaph Keikara Muhumuza, Karl Lundengard, Jonas Osterberg,

Sergei Silvestrov, John Magero Mango and Godwin Kakuba.

Optimization of the Wishart joint eigenvalue probability density

distribution based on the Vandermonde determinant.

Accepted for publication in Algebraic structures and Applications.

SPAS2017, Vasteras and Stockholm, Sweden, October 4 – 6, 2017,

Sergei Silvestrov, Anatoliy Malyarenko, Milica Rancic (Eds),

Springer International Publishing, 2019.

77

78

2.1 EXTREME POINTS OF THE VANDERMONDEDETERMINANT AND RELATED DETERMINANTS IN 3D

2.1 Extreme points of the Vandermonde determi-nant and related determinants on various sur-faces in three dimensions

In this chapter we will discuss how to optimize the determinant of the Van-dermonde matrix and some related determinants over various surfaces inthree dimensions and the results will be visualized.

2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions

This section is based on Section 1.1 of Paper A

In this section we plot the values of the determinant

v3(x3) = (x3 − x2)(x3 − x1)(x2 − x1),

and also the generalized Vandermonde determinant g3(x3,a3) for three dif-ferent choices of a3 over the unit sphere x2

1 +x22 +x2

3 = 1 in R3. Our plots areover the unit sphere but the determinant exhibits the same general behaviorover centered spheres of any radius. This follows directly from (1.4) andthat exactly one element from each row appears in the determinant. Forany scalar c we get

gn(cxn,an) =

(n∏i=1

cai

)gn(xn,an),

which for vn becomes

vn(cxn) = cn(n−1)

2 vn(xn), (32)

and so the values over different radii differ only by a constant factor.In Figure 2.1 value of v3(x3) has been plotted over the unit sphere and

the curves where the determinant vanishes are traced as black lines.The coordinates in Figure 2.1 (b) are related to x3 by

x3 =

2 0 1−1 1 1−1 −1 1

1/√

6 0 0

0 1/√

2 0

0 0 1/√

3

t, (33)

where the columns in the product of the two matrices are the basis vectors inR3. The unit sphere in R3 can also be described using spherical coordinates.In Figure 2.1 (c) the following parametrization was used.

t(θ, φ) =

cos(φ) sin(θ)sin(φ)

cos(φ) cos(θ)

. (34)

77


2.1 Extreme points of the Vandermonde determi-nant and related determinants on various sur-faces in three dimensions

In this chapter we will discuss how to optimize the determinant of the Van-dermonde matrix and some related determinants over various surfaces inthree dimensions and the results will be visualized.

2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions


In this section we plot the values of the determinant

v3(x3) = (x3 − x2)(x3 − x1)(x2 − x1),

and also the generalized Vandermonde determinant g3(x3,a3) for three dif-ferent choices of a3 over the unit sphere x2

1 +x22 +x2

3 = 1 in R3. Our plots areover the unit sphere but the determinant exhibits the same general behaviorover centered spheres of any radius. This follows directly from (1.4) andthat exactly one element from each row appears in the determinant. Forany scalar c we get

gn(cxn,an) =

(n∏i=1

cai

)gn(xn,an),

which for vn becomes

vn(cxn) = cn(n−1)

2 vn(xn), (32)

and so the values over different radii differ only by a constant factor.In Figure 2.1 value of v3(x3) has been plotted over the unit sphere and

the curves where the determinant vanishes are traced as black lines.The coordinates in Figure 2.1 (b) are related to x3 by

x3 =

2 0 1−1 1 1−1 −1 1

1/√

6 0 0

0 1/√

2 0

0 0 1/√

3

t, (33)

where the columns in the product of the two matrices are the basis vectors inR3. The unit sphere in R3 can also be described using spherical coordinates.In Figure 2.1 (c) the following parametrization was used.

t(θ, φ) =

cos(φ) sin(θ)sin(φ)

cos(φ) cos(θ)

. (34)

77

79


(a) Plot with respect tothe regular x-basis.

(b) Plot with respect tothe t-basis, see (33).

(c) Plot with respect toparametrization (34).

Figure 2.1: Plot of v3(x3) over the unit sphere.

We will use this t-basis and spherical parametrization throughout this sec-tion.

From the plots in Figure 2.1 it can be seen that the number of extremepoints for v3 over the unit sphere seem to be 6 = 3!. It can also been seenthat all extreme points seem to lie in the plane through the origin thatis orthogonal to an apparent symmetry axis in the direction (1, 1, 1), thedirection of t3. We will see later that the extreme points for vn indeed lie in

the hyperplanen∑i=1

xi = 0 for all n, see Theorem 2.2, and the total number

of extreme points for vn equals n!, see Remark 2.1.The black lines where v3(x3) vanishes are actually the intersections be-

tween the sphere and the three planes x3 − x1 = 0, x3 − x2 = 0 andx2 − x1 = 0, as these differences appear as factors in v3(x3).

We will see later on that the extreme points are the six points acquiredfrom permuting the coordinates in

x3 =1√2

(−1, 0, 1) .

For reasons that will become clear in Section 2.2.1 it is also useful to thinkabout these coordinates as the roots of the polynomial

P3(x) = x3 − 1

2x.

So far we have only considered the behavior of v3(x3), that is g3(x3,a3)with a3 = (0, 1, 2). We now consider three generalized Vandermonde de-terminants, namely g3 with a3 = (0, 1, 3), a3 = (0, 2, 3) and a3 = (1, 2, 3).These three determinants show increasingly more structure and they all havea neat formula in terms of v3 and the elementary symmetric polynomials

ekn = ek(x1, · · · , xn) =∑

1≤i1<i2<···<ik≤nxi1xi2 · · ·xik ,

78




(c) Plot with respect toparametrization (34).

Figure 2.1: Plot of v3(x3) over the unit sphere.

We will use this t-basis and spherical parametrization throughout this sec-tion.

From the plots in Figure 2.1 it can be seen that the number of extremepoints for v3 over the unit sphere seem to be 6 = 3!. It can also been seenthat all extreme points seem to lie in the plane through the origin thatis orthogonal to an apparent symmetry axis in the direction (1, 1, 1), thedirection of t3. We will see later that the extreme points for vn indeed lie in

the hyperplanen∑i=1

xi = 0 for all n, see Theorem 2.2, and the total number

of extreme points for vn equals n!, see Remark 2.1.The black lines where v3(x3) vanishes are actually the intersections be-

tween the sphere and the three planes x3 − x1 = 0, x3 − x2 = 0 andx2 − x1 = 0, as these differences appear as factors in v3(x3).

We will see later on that the extreme points are the six points acquiredfrom permuting the coordinates in

x3 =1√2

(−1, 0, 1) .

For reasons that will become clear in Section 2.2.1 it is also useful to thinkabout these coordinates as the roots of the polynomial

P3(x) = x3 − 1

2x.

So far we have only considered the behavior of v3(x3), that is g3(x3,a3)with a3 = (0, 1, 2). We now consider three generalized Vandermonde de-terminants, namely g3 with a3 = (0, 1, 3), a3 = (0, 2, 3) and a3 = (1, 2, 3).These three determinants show increasingly more structure and they all havea neat formula in terms of v3 and the elementary symmetric polynomials

ekn = ek(x1, · · · , xn) =∑

1≤i1<i2<···<ik≤nxi1xi2 · · ·xik ,

78

80




(c) Plot with respect toangles given in (34).

Figure 2.2: Plot of g3(x3, (0, 1, 3)) over the unit sphere.

where we will simply use ek whenever n is clear from the context.

In Figure 2.2 we see the determinant

g3(x3, (0, 1, 3)) =

∣∣∣∣∣∣1 1 1x1 x2 x3

x31 x3

2 x33

∣∣∣∣∣∣ = v3(x3)e1,

plotted over the unit sphere. The expression v3(x3)e1 is easy to derive, thev3(x3) is there since the determinant must vanish whenever any two columnsare equal, which is exactly what the Vandermonde determinant expresses.The e1 follows by a simple polynomial division. As can be seen in the plotswe have an extra black circle where the determinant vanishes compared toFigure 2.1. This circle lies in the plane e1 = x1 + x2 + x3 = 0 where wepreviously found the extreme points of v3(x3) and thus doubles the numberof extreme points to 2 · 3!.

A similar treatment can be made of the remaining two generalized de-terminants that we are interested in, plotted in the following two figures.





79






where we will simply use ek whenever n is clear from the context.

In Figure 2.2 we see the determinant

g3(x3, (0, 1, 3)) =

∣∣∣∣∣∣1 1 1x1 x2 x3

x31 x3

2 x33

∣∣∣∣∣∣ = v3(x3)e1,

plotted over the unit sphere. The expression v3(x3)e1 is easy to derive, thev3(x3) is there since the determinant must vanish whenever any two columnsare equal, which is exactly what the Vandermonde determinant expresses.The e1 follows by a simple polynomial division. As can be seen in the plotswe have an extra black circle where the determinant vanishes compared toFigure 2.1. This circle lies in the plane e1 = x1 + x2 + x3 = 0 where wepreviously found the extreme points of v3(x3) and thus doubles the numberof extreme points to 2 · 3!.

A similar treatment can be made of the remaining two generalized de-terminants that we are interested in, plotted in the following two figures.





79

81






a3 g3(x3,a3)

(0, 1, 2) v3(x3)e0 = (x3 − x2)(x3 − x1)(x2 − x1)(0, 1, 3) v3(x3)e1 = (x3 − x2)(x3 − x1)(x2 − x1)(x1 + x2 + x3)(0, 2, 3) v3(x3)e2 = (x3 − x2)(x3 − x1)(x2 − x1)(x1x2 + x1x3 + x2x3)(1, 2, 3) v3(x3)e3 = (x3 − x2)(x3 − x1)(x2 − x1)x1x2x3

Table 2.1: Table of some determinants of generalized Vandermonde matrices.

The four determinants treated so far are collected in Table 2.1. Deriva-tion of these determinants is straight forward. We note that all but one ofthem vanish on a set of planes through the origin. For a = (0, 2, 3) we havethe usual Vandermonde planes but the intersection of e2 = 0 and the unitsphere occur at two circles.

x1x2 + x1x3 + x2x3 =1

2

((x1 + x2 + x3)2 − (x2

1 + x22 + x2

3))

=1

2

((x1 + x2 + x3)2 − 1

)=

1

2(x1 + x2 + x3 + 1) (x1 + x2 + x3 − 1) ,

and so g3(x3, (0, 2, 3)) vanish on the sphere on two circles lying on the planesx1 + x2 + x3 + 1 = 0 and x1 + x2 + x3 − 1 = 0. These circles can be seen inFigure 2.3 as the two black circles perpendicular to the direction (1, 1, 1).

Note also that while v3 and g3(x3, (0, 1, 3)) have the same absolute valueon all their respective local extreme points (by symmetry) we have that bothg3(x3, (0, 2, 3)) and g3(x3, (1, 2, 3)) have different absolute values for some oftheir respective extreme points.

80






a3 g3(x3,a3)

(0, 1, 2) v3(x3)e0 = (x3 − x2)(x3 − x1)(x2 − x1)(0, 1, 3) v3(x3)e1 = (x3 − x2)(x3 − x1)(x2 − x1)(x1 + x2 + x3)(0, 2, 3) v3(x3)e2 = (x3 − x2)(x3 − x1)(x2 − x1)(x1x2 + x1x3 + x2x3)(1, 2, 3) v3(x3)e3 = (x3 − x2)(x3 − x1)(x2 − x1)x1x2x3

Table 2.1: Table of some determinants of generalized Vandermonde matrices.

The four determinants treated so far are collected in Table 2.1. Deriva-tion of these determinants is straight forward. We note that all but one ofthem vanish on a set of planes through the origin. For a = (0, 2, 3) we havethe usual Vandermonde planes but the intersection of e2 = 0 and the unitsphere occur at two circles.

x1x2 + x1x3 + x2x3 =1

2

((x1 + x2 + x3)2 − (x2

1 + x22 + x2

3))

=1

2

((x1 + x2 + x3)2 − 1

)=

1

2(x1 + x2 + x3 + 1) (x1 + x2 + x3 − 1) ,

and so g3(x3, (0, 2, 3)) vanish on the sphere on two circles lying on the planesx1 + x2 + x3 + 1 = 0 and x1 + x2 + x3 − 1 = 0. These circles can be seen inFigure 2.3 as the two black circles perpendicular to the direction (1, 1, 1).

Note also that while v3 and g3(x3, (0, 1, 3)) have the same absolute valueon all their respective local extreme points (by symmetry) we have that bothg3(x3, (0, 2, 3)) and g3(x3, (1, 2, 3)) have different absolute values for some oftheir respective extreme points.

80

82


2.1.2 Extreme points of the Vandermonde determinant onthe three-dimensional unit sphere


It is fairly simple to describe v3(x3) on the circle that is formed by theintersection of the unit sphere and the plane x1 + x2 + x3 = 0. UsingRodrigues’ rotation formula to rotate a point, x, around the axis 1√

3(1, 1, 1)

with the angle θ will give the rotation matrix

Rθ =1

3

2 cos(θ) + 1 1− cos(θ)−√

3 sin(θ) 1− cos(θ)+√

3 sin(θ)

1− cos(θ)+√

3 sin(θ) 2 cos(θ) + 1 1− cos(θ)−√

3 sin(θ)

1− cos(θ)−√


3 sin(θ) 2 cos(θ) + 1

.A point which already lies on S2 can then be rotated to any other point

on S2 by letting Rθ act on the point. Choosing the point x = 1√2

(−1, 0, 1)

gives the Vandermonde determinant a convenient form on the circle since:

Rθx =1√6

−√3 cos(θ)− sin(θ)−2 sin(θ)√

3 cos(θ) + sin(θ)

,which gives

2v3(Rθx) = 2(√

3 cos(θ) + sin(θ))

(√3 cos(θ) + sin(θ) + 2 sin(θ)

)(−2 sin(θ) +

√3 cos(θ) + sin(θ)

)=

1√2

(4 cos(θ)3 − 3 cos(θ)

)=

1√2

cos(3θ).

Note that the final equality follows from cos(nθ) = Tn(cos(θ)) where Tn isthe nth Chebyshev polynomial of the first kind. From formula (55) if followsthat P3(x) = T3(x) but for higher dimensions the relationship between theChebyshev polynomials and Pn is not as simple.Finding the maximum points for v3(x3) on this form is simple. The Van-dermonde determinant will be maximal when 3θ = 2nπ where n is someinteger. This gives three local maxima corresponding to θ1 = 0, θ2 = 2π

3and θ3 = 4π

3 . These points correspond to cyclic permutation of the coordi-nates of x = 1√

2(−1, 0, 1). Analogously the minimas for v3(x3) can be shown

to be a transposition followed by cyclic permutation of the coordinates of x.Thus any permutation of the coordinates of x correspond to a local extremepoint just like it was stated on page 78.

81


2.1.2 Extreme points of the Vandermonde determinant onthe three-dimensional unit sphere


It is fairly simple to describe v3(x3) on the circle that is formed by theintersection of the unit sphere and the plane x1 + x2 + x3 = 0. UsingRodrigues’ rotation formula to rotate a point, x, around the axis 1√

3(1, 1, 1)

with the angle θ will give the rotation matrix

Rθ =1

3

2 cos(θ) + 1 1− cos(θ)−√


3 sin(θ)

1− cos(θ)+√

3 sin(θ) 2 cos(θ) + 1 1− cos(θ)−√

3 sin(θ)

1− cos(θ)−√


3 sin(θ) 2 cos(θ) + 1

.A point which already lies on S2 can then be rotated to any other point

on S2 by letting Rθ act on the point. Choosing the point x = 1√2

(−1, 0, 1)

gives the Vandermonde determinant a convenient form on the circle since:

Rθx =1√6

−√3 cos(θ)− sin(θ)−2 sin(θ)√

3 cos(θ) + sin(θ)

,which gives

2v3(Rθx) = 2(√

3 cos(θ) + sin(θ))

(√3 cos(θ) + sin(θ) + 2 sin(θ)

)(−2 sin(θ) +

√3 cos(θ) + sin(θ)

)=

1√2

(4 cos(θ)3 − 3 cos(θ)

)=

1√2

cos(3θ).

Note that the final equality follows from cos(nθ) = Tn(cos(θ)) where Tn isthe nth Chebyshev polynomial of the first kind. From formula (55) if followsthat P3(x) = T3(x) but for higher dimensions the relationship between theChebyshev polynomials and Pn is not as simple.Finding the maximum points for v3(x3) on this form is simple. The Van-dermonde determinant will be maximal when 3θ = 2nπ where n is someinteger. This gives three local maxima corresponding to θ1 = 0, θ2 = 2π

3and θ3 = 4π

3 . These points correspond to cyclic permutation of the coordi-nates of x = 1√

2(−1, 0, 1). Analogously the minimas for v3(x3) can be shown

to be a transposition followed by cyclic permutation of the coordinates of x.Thus any permutation of the coordinates of x correspond to a local extremepoint just like it was stated on page 78.

81

83


2.1.3 Optimisation using Grobner bases

This section is based on Section 4 of Paper B

In this section we will find the extreme points of the Vandermonde de-terminant on a few different surfaces. This will be done using Lagrangemultipliers and Grobner bases but first we will make an observation aboutthe Vandermonde determinant that will be useful later.

Lemma 2.1. The Vandermonde determinant is a homogeneous polynomialof degree n(n−1)

2 .

Proof. Considering the expression for the Vandermonde determinant in The-

orem 1.2 the number of factor of vn(x) isn∑i=1

i− 1 =n(n− 1)

2. Thus

vn(cx) = cn(n−1)

2 vn(x). (35)

Grobner bases together with algorithms to find them, and algorithmsfor solving a polynomial equation is an important tool that arises in manyapplications. One such application is the optimization of polynomials overaffine varieties through the method of Lagrange multipliers. We will heregive some main points and informal discussion on these methods as an in-troduction and describe some notation.

Definition 2.1. ([60]) Let f1, · · · , fm be polynomials in R[x1, · · · , xn]. Theaffine variety V (f1, · · · , fm) defined by f1, · · · , fm is the set of all points(x1, · · · , xn) ∈ Rn such that fi(x1, · · · , xn) = 0 for all 1 ≤ i ≤ m.

When n = 3 we will sometimes use the variables x, y, z instead ofx1, x2, x3. Affine varieties are this way the common zeros of a set of multi-variate polynomials. Such sets of polynomials will generate a greater set ofpolynomials [60] by

〈f1, · · · , fm〉 ≡

m∑i=1

hifi : h1, · · · , hm ∈ R[x1, · · · , xn]

,

and this larger set will define the same variety. But it will also define anideal (a set of polynomials that contains the zero-polynomial and is closedunder addition, and absorbs multiplication by any other polynomial) byI(f1, · · · , fm) = 〈f1, · · · , fm〉. A Grobner basis for this ideal is then a finiteset of polynomials g1, · · · , gk such that the ideal generated by the leadingterms of the polynomials g1, · · · , gk is the same ideal as that generated byall the leading terms of polynomials in I = 〈f1, · · · , fm〉.

82


2.1.3 Optimisation using Grobner bases


In this section we will find the extreme points of the Vandermonde de-terminant on a few different surfaces. This will be done using Lagrangemultipliers and Grobner bases but first we will make an observation aboutthe Vandermonde determinant that will be useful later.

Lemma 2.1. The Vandermonde determinant is a homogeneous polynomialof degree n(n−1)

2 .

Proof. Considering the expression for the Vandermonde determinant in The-

orem 1.2 the number of factor of vn(x) isn∑i=1

i− 1 =n(n− 1)

2. Thus

vn(cx) = cn(n−1)

2 vn(x). (35)

Grobner bases together with algorithms to find them, and algorithmsfor solving a polynomial equation is an important tool that arises in manyapplications. One such application is the optimization of polynomials overaffine varieties through the method of Lagrange multipliers. We will heregive some main points and informal discussion on these methods as an in-troduction and describe some notation.

Definition 2.1. ([60]) Let f1, · · · , fm be polynomials in R[x1, · · · , xn]. Theaffine variety V (f1, · · · , fm) defined by f1, · · · , fm is the set of all points(x1, · · · , xn) ∈ Rn such that fi(x1, · · · , xn) = 0 for all 1 ≤ i ≤ m.

When n = 3 we will sometimes use the variables x, y, z instead ofx1, x2, x3. Affine varieties are this way the common zeros of a set of multi-variate polynomials. Such sets of polynomials will generate a greater set ofpolynomials [60] by

〈f1, · · · , fm〉 ≡

m∑i=1

hifi : h1, · · · , hm ∈ R[x1, · · · , xn]

,

and this larger set will define the same variety. But it will also define anideal (a set of polynomials that contains the zero-polynomial and is closedunder addition, and absorbs multiplication by any other polynomial) byI(f1, · · · , fm) = 〈f1, · · · , fm〉. A Grobner basis for this ideal is then a finiteset of polynomials g1, · · · , gk such that the ideal generated by the leadingterms of the polynomials g1, · · · , gk is the same ideal as that generated byall the leading terms of polynomials in I = 〈f1, · · · , fm〉.

82

84


In this paper we consider the optimization of the Vandermonde deter-minant vn(x) over surfaces defined by a polynomial equation on the form

sn(x1, · · · , xn ; p; a1, · · · , an) ≡n∑i=1

ai|xi|p = 1, (36)

where we will select the constants ai and p to get ellipsoids in three di-mensions, cylinders in three dimensions, and spheres under the p-norm inn dimensions. The cases of the ellipsoids and the cylinders are suitable forsolution by Grobner basis methods, but due to the existing symmetries forthe spheres other methods are more suitable, as provided in Section 2.3.3.

From (35) and the convexity of the interior of the sets defined by (36),under a suitable choice of the constant p and non-negative ai, it is easy

to see that the optimal value of vn onn∑i=1

ai|xi|p ≤ 1 will be attained on

n∑i=1

ai|xi|p = 1. And so, by the method of Lagrange multipliers we have that

the minimal/maximal values of vn(x1, · · · , xn) on sn(x1, · · · , xn) ≤ 1 will beattained at points such that ∂vn

∂xi−λ∂sn∂xi

= 0 for 1 ≤ i ≤ n and some constantλ and sn(x1, · · · , xn)− 1 = 0, [243].

For p = 2 the resulting set of equations will form a set of polynomials inλ, x1, · · · , xn. These polynomials will define an ideal over R[λ, x1, · · · , xn],and by finding a Grobner basis for this ideal we can use the especially niceproperties of Grobner bases to find analytical solutions to these problems,that is, to find roots for the polynomials in the computed basis.

2.1.4 Extreme points on the ellipsoid in three dimensions


In this section we will find the extreme points of the Vandermonde determi-nant on the three dimensional ellipsoid given by

ax2 + by2 + cz2 = 1 (37)

where a > 0, b > 0, c > 0.

Using the method of Lagrange multipliers together with (37) and somerewriting gives that all stationary points of the Vandermonde determinantlie in the variety

V = V(ax2 + by2 + cz2 − 1, ax+ by + cz,

ax(z − x)(y − x)− by(z − y)(y − x) + cz(z − y)(z − x)).

83


In this paper we consider the optimization of the Vandermonde deter-minant vn(x) over surfaces defined by a polynomial equation on the form

sn(x1, · · · , xn ; p; a1, · · · , an) ≡n∑i=1

ai|xi|p = 1, (36)

where we will select the constants ai and p to get ellipsoids in three di-mensions, cylinders in three dimensions, and spheres under the p-norm inn dimensions. The cases of the ellipsoids and the cylinders are suitable forsolution by Grobner basis methods, but due to the existing symmetries forthe spheres other methods are more suitable, as provided in Section 2.3.3.

From (35) and the convexity of the interior of the sets defined by (36),under a suitable choice of the constant p and non-negative ai, it is easy

to see that the optimal value of vn onn∑i=1

ai|xi|p ≤ 1 will be attained on

n∑i=1

ai|xi|p = 1. And so, by the method of Lagrange multipliers we have that

the minimal/maximal values of vn(x1, · · · , xn) on sn(x1, · · · , xn) ≤ 1 will beattained at points such that ∂vn

∂xi−λ∂sn∂xi

= 0 for 1 ≤ i ≤ n and some constantλ and sn(x1, · · · , xn)− 1 = 0, [243].

For p = 2 the resulting set of equations will form a set of polynomials inλ, x1, · · · , xn. These polynomials will define an ideal over R[λ, x1, · · · , xn],and by finding a Grobner basis for this ideal we can use the especially niceproperties of Grobner bases to find analytical solutions to these problems,that is, to find roots for the polynomials in the computed basis.

2.1.4 Extreme points on the ellipsoid in three dimensions


In this section we will find the extreme points of the Vandermonde determi-nant on the three dimensional ellipsoid given by

ax2 + by2 + cz2 = 1 (37)

where a > 0, b > 0, c > 0.

Using the method of Lagrange multipliers together with (37) and somerewriting gives that all stationary points of the Vandermonde determinantlie in the variety

V = V(ax2 + by2 + cz2 − 1, ax+ by + cz,

ax(z − x)(y − x)− by(z − y)(y − x) + cz(z − y)(z − x)).

83

85


Computing a Grobner basis for V using the lexicographic order x > y > zgive the following three basis polynomials:

g1(z) =(a+ b)(a− b)2

−(4(a+ b)2(a+ c)(b+ c) + 3c2(a2 + ab+ b2) + 3c(a3 + b3)

)z2

+ 3c(a+ b+ c)(4(a+ b)(a+ c)(b+ c) + (a2 + b2)c+ (a+ b)c2

)z4

− c2(b+ c)(a+ c)(a+ b+ c)2z6, (38)

g2(y, z) =(2(a+ b)2(a+ c)(b+ c) + c(a2 + 2b2)(a+ b+ c) + 2bc2(a+ b)

)z

+ q1z5 − q2z

3 − b(a− b)(a+ b)(a+ b+ 3c)y, (39)

g3(x, z) =(2(a+ b)2(a+ c)(b+ c) + c(2a2 + b2)(a+ b+ c) + 2ac2(a+ b)

)z

− q1z5 + q2z

3 − a(a− b)(a+ b)(a+ b+ 3c)x, (40)

q1 = 9 c2(b+ c)(a+ c)(a+ b+ c)2,

q2 = 3c(a+ b+ c)(3a2b+ 4a2c+ 3ab2 + 6abc+ 4ac2 + 4b2c+ 4bc2).

This basis was calculated using software for symbolic computation [200].Since g1 only depends on z, and g2 and g3 are first degree polynomials in

y and x respectively, the stationary points can be found by finding the rootsof g1 and then calculate the corresponding x and y coordinates. A generalformula can be found in this case (since g1 only contains even powers of zit can be treated as a third degree polynomial) but it is quite cumbersomeand we will therefore not give it explicitly.

Lemma 2.2. The extreme points of v3 on an ellipsoid will have real coor-dinates.

Proof. The discriminant is a useful tool for determining how many real rootslow-level polynomials have. Following Irving [135] the discriminant, ∆(p),of a third degree polynomial p(x) = c0 + c1x+ c2x

2 + c3x3 is

∆ = 18c1c2c3c4 − 4c32c4 + c2

2c23 − 4c1c

33 − 27c2

1c24

and if p(x) is non-negative then all roots will be real (but not necessarilydistinct). Since the first basis polynomial g1 only contains terms with evenexponents and is of degree 6 the polynomial g1 defined by g1(z2) = g1(z)will be a polynomial of degree 3 whose roots are the square roots of g1.Calculating the discriminant of g1 gives

∆(g1) = 9(a− b)2(a+ b+ 3c)2(a+ b+ c)4abc3(32(a3b2 + a3c2 + a2b3 + a2c3 + b3c2 + b2c3) + 61abc(a+ b+ c)2

).

Since a, b and c are all positive numbers it is clear that ∆(g1) is non-negative. Furthermore, since a, b and c are positive numbers all terms in g1

with odd powers have negative coefficients and all terms with even powershave positive coefficients. Thus if w < 0 then g1(w) > 0 and thus all rootsmust be positive.

84


Computing a Grobner basis for V using the lexicographic order x > y > zgive the following three basis polynomials:

g1(z) =(a+ b)(a− b)2

−(4(a+ b)2(a+ c)(b+ c) + 3c2(a2 + ab+ b2) + 3c(a3 + b3)

)z2

+ 3c(a+ b+ c)(4(a+ b)(a+ c)(b+ c) + (a2 + b2)c+ (a+ b)c2

)z4

− c2(b+ c)(a+ c)(a+ b+ c)2z6, (38)

g2(y, z) =(2(a+ b)2(a+ c)(b+ c) + c(a2 + 2b2)(a+ b+ c) + 2bc2(a+ b)

)z

+ q1z5 − q2z

3 − b(a− b)(a+ b)(a+ b+ 3c)y, (39)

g3(x, z) =(2(a+ b)2(a+ c)(b+ c) + c(2a2 + b2)(a+ b+ c) + 2ac2(a+ b)

)z

− q1z5 + q2z

3 − a(a− b)(a+ b)(a+ b+ 3c)x, (40)

q1 = 9 c2(b+ c)(a+ c)(a+ b+ c)2,

q2 = 3c(a+ b+ c)(3a2b+ 4a2c+ 3ab2 + 6abc+ 4ac2 + 4b2c+ 4bc2).

This basis was calculated using software for symbolic computation [200].Since g1 only depends on z, and g2 and g3 are first degree polynomials in

y and x respectively, the stationary points can be found by finding the rootsof g1 and then calculate the corresponding x and y coordinates. A generalformula can be found in this case (since g1 only contains even powers of zit can be treated as a third degree polynomial) but it is quite cumbersomeand we will therefore not give it explicitly.

Lemma 2.2. The extreme points of v3 on an ellipsoid will have real coor-dinates.

Proof. The discriminant is a useful tool for determining how many real rootslow-level polynomials have. Following Irving [135] the discriminant, ∆(p),of a third degree polynomial p(x) = c0 + c1x+ c2x

2 + c3x3 is

∆ = 18c1c2c3c4 − 4c32c4 + c2

2c23 − 4c1c

33 − 27c2

1c24

and if p(x) is non-negative then all roots will be real (but not necessarilydistinct). Since the first basis polynomial g1 only contains terms with evenexponents and is of degree 6 the polynomial g1 defined by g1(z2) = g1(z)will be a polynomial of degree 3 whose roots are the square roots of g1.Calculating the discriminant of g1 gives

∆(g1) = 9(a− b)2(a+ b+ 3c)2(a+ b+ c)4abc3(32(a3b2 + a3c2 + a2b3 + a2c3 + b3c2 + b2c3) + 61abc(a+ b+ c)2

).

Since a, b and c are all positive numbers it is clear that ∆(g1) is non-negative. Furthermore, since a, b and c are positive numbers all terms in g1

with odd powers have negative coefficients and all terms with even powershave positive coefficients. Thus if w < 0 then g1(w) > 0 and thus all rootsmust be positive.

84

86


Figure 2.5: Illustration of the ellipsoid defined byx2

9+y2

4+z2 = 0 with the ex-

treme points of the Vandermonde determinant marked. Displayedin Cartesian coordinates on the right and in ellipsoidal coordinateson the left.

An illustration of an ellipsoid and the extreme points of the Vandermondedeterminant on its surface is shown in Figure 2.5.

2.1.5 Extreme points on the cylinder in three dimensions


In this section we will examine the local extreme points on an infinitely longcylinder aligned with the x-axis in 3 dimensions. In this case we do not needto use Grobner basis techniques since the problem can be reduced to a onedimensional polynomial equation.

The cylinder is defined by

by2 + cz2 = 1, where b > 0, c > 0. (41)

Using the method of Lagrange multipliers gives the equation system

∂v3

∂x= 0,

∂v3

∂y= 2λby,

∂v3

∂z= 2λcz.

Taking the sum of each expression gives

by + cz = 0⇔ y = −cbz. (42)

Combining (41) and (42) gives(cb

+ 1)cz2 = 1⇒ z = ±

√b

c

1√b+ c

⇒ y = ∓√c

b

1√b+ c

.

85


Figure 2.5: Illustration of the ellipsoid defined byx2

9+y2

4+z2 = 0 with the ex-

treme points of the Vandermonde determinant marked. Displayedin Cartesian coordinates on the right and in ellipsoidal coordinateson the left.

An illustration of an ellipsoid and the extreme points of the Vandermondedeterminant on its surface is shown in Figure 2.5.

2.1.5 Extreme points on the cylinder in three dimensions


In this section we will examine the local extreme points on an infinitely longcylinder aligned with the x-axis in 3 dimensions. In this case we do not needto use Grobner basis techniques since the problem can be reduced to a onedimensional polynomial equation.

The cylinder is defined by

by2 + cz2 = 1, where b > 0, c > 0. (41)

Using the method of Lagrange multipliers gives the equation system

∂v3

∂x= 0,

∂v3

∂y= 2λby,

∂v3

∂z= 2λcz.

Taking the sum of each expression gives

by + cz = 0⇔ y = −cbz. (42)

Combining (41) and (42) gives(cb

+ 1)cz2 = 1⇒ z = ±

√b

c

1√b+ c

⇒ y = ∓√c

b

1√b+ c

.

85

87


Figure 2.6: Illustration of the cylinder defined by y2 +16

25z2 = 1 with the ex-

treme points of the Vandermonde determinant marked. Displayedin Cartesian coordinates on the right and in cylindrical coordinateson the left.

Thus the plane defined by (42) intersects with the cylinder along thelines

`1 =

(x,

√c

b

1√b+ c

,−√b

c

1√b+ c

)∣∣∣∣x ∈ R

= (x, r,−s)|x ∈ R ,

`2 =

(x,−

√c

b

1√b+ c

,

√b

c

1√b+ c

)∣∣∣∣x ∈ R

= (x,−r, s)|x ∈ R .

Finding the stationary points for v3 along `1:

v3 (x, r,−s) =

(x2 +

1√b+ c

(√b

c−√c

b

)x+

1

b+ c

)(r + s) ,

∂v3

∂x(x, r,−s) =

(2x+

1√b+ c

(√b

c−√c

b

))(r + s) .

From this it follows that

∂v3

∂x(x, r,−s) = 0⇔ x =

1

2√b+ c

(√c

b−√b

c

).

Thus

x1 =1√b+ c

(1

2

(√c

b−√b

c

),

√c

b,−√b

c

)(43)

is the only stationary point on `1. It can similarly be shown that x2 = −x1

is the only stationary point on `2.

The location of these points on the cylinder are shown in Figure 2.6.

86


Figure 2.6: Illustration of the cylinder defined by y2 +16

25z2 = 1 with the ex-

treme points of the Vandermonde determinant marked. Displayedin Cartesian coordinates on the right and in cylindrical coordinateson the left.

Thus the plane defined by (42) intersects with the cylinder along thelines

`1 =

(x,

√c

b

1√b+ c

,−√b

c

1√b+ c

)∣∣∣∣x ∈ R

= (x, r,−s)|x ∈ R ,

`2 =

(x,−

√c

b

1√b+ c

,

√b

c

1√b+ c

)∣∣∣∣x ∈ R

= (x,−r, s)|x ∈ R .

Finding the stationary points for v3 along `1:

v3 (x, r,−s) =

(x2 +

1√b+ c

(√b

c−√c

b

)x+

1

b+ c

)(r + s) ,

∂v3

∂x(x, r,−s) =

(2x+

1√b+ c

(√b

c−√c

b

))(r + s) .

From this it follows that

∂v3

∂x(x, r,−s) = 0⇔ x =

1

2√b+ c

(√c

b−√b

c

).

Thus

x1 =1√b+ c

(1

2

(√c

b−√b

c

),

√c

b,−√b

c

)(43)

is the only stationary point on `1. It can similarly be shown that x2 = −x1

is the only stationary point on `2.

The location of these points on the cylinder are shown in Figure 2.6.

86

88


2.1.6 Optimizing the Vandermonde determinant on a surfacedefined by a homogeneous polynomial


When using Lagrange multipliers it can be desirable to not have to considerthe λ-parameter (the scaling between the gradient and direction given bythe constraint). We demonstrate a simple way to remove this parameterwhen the surface is defined by an homogeneous polynomial.

Lemma 2.3. Let g : R → R be a homogeneous polynomial such thatg(cx) = ckg(x) with k 6= n(n−1)

2 . If g(x) = 1, x ∈ Cn defines a continu-ous bounded surface then any point on the surface that is a stationary pointfor the Vandermonde determinant, z ∈ Cn, can be written as z = cy where

∂vn∂xi

∣∣∣∣x=y

=∂g

∂xi

∣∣∣∣x=y

, i ∈ 1, 2, . . . , n (44)

and c = g(y)−1k .

Proof. By the method of Lagrange multipliers the point y ∈ x ∈ Rn|g(x) =1 is a stationary point for the Vandermonde determinant if

∂vn∂xk

∣∣∣∣x=y

= λ∂g

∂xk

∣∣∣∣x=y

, k ∈ 1, 2, . . . , n

for some λ ∈ R.

The stationary points on the surface given by g(cx) = ck are given by

cn(n−1)

2∂vn∂xk

∣∣∣∣x=y

= ckλ∂g

∂xk

∣∣∣∣x=y

, k ∈ 1, 2, . . . , n

and if c is chosen such that λ = cn(1−n)

2 ck then the stationary points aredefined by

∂vn∂xk

=∂g

∂xk, k ∈ 1, 2, . . . , n.

Suppose that y ∈ x ∈ Rn|g(x) = ck is a stationary point for vn then

the point given by z = cy where c = g(y)−1k will be a stationary point

for the Vandermonde determinant and will lie on the surface defined byg(x) = 1.

Lemma 2.4. If z is a stationary point for the Vandermonde determinanton the surface g(x) = 1 where g(x) is a homogeneous polynomial then −z iseither a stationary point or does not lie on the surface.

87


2.1.6 Optimizing the Vandermonde determinant on a surfacedefined by a homogeneous polynomial


When using Lagrange multipliers it can be desirable to not have to considerthe λ-parameter (the scaling between the gradient and direction given bythe constraint). We demonstrate a simple way to remove this parameterwhen the surface is defined by an homogeneous polynomial.

Lemma 2.3. Let g : R → R be a homogeneous polynomial such thatg(cx) = ckg(x) with k 6= n(n−1)

2 . If g(x) = 1, x ∈ Cn defines a continu-ous bounded surface then any point on the surface that is a stationary pointfor the Vandermonde determinant, z ∈ Cn, can be written as z = cy where

∂vn∂xi

∣∣∣∣x=y

=∂g

∂xi

∣∣∣∣x=y

, i ∈ 1, 2, . . . , n (44)

and c = g(y)−1k .

Proof. By the method of Lagrange multipliers the point y ∈ x ∈ Rn|g(x) =1 is a stationary point for the Vandermonde determinant if

∂vn∂xk

∣∣∣∣x=y

= λ∂g

∂xk

∣∣∣∣x=y

, k ∈ 1, 2, . . . , n

for some λ ∈ R.

The stationary points on the surface given by g(cx) = ck are given by

cn(n−1)

2∂vn∂xk

∣∣∣∣x=y

= ckλ∂g

∂xk

∣∣∣∣x=y

, k ∈ 1, 2, . . . , n

and if c is chosen such that λ = cn(1−n)

2 ck then the stationary points aredefined by

∂vn∂xk

=∂g

∂xk, k ∈ 1, 2, . . . , n.

Suppose that y ∈ x ∈ Rn|g(x) = ck is a stationary point for vn then

the point given by z = cy where c = g(y)−1k will be a stationary point

for the Vandermonde determinant and will lie on the surface defined byg(x) = 1.

Lemma 2.4. If z is a stationary point for the Vandermonde determinanton the surface g(x) = 1 where g(x) is a homogeneous polynomial then −z iseither a stationary point or does not lie on the surface.

87

89


Proof. Since g(−x) = (−1)kg(x) is either 1 or −1 then |vn(x)| = |vn(−x)|for any point, including z and the points in a neighbourhood around it whichmeans that if g(−x) = g(x) then the stationary points are preserved andotherwise the point will lie on the surface defined by g(x) = −1 instead ofg(x) = 1.

A well-known example of homogeneous polynomials are quadratic forms.If we let

g(x) = x>aSx

then g(x) is a quadratic form which in turn is a homogeneous polynomialwith k = 2. If S is a positive definite matrix then g(x) = 1 defines anellipsoid. Here we will demonstrate the use of Lemma 2.3 to find the extremepoints on a rotated ellipsoid.

Consider the ellipsoid defined by

1

9x2 +

5

8y2 +

3

4yz +

5

8z2 = 1 (45)

then by Lemma 2.3 we can instead consider the points in the variety

V = V(− 2xy + 2xz + y2 − z2 − 2

9x,

− x2 + 2xy − 2yz + z2 − 5

4y − 3

4z,

− 2xz − y2 + 2yz + x2 − 3

4y − 5

4z).

Finding the Grobner basis of V gives

g1(z) = z(6z + 1)(260642z2 − 27436z + 697),

g2(y, z) = − 1138484256z3 − 127275604z2 + 16689841z + 6277879y,

g3(x, z) = 10246358304z3 + 1145480436z2 − 93707658z + 6277879x.

This system is not difficult to solve and the resulting points are:

p0 = (0, 0, 0),

p1 =

(0,

1

6,−1

6

),

p2 =

(45√

2

361,− 1

19− 5√

2

722,

1

19− 5√

2

722

),

p3 =

(45√

2

361,− 1

19+

5√

2

722,

1

19+

5√

2

722

).

The point p0 is an artifact of the rewrite and does not lie on any ellipsoidand can therefore be discarded. By Lemma 2.4 there are also three more

88


Proof. Since g(−x) = (−1)kg(x) is either 1 or −1 then |vn(x)| = |vn(−x)|for any point, including z and the points in a neighbourhood around it whichmeans that if g(−x) = g(x) then the stationary points are preserved andotherwise the point will lie on the surface defined by g(x) = −1 instead ofg(x) = 1.

A well-known example of homogeneous polynomials are quadratic forms.If we let

g(x) = x>aSx

then g(x) is a quadratic form which in turn is a homogeneous polynomialwith k = 2. If S is a positive definite matrix then g(x) = 1 defines anellipsoid. Here we will demonstrate the use of Lemma 2.3 to find the extremepoints on a rotated ellipsoid.

Consider the ellipsoid defined by

1

9x2 +

5

8y2 +

3

4yz +

5

8z2 = 1 (45)

then by Lemma 2.3 we can instead consider the points in the variety

V = V(− 2xy + 2xz + y2 − z2 − 2

9x,

− x2 + 2xy − 2yz + z2 − 5

4y − 3

4z,

− 2xz − y2 + 2yz + x2 − 3

4y − 5

4z).

Finding the Grobner basis of V gives

g1(z) = z(6z + 1)(260642z2 − 27436z + 697),

g2(y, z) = − 1138484256z3 − 127275604z2 + 16689841z + 6277879y,

g3(x, z) = 10246358304z3 + 1145480436z2 − 93707658z + 6277879x.

This system is not difficult to solve and the resulting points are:

p0 = (0, 0, 0),

p1 =

(0,

1

6,−1

6

),

p2 =

(45√

2

361,− 1

19− 5√

2

722,

1

19− 5√

2

722

),

p3 =

(45√

2

361,− 1

19+

5√

2

722,

1

19+

5√

2

722

).

The point p0 is an artifact of the rewrite and does not lie on any ellipsoidand can therefore be discarded. By Lemma 2.4 there are also three more

88

90

2.2. EXTREME POINTS OF THE VANDERMONDEDETERMINANT ON THE SPHERE

Figure 2.7: Illustration of the ellipsoid defined by (45) with the extreme pointsof the Vandermonde determinant marked. Displayed in Cartesiancoordinates on the right and in ellipsoidal coordinates on the left.

stationary points p4 = −p1, p5 = −p2 and p6 = −p3. Rescaling each ofthese points according to Lemma 2.3 gives qi =

√g(pi) which are all points

on the ellipsoid defined by g(x) = 1. The result is illustrated in Figure 2.7.

Note that this example gives a simple case with a Grobner basis that issmall and easy to find. Using this technique for other polynomials and inhigher dimensions can require significant computational resources.

2.2 Extreme points of the Vandermonde determi-nant on the sphere

In this section we will consider the extreme points of the Vandermondedeterminant on the n-dimensional unit sphere in Rn. We want both to findan analytical solution and to identify some properties of the determinantthat can help us to visualize it in some area around the extreme points indimensions n > 3.

2.2.1 The extreme points on the sphere given by roots of apolynomial


The extreme points of the Vandermonde determinant on the unit sphere inRn are known and given by Theorem 2.3 where we present a special case ofTheorem 6.7.3 in [269]. We will also provide a proof that is more explicitthan the one in [269] and that exposes more of the rich symmetric prop-erties of the Vandermonde determinant. For the sake of convenience someproperties related to the extreme points of the Vandermonde determinantdefined by real vectors xn will be presented before Theorem 2.3.

89


Figure 2.7: Illustration of the ellipsoid defined by (45) with the extreme pointsof the Vandermonde determinant marked. Displayed in Cartesiancoordinates on the right and in ellipsoidal coordinates on the left.

stationary points p4 = −p1, p5 = −p2 and p6 = −p3. Rescaling each ofthese points according to Lemma 2.3 gives qi =

√g(pi) which are all points

on the ellipsoid defined by g(x) = 1. The result is illustrated in Figure 2.7.

Note that this example gives a simple case with a Grobner basis that issmall and easy to find. Using this technique for other polynomials and inhigher dimensions can require significant computational resources.

2.2 Extreme points of the Vandermonde determi-nant on the sphere

In this section we will consider the extreme points of the Vandermondedeterminant on the n-dimensional unit sphere in Rn. We want both to findan analytical solution and to identify some properties of the determinantthat can help us to visualize it in some area around the extreme points indimensions n > 3.

2.2.1 The extreme points on the sphere given by roots of apolynomial


The extreme points of the Vandermonde determinant on the unit sphere inRn are known and given by Theorem 2.3 where we present a special case ofTheorem 6.7.3 in [269]. We will also provide a proof that is more explicitthan the one in [269] and that exposes more of the rich symmetric prop-erties of the Vandermonde determinant. For the sake of convenience someproperties related to the extreme points of the Vandermonde determinantdefined by real vectors xn will be presented before Theorem 2.3.

89

91


Theorem 2.1. For any 1 ≤ k ≤ n

∂vn∂xk

=

n∑i=1i 6=k

vn(xn)

xk − xi. (46)

This theorem will be proven after introducing the following useful lemma:

Lemma 2.5. For any 1 ≤ k ≤ n− 1

∂vn∂xk

= − vn(xn)

xn − xk+

[n−1∏i=1

(xn − xi)

]∂vn−1

∂xk(47)

and∂vn∂xn

=n−1∑i=1

vn(xn)

xn − xi. (48)

Proof. Note that the determinant can be described recursively

vn(xn) =

[n−1∏i=1

(xn − xi)

] ∏1≤i<j≤n−1

(xj − xi)

=

[n−1∏i=1

(xn − xi)

]vn−1(xn−1). (49)

Formula (47) follows immediately from applying the differentiation formulafor products on (49). Formula (48) follows from (49), the differentiation rulefor products and that vn−1(xn−1) is independent of xn.

∂vn∂xn

=vn−1(xn−1)

xn − x1

n−1∏i=1

(xn − xi)

+ (xn − x1)∂

∂xn

(vn−1(xn−1)

xn − x1

n−1∏i=1

(xn − xi)

)

=vn(xn)

xn − x1+

vn(xn)

xn − x2

+ (xn − x1)(xn − x2)∂

∂xn

(vn(xn)

(xn − x1)(xn − x2)

)=n−1∑i=1

vn(xn)

xn − xi+

[n−1∏i=1

(xn − xi)

]∂vn−1

∂xn=

n−1∑i=1

vn(xn)

xn − xi.

Proof of Theorem 2.1. Using Lemma 2.5 we can see that when k = n (46)follows immediately from (48). The case 1 ≤ k < n will be proved usinginduction. Using (47) gives

∂vn∂xk

= − vn(xn)

xn − xk+

[n−1∏i=1

(xn − xi)

]∂vn−1

∂xk.

90


Theorem 2.1. For any 1 ≤ k ≤ n

∂vn∂xk

=

n∑i=1i 6=k

vn(xn)

xk − xi. (46)

This theorem will be proven after introducing the following useful lemma:

Lemma 2.5. For any 1 ≤ k ≤ n− 1

∂vn∂xk

= − vn(xn)

xn − xk+

[n−1∏i=1

(xn − xi)

]∂vn−1

∂xk(47)

and∂vn∂xn

=n−1∑i=1

vn(xn)

xn − xi. (48)

Proof. Note that the determinant can be described recursively

vn(xn) =

[n−1∏i=1

(xn − xi)

] ∏1≤i<j≤n−1

(xj − xi)

=

[n−1∏i=1

(xn − xi)

]vn−1(xn−1). (49)

Formula (47) follows immediately from applying the differentiation formulafor products on (49). Formula (48) follows from (49), the differentiation rulefor products and that vn−1(xn−1) is independent of xn.

∂vn∂xn

=vn−1(xn−1)

xn − x1

n−1∏i=1

(xn − xi)

+ (xn − x1)∂

∂xn

(vn−1(xn−1)

xn − x1

n−1∏i=1

(xn − xi)

)

=vn(xn)

xn − x1+

vn(xn)

xn − x2

+ (xn − x1)(xn − x2)∂

∂xn

(vn(xn)

(xn − x1)(xn − x2)

)=n−1∑i=1

vn(xn)

xn − xi+

[n−1∏i=1

(xn − xi)

]∂vn−1

∂xn=

n−1∑i=1

vn(xn)

xn − xi.

Proof of Theorem 2.1. Using Lemma 2.5 we can see that when k = n (46)follows immediately from (48). The case 1 ≤ k < n will be proved usinginduction. Using (47) gives

∂vn∂xk

= − vn(xn)

xn − xk+

[n−1∏i=1

(xn − xi)

]∂vn−1

∂xk.

90

92


Supposing that formula (46) is true for n− 1 results in

∂vn∂xk

=− vn(xn)

xn − xk+

[n−1∏i=1

(xn − xi)

]n−1∑i=1i 6=k

vn−1(xn−1)

xk − xi

=vn(xn)

xk − xn+n−1∑i=1i 6=k

vn(xn)

xk − xi=

n∑i=1i 6=k

vn(xn)

xk − xi.

Showing that (46) is true for n = 2 completes the proof

∂v2

∂x1=

∂

∂x1(x2 − x1) = −1 =

x2 − x1

x1 − x2=

2∑i=1i 6=1

v2(x2)

x1 − xi

∂v2

∂x2=

∂

∂x2(x2 − x1) = 1 =

x2 − x1

x2 − x1=

2∑i=1i 6=2

v2(x2)

x2 − xi.

Theorem 2.2. The extreme points of vn(xn) on the unit sphere can all befound in the hyperplane defined by

n∑i=1

xi = 0. (50)

This theorem will be proved after the introduction of the following usefullemma:

Lemma 2.6. For any n ≥ 2 the sum of the partial derivatives of vn(xn)will be zero.

n∑k=1

∂vn∂xk

= 0. (51)

Proof. This lemma is easily proven using Lemma 2.5 and induction:

n∑k=1

∂vn∂xk

=

n−1∑k=1

(− vn(xn)

xn − xk+

[n−1∏i=1

(xn − xi)

]∂vn−1

∂xk

)+

n−1∑i=1

vn(xn)

xn − xi

=

[n−1∏i=1

(xn − xi)

]n−1∑k=1

∂vn−1

∂xk.

Thus if equation (51) is true for n− 1 it is also true for n. Showing that theequation holds for n = 2 is very simple

∂v2

∂x1+∂v2

∂x2= −1 + 1 = 0.

91


Supposing that formula (46) is true for n− 1 results in

∂vn∂xk

=− vn(xn)

xn − xk+

[n−1∏i=1

(xn − xi)

]n−1∑i=1i 6=k

vn−1(xn−1)

xk − xi

=vn(xn)

xk − xn+n−1∑i=1i 6=k

vn(xn)

xk − xi=

n∑i=1i 6=k

vn(xn)

xk − xi.

Showing that (46) is true for n = 2 completes the proof

∂v2

∂x1=

∂

∂x1(x2 − x1) = −1 =

x2 − x1

x1 − x2=

2∑i=1i 6=1

v2(x2)

x1 − xi

∂v2

∂x2=

∂

∂x2(x2 − x1) = 1 =

x2 − x1

x2 − x1=

2∑i=1i 6=2

v2(x2)

x2 − xi.

Theorem 2.2. The extreme points of vn(xn) on the unit sphere can all befound in the hyperplane defined by

n∑i=1

xi = 0. (50)

This theorem will be proved after the introduction of the following usefullemma:

Lemma 2.6. For any n ≥ 2 the sum of the partial derivatives of vn(xn)will be zero.

n∑k=1

∂vn∂xk

= 0. (51)

Proof. This lemma is easily proven using Lemma 2.5 and induction:

n∑k=1

∂vn∂xk

=

n−1∑k=1

(− vn(xn)

xn − xk+

[n−1∏i=1

(xn − xi)

]∂vn−1

∂xk

)+

n−1∑i=1

vn(xn)

xn − xi

=

[n−1∏i=1

(xn − xi)

]n−1∑k=1

∂vn−1

∂xk.

Thus if equation (51) is true for n− 1 it is also true for n. Showing that theequation holds for n = 2 is very simple

∂v2

∂x1+∂v2

∂x2= −1 + 1 = 0.

91

93


Proof of Theorem 2.2. Using the method of Lagrange multipliers it followsthat any xn on the unit sphere that is an extreme point of the Vandermondedeterminant will also be a stationary point for the Lagrange function

Λn(xn, λ) = v(xn)− λ

(n∑i=1

x2i − 1

)

for some λ. Explicitly this requirement becomes

∂Λn∂xk

= 0 for all 1 ≤ k ≤ n, (52)

∂Λn∂λ

= 0. (53)

Equation (53) corresponds to the restriction to the unit sphere and is there-fore immediately satisfied. Since all the partial derivatives of the Lagrangefunction should be equal to zero it is obvious that the sum of the partialderivatives will also be equal to zero. Combining this with Lemma 2.6 gives

n∑k=1

∂Λn∂xk

=

n∑k=1

(∂vn∂xk− 2λxk

)= −2λ

n∑k=1

xk = 0. (54)

There are two ways to satisfy condition (54) either λ = 0 or

n∑k=1

xk = 0.

When λ = 0 equation (52) reduces to

∂vn∂xk

= 0 for all 1 ≤ k ≤ n,

and by equation (32) this can only be true if vn(xn) = 0, which is of no

interest to us, and so all extreme points must lie in the hyperplanen∑k=1

xk =

0.

Theorem 2.3. A point on the unit sphere in Rn, xn = (x1, x2, . . . xn),is an extreme point of the Vandermonde determinant if and only if all xi,i ∈ 1, 2, . . . n, are distinct roots of the rescaled Hermite polynomial

Pn(x) = (2n(n− 1))−n2 Hn

(√n(n− 1)

2x

). (55)

92


Proof of Theorem 2.2. Using the method of Lagrange multipliers it followsthat any xn on the unit sphere that is an extreme point of the Vandermondedeterminant will also be a stationary point for the Lagrange function

Λn(xn, λ) = v(xn)− λ

(n∑i=1

x2i − 1

)

for some λ. Explicitly this requirement becomes

∂Λn∂xk

= 0 for all 1 ≤ k ≤ n, (52)

∂Λn∂λ

= 0. (53)

Equation (53) corresponds to the restriction to the unit sphere and is there-fore immediately satisfied. Since all the partial derivatives of the Lagrangefunction should be equal to zero it is obvious that the sum of the partialderivatives will also be equal to zero. Combining this with Lemma 2.6 gives

n∑k=1

∂Λn∂xk

=

n∑k=1

(∂vn∂xk− 2λxk

)= −2λ

n∑k=1

xk = 0. (54)

There are two ways to satisfy condition (54) either λ = 0 or

n∑k=1

xk = 0.

When λ = 0 equation (52) reduces to

∂vn∂xk

= 0 for all 1 ≤ k ≤ n,

and by equation (32) this can only be true if vn(xn) = 0, which is of no

interest to us, and so all extreme points must lie in the hyperplanen∑k=1

xk =

0.

Theorem 2.3. A point on the unit sphere in Rn, xn = (x1, x2, . . . xn),is an extreme point of the Vandermonde determinant if and only if all xi,i ∈ 1, 2, . . . n, are distinct roots of the rescaled Hermite polynomial

Pn(x) = (2n(n− 1))−n2 Hn

(√n(n− 1)

2x

). (55)

92

94


Remark 2.1. Note that if xn = (x1, x2, . . . xn) is an extreme point of theVandermonde determinant then any other point whose coordinates are apermutation of the coordinates of xn is also an extreme point. This followsfrom the determinant function being, by definition, alternating with respectto the columns of the matrix and the xis defines the columns of the Vander-monde matrix. Thus any permutation of the xis will give the same value for|vn(xn)|. Since there are n! permutations there will be at least n! extremepoints. The roots of the polynomial (55) define the set of xis fully and thusthere are exactly n! extreme points, n!/2 positive and n!/2 negative.

Remark 2.2. All terms in Pn(x) are of even order if n is even and of oddorder when n is odd. This means that the roots of Pn(x) will be symmetricalin the sense that if xi is a root then −xi is also a root.

Proof of Theorem 2.3. By the method of Lagrange multipliers condition (52)must be satisfied for any extreme point. If xn is a fixed extreme point sothat

vn(xn) = vmax,

then (52) can be written explicitly, using (46), as

∂Λn∂xk

=n∑i=1i 6=k

vmaxxk − xi

− 2λxk = 0 for all 1 ≤ k ≤ n,

or alternatively by introducing a new multiplier ρ as

n∑i=1i 6=k

1

xk − xi=

2λ

vmaxxk =

ρ

nxk for all 1 ≤ k ≤ n. (56)

By forming the polynomial f(x) = (x− x1)(x− x2) · · · (x− xn) and notingthat

f ′(xk) =n∑j=1

n∏i=1i 6=j

(x− xi)∣∣∣∣x=xk

=n∏i=1i 6=k

(xk − xi),

f ′′(xk) =n∑l=1

n∑j=1j 6=l

n∏i=1i 6=ji 6=l


=n∑j=1j 6=k

n∏i=1i 6=ji 6=k

(xk − xi) +n∑l=1l 6=k

n∏i=1i 6=li 6=k

(xk − xi)

= 2n∑j=1j 6=k

n∏i=1i 6=ji 6=k

(xk − xi),

93


Remark 2.1. Note that if xn = (x1, x2, . . . xn) is an extreme point of theVandermonde determinant then any other point whose coordinates are apermutation of the coordinates of xn is also an extreme point. This followsfrom the determinant function being, by definition, alternating with respectto the columns of the matrix and the xis defines the columns of the Vander-monde matrix. Thus any permutation of the xis will give the same value for|vn(xn)|. Since there are n! permutations there will be at least n! extremepoints. The roots of the polynomial (55) define the set of xis fully and thusthere are exactly n! extreme points, n!/2 positive and n!/2 negative.

Remark 2.2. All terms in Pn(x) are of even order if n is even and of oddorder when n is odd. This means that the roots of Pn(x) will be symmetricalin the sense that if xi is a root then −xi is also a root.

Proof of Theorem 2.3. By the method of Lagrange multipliers condition (52)must be satisfied for any extreme point. If xn is a fixed extreme point sothat

vn(xn) = vmax,

then (52) can be written explicitly, using (46), as

∂Λn∂xk

=n∑i=1i 6=k

vmaxxk − xi

− 2λxk = 0 for all 1 ≤ k ≤ n,

or alternatively by introducing a new multiplier ρ as

n∑i=1i 6=k

1

xk − xi=

2λ

vmaxxk =

ρ

nxk for all 1 ≤ k ≤ n. (56)

By forming the polynomial f(x) = (x− x1)(x− x2) · · · (x− xn) and notingthat

f ′(xk) =n∑j=1

n∏i=1i 6=j


=n∏i=1i 6=k

(xk − xi),

f ′′(xk) =n∑l=1

n∑j=1j 6=l

n∏i=1i 6=ji 6=l


=n∑j=1j 6=k

n∏i=1i 6=ji 6=k

(xk − xi) +n∑l=1l 6=k

n∏i=1i 6=li 6=k

(xk − xi)

= 2n∑j=1j 6=k

n∏i=1i 6=ji 6=k

(xk − xi),

93

95


we can rewrite (56) as

1

2

f ′′(xk)

f ′(xk)=ρ

nxk,

or

f ′′(xk)−2ρ

nxkf

′(xk) = 0.

And since the last equation must vanish for all k we must have

f ′′(x)− 2ρ

nxf ′(x) = cf(x), (57)

for some constant c. To find c the xn-terms of the right and left part ofequation (57) are compared to each other,

c · cnxn = −2ρ

nxncnx

n−1 = −2ρ · cnxn ⇒ c = −2ρ.

Thus the following differential equation for f(x) must be satisfied

f ′′(x)− 2ρ

nxf ′(x) + 2ρf(x) = 0. (58)

Choosing x = az gives

f ′′(az)− 2ρ

(n− 1)a2zf ′(az) + 2ρf(az)

=1

a2

d2f

dz2(az)− 2ρ

naz

1

a

df

dz(az) + 2ρf(az) = 0.

By setting g(z) = f(az) and choosing a =√

nρ a differential equation that

matches the definition for the Hermite polynomials is found:

g′′(z)− 2zg′(z) + 2ng(z) = 0. (59)

By definition the solution to (59) is g(z) = bHn(z) where b is a constant.An exact expression for the constant a can be found using Lemma 2.7 (forthe sake of convenience the lemma is stated and proved after this theorem).We get

n∑i=1

x2i =

n∑i=1

a2z2i = 1⇒ a2n(n− 1)

2= 1,

and so

a =

√2

n(n− 1).

94


we can rewrite (56) as

1

2

f ′′(xk)

f ′(xk)=ρ

nxk,

or

f ′′(xk)−2ρ

nxkf

′(xk) = 0.

And since the last equation must vanish for all k we must have

f ′′(x)− 2ρ

nxf ′(x) = cf(x), (57)

for some constant c. To find c the xn-terms of the right and left part ofequation (57) are compared to each other,

c · cnxn = −2ρ

nxncnx

n−1 = −2ρ · cnxn ⇒ c = −2ρ.

Thus the following differential equation for f(x) must be satisfied

f ′′(x)− 2ρ

nxf ′(x) + 2ρf(x) = 0. (58)

Choosing x = az gives

f ′′(az)− 2ρ

(n− 1)a2zf ′(az) + 2ρf(az)

=1

a2

d2f

dz2(az)− 2ρ

naz

1

a

df

dz(az) + 2ρf(az) = 0.

By setting g(z) = f(az) and choosing a =√

nρ a differential equation that

matches the definition for the Hermite polynomials is found:

g′′(z)− 2zg′(z) + 2ng(z) = 0. (59)

By definition the solution to (59) is g(z) = bHn(z) where b is a constant.An exact expression for the constant a can be found using Lemma 2.7 (forthe sake of convenience the lemma is stated and proved after this theorem).We get

n∑i=1

x2i =

n∑i=1

a2z2i = 1⇒ a2n(n− 1)

2= 1,

and so

a =

√2

n(n− 1).

94

96


Thus condition (52) is satisfied when xi are the roots of

Pn(x) = bHn (z) = bHn

(√n(n− 1)

2x

).

Choosing b = (2n(n− 1))−n2 gives Pn(x) with leading coefficient 1. This can

be confirmed by calculating the leading coefficient of P (x) using the explicitexpression for the Hermite polynomial (61). This completes the proof.

Lemma 2.7. Let xi, i = 1, 2, . . . , n be roots of the Hermite polynomialHn(x). Then

n∑i=1

x2i =

n(n− 1)

2.

Proof. By letting ek(x1, . . . xn) denote the elementary symmetric polynomi-als Hn(x) can be written as

Hn(x) = An(x− x1) · · · (x− xn)

= An(xn − e1(x1, . . . , xn)xn−1 + e2(x1, . . . , xn)xn−2 + q(x))

where q(x) is a polynomial of degree n− 3. Noting that

n∑i=1

x2i = (x1 + . . .+ xn)2 − 2

∑1≤i<j≤n

xixj

= e1(x1, . . . , xn)2 − 2e2(x1, . . . , xn), (60)

it is clear the the sum of the square of the roots can be described using thecoefficients for xn, xn−1 and xn−2. The explicit expression for Hn(x) is [269]

Hn(x) = n!

bn2 c∑i=0

(−1)i

i!

(2x)n−2i

(n− 2i)!

= 2nxn − 2n−2n(n− 1)xn−2 + n!

bn2 c∑i=3

(−1)i

i!

(2x)n−2i

(n− 2i)!. (61)

Comparing the coefficients in the two expressions for Hn(x) gives

An = 2n,

Ane1(x1, . . . , xn) = 0,

Ane2(x1, . . . , xn) = −n(n− 1)2n−2.

Thus by (60)n∑i=1

x2i =

n(n− 1)

2.

95


Thus condition (52) is satisfied when xi are the roots of

Pn(x) = bHn (z) = bHn

(√n(n− 1)

2x

).

Choosing b = (2n(n− 1))−n2 gives Pn(x) with leading coefficient 1. This can

be confirmed by calculating the leading coefficient of P (x) using the explicitexpression for the Hermite polynomial (61). This completes the proof.

Lemma 2.7. Let xi, i = 1, 2, . . . , n be roots of the Hermite polynomialHn(x). Then

n∑i=1

x2i =

n(n− 1)

2.

Proof. By letting ek(x1, . . . xn) denote the elementary symmetric polynomi-als Hn(x) can be written as

Hn(x) = An(x− x1) · · · (x− xn)

= An(xn − e1(x1, . . . , xn)xn−1 + e2(x1, . . . , xn)xn−2 + q(x))

where q(x) is a polynomial of degree n− 3. Noting that

n∑i=1

x2i = (x1 + . . .+ xn)2 − 2

∑1≤i<j≤n

xixj

= e1(x1, . . . , xn)2 − 2e2(x1, . . . , xn), (60)

it is clear the the sum of the square of the roots can be described using thecoefficients for xn, xn−1 and xn−2. The explicit expression for Hn(x) is [269]

Hn(x) = n!

bn2 c∑i=0

(−1)i

i!

(2x)n−2i

(n− 2i)!

= 2nxn − 2n−2n(n− 1)xn−2 + n!

bn2 c∑i=3

(−1)i

i!

(2x)n−2i

(n− 2i)!. (61)

Comparing the coefficients in the two expressions for Hn(x) gives

An = 2n,

Ane1(x1, . . . , xn) = 0,

Ane2(x1, . . . , xn) = −n(n− 1)2n−2.

Thus by (60)n∑i=1

x2i =

n(n− 1)

2.

95

97


Theorem 2.4. The coefficients, ak, for the term xk in Pn(x) given by (55)are given by the following relations

an = 1, an−1 = 0, an−2 =1

2,

ak = − (k + 1)(k + 2)

n(n− 1)(n− k)ak+2, 1 ≤ k ≤ n− 3. (62)

Proof. Equation (58) tells us that

Pn(x) =1

2ρP ′′n (x)− 1

nxP ′n(x). (63)

That an = 1 follows from the definition of Pn and an−1 = 0 follows from theHermite polynomials only having terms of odd powers when n is odd andeven powers when n is even. That an−2 = 1

2 can be easily shown using thedefinition of Pn and the explicit formula for the Hermite polynomials (61).The value of the ρ can be found by comparing the xn−2 terms in (63)

an−2 =1

2ρn(n− 1)an +

1

n(n− 2)an−2.

From this follows1

2ρ=

−1

n2(n− 1).

Comparing the xn−l terms in (63) gives the following relation

an−l =1

2ρ(n− l + 2)(n− l)an−l+2 + (n− l)an−l

1

n

which is equivalent to

an−l = an−l+2−(n− l + 2)(n− l + 1)

ln2(n− 1).

Letting k = n− l gives (62).

2.2.2 Further visual exploration on the sphere


Visualization of the determinant v3(x3) on the unit sphere is straightforward,as well as visualizations for g3(x3,a) for different a. In three dimensions allpoints on the sphere can be mapped to the plane. In higher dimensionswe need to reduce the set of visualized points somehow. In this sectionwe provide visualizations for v4, . . . , v7 by using symmetry properties of theVandermonde determinant.

96


Theorem 2.4. The coefficients, ak, for the term xk in Pn(x) given by (55)are given by the following relations

an = 1, an−1 = 0, an−2 =1

2,

ak = − (k + 1)(k + 2)

n(n− 1)(n− k)ak+2, 1 ≤ k ≤ n− 3. (62)

Proof. Equation (58) tells us that

Pn(x) =1

2ρP ′′n (x)− 1

nxP ′n(x). (63)

That an = 1 follows from the definition of Pn and an−1 = 0 follows from theHermite polynomials only having terms of odd powers when n is odd andeven powers when n is even. That an−2 = 1

2 can be easily shown using thedefinition of Pn and the explicit formula for the Hermite polynomials (61).The value of the ρ can be found by comparing the xn−2 terms in (63)

an−2 =1

2ρn(n− 1)an +

1

n(n− 2)an−2.

From this follows1

2ρ=

−1

n2(n− 1).

Comparing the xn−l terms in (63) gives the following relation

an−l =1

2ρ(n− l + 2)(n− l)an−l+2 + (n− l)an−l

1

n

which is equivalent to

an−l = an−l+2−(n− l + 2)(n− l + 1)

ln2(n− 1).

Letting k = n− l gives (62).

2.2.2 Further visual exploration on the sphere


Visualization of the determinant v3(x3) on the unit sphere is straightforward,as well as visualizations for g3(x3,a) for different a. In three dimensions allpoints on the sphere can be mapped to the plane. In higher dimensionswe need to reduce the set of visualized points somehow. In this sectionwe provide visualizations for v4, . . . , v7 by using symmetry properties of theVandermonde determinant.

96

98


Four dimensions

By Theorem 2.2 we know that the extreme points of v4(x4) on the sphereall lie in the hyperplane x1 + x2 + x3 + x4 = 0. The intersection of thishyperplane with the unit sphere in R4 can be described as a unit sphere inR3, under a suitable basis, and can then be easily visualized.

This can be realized using the transformation

x =

−1 −1 0−1 1 01 0 −11 0 1

1/√

4 0 0

0 1/√

2 0

0 0 1/√

2

t. (64)

(a) Plot with t-basis given by (64). (b) Plot with θ and φ given by (34).

Figure 2.8: Plot of v4(x4) over points on the unit sphere.

The results of plotting the v4(x4) after performing this transformationcan be seen in Figure 2.8. All 24 = 4! extreme points are clearly visible.

From Figure 2.8 we see that whenever we have a local maxima we havea local maxima at the opposite side of the sphere as well, and the same forminima. This is due to the occurrence of the exponents in the rows of Vn.From equation (32) we have

vn((−1)xn) = (−1)n(n−1)

2 vn(xn),

and so opposite points are both maxima or both minima if n = 4k orn = 4k + 1 for some k ∈ Z+ and opposite points are of different typesif n = 4k − 2 or n = 4k − 1 for some k ∈ Z+.

By Theorem 2.3 the extreme points on the unit sphere for v4(x4) isdescribed by the roots of this polynomial

P4(x) = x4 − 1

2x2 +

1

48.

97


Four dimensions

By Theorem 2.2 we know that the extreme points of v4(x4) on the sphereall lie in the hyperplane x1 + x2 + x3 + x4 = 0. The intersection of thishyperplane with the unit sphere in R4 can be described as a unit sphere inR3, under a suitable basis, and can then be easily visualized.

This can be realized using the transformation

x =

−1 −1 0−1 1 01 0 −11 0 1

1/√

4 0 0

0 1/√

2 0

0 0 1/√

2

t. (64)



The results of plotting the v4(x4) after performing this transformationcan be seen in Figure 2.8. All 24 = 4! extreme points are clearly visible.

From Figure 2.8 we see that whenever we have a local maxima we havea local maxima at the opposite side of the sphere as well, and the same forminima. This is due to the occurrence of the exponents in the rows of Vn.From equation (32) we have

vn((−1)xn) = (−1)n(n−1)

2 vn(xn),

and so opposite points are both maxima or both minima if n = 4k orn = 4k + 1 for some k ∈ Z+ and opposite points are of different typesif n = 4k − 2 or n = 4k − 1 for some k ∈ Z+.


P4(x) = x4 − 1

2x2 +

1

48.

97

99


The roots of P4(x) are:

x41 = −1

2

√1 +

√2

3, x42 = −1

2

√1−

√2

3,

x43 =1

2

√1−

√2

3, x44 =

1

2

√1 +

√2

3.

Five dimensions

By Theorem 2.3 or 2.4 we see that the polynomials providing the coordinatesof the extreme points have all even or all odd powers. From this it is easyto see that all coordinates of the extreme points must come in pairs xi,−xi.Furthermore, by Theorem 2.2 we know that the extreme points of v5(x5) onthe sphere all lie in the hyperplane x1 + x2 + x3 + x4 + x5 = 0.

We use this to visualize v5(x5) by selecting a subspace of R5 that containsall points that have coordinates which are symmetrically placed on the realline, (x1, x2, 0,−x2,−x1).

The coordinates in Figure 2.9 (a) are related to x5 by

x5 =

−1 0 10 −1 10 0 10 1 11 0 1

1/√

2 0 0

0 1/√

2 0

0 0 1/√

5

t. (65)



The result, see Figure 2.9, is a visualization of a subspace containing8 of the 120 extreme points. Note that to satisfy the condition that the

98



x41 = −1

2

√1 +

√2

3, x42 = −1

2

√1−

√2

3,

x43 =1

2

√1−

√2

3, x44 =

1

2

√1 +

√2

3.

Five dimensions

By Theorem 2.3 or 2.4 we see that the polynomials providing the coordinatesof the extreme points have all even or all odd powers. From this it is easyto see that all coordinates of the extreme points must come in pairs xi,−xi.Furthermore, by Theorem 2.2 we know that the extreme points of v5(x5) onthe sphere all lie in the hyperplane x1 + x2 + x3 + x4 + x5 = 0.

We use this to visualize v5(x5) by selecting a subspace of R5 that containsall points that have coordinates which are symmetrically placed on the realline, (x1, x2, 0,−x2,−x1).


x5 =

−1 0 10 −1 10 0 10 1 11 0 1

1/√

2 0 0

0 1/√

2 0

0 0 1/√

5

t. (65)



The result, see Figure 2.9, is a visualization of a subspace containing8 of the 120 extreme points. Note that to satisfy the condition that the

98

100


coordinates should be symmetrically distributed pairs can be fulfilled in twoother subspaces with points that can be described in the following ways:(x1, x2, 0,−x1,−x2) and (x2,−x2, 0, x1,−x1). This means that a transfor-mation similar to (65) can describe 3 · 8 = 24 different extreme points.

The transformation (65) corresponds to choosing x3 = 0. Choosinganother coordinate to be zero will give a different subspace of R5 whichbehaves identically to the visualized one. This multiplies the number ofextreme points by five to the expected 5 · 4! = 120. By Theorem 2.3 theextreme points on the unit sphere for v5(x5) is described by the roots of thispolynomial

P5(x) = x5 − 1

2x3 +

3

80x.


x51 = −x55, x52 = −x54, x53 = 0,

x54 =1

2

√1−

√2

5, x55 =

1

2

√1 +

√2

5.

Six dimensions

As for v5(x5) we use symmetry to visualize v6(x6). We select a subspace ofR6 with all symmetrical points (x1, x2, x3,−x3,−x2,−x1) on the sphere.


x6 =

−1 0 00 −1 00 0 −10 0 10 1 01 0 0

1/√

2 0 0

0 1/√

2 0

0 0 1/√

2

t. (66)

In Figure 2.10 there are 48 visible extreme points. The remaining ex-treme points can be found using arguments analogous the five-dimensionalcase.


P6(x) = x6 − 1

2x4 +

1

20x2 − 1

1800.

99


coordinates should be symmetrically distributed pairs can be fulfilled in twoother subspaces with points that can be described in the following ways:(x1, x2, 0,−x1,−x2) and (x2,−x2, 0, x1,−x1). This means that a transfor-mation similar to (65) can describe 3 · 8 = 24 different extreme points.

The transformation (65) corresponds to choosing x3 = 0. Choosinganother coordinate to be zero will give a different subspace of R5 whichbehaves identically to the visualized one. This multiplies the number ofextreme points by five to the expected 5 · 4! = 120. By Theorem 2.3 theextreme points on the unit sphere for v5(x5) is described by the roots of thispolynomial

P5(x) = x5 − 1

2x3 +

3

80x.


x51 = −x55, x52 = −x54, x53 = 0,

x54 =1

2

√1−

√2

5, x55 =

1

2

√1 +

√2

5.

Six dimensions

As for v5(x5) we use symmetry to visualize v6(x6). We select a subspace ofR6 with all symmetrical points (x1, x2, x3,−x3,−x2,−x1) on the sphere.


x6 =

−1 0 00 −1 00 0 −10 0 10 1 01 0 0

1/√

2 0 0

0 1/√

2 0

0 0 1/√

2

t. (66)

In Figure 2.10 there are 48 visible extreme points. The remaining ex-treme points can be found using arguments analogous the five-dimensionalcase.


P6(x) = x6 − 1

2x4 +

1

20x2 − 1

1800.

99

101





x61 =− x66, x62 = −x65, x63 = −x64,

x64 =(−1)

34

2√

15

(10i− 3

√10

(z6w

136 + z6w

136

)) 12

=1

2√

15

√10− 2

√10(√

3l6 − k6

), (67)

x65 =(−1)

14

2√

15

(−10i− 3

√10

(z6w

136 + z6w

136

)) 12

=1

2√

15

√10− 2

√10(√

3l6 + k6

), (68)

x66 =

(1

30

(3√

10

(w

136 + w

136

)+ 5

)) 12

=

√1

30

(2√

10 · k6 + 5), (69)

z6 =√

3 + i, w6 = 2 + i√

6

k6 = cos

(1

3arctan

(√3

2

)), l6 = sin

(1

3arctan

(√3

2

)).

100





x61 =− x66, x62 = −x65, x63 = −x64,

x64 =(−1)

34

2√

15

(10i− 3

√10

(z6w

136 + z6w

136

)) 12

=1

2√

15

√10− 2

√10(√

3l6 − k6

), (67)

x65 =(−1)

14

2√

15

(−10i− 3

√10

(z6w

136 + z6w

136

)) 12

=1

2√

15

√10− 2

√10(√

3l6 + k6

), (68)

x66 =

(1

30

(3√

10

(w

136 + w

136

)+ 5

)) 12

=

√1

30

(2√

10 · k6 + 5), (69)

z6 =√

3 + i, w6 = 2 + i√

6

k6 = cos

(1

3arctan

(√3

2

)), l6 = sin

(1

3arctan

(√3

2

)).

100

102


Seven dimensions

As for v6(x6) we use symmetry to visualize v7(x7). We select a subspace ofR7 that contains all symmetrical points (x1, x2, x3, 0,−x3,−x2,−x1) on thesphere.


x7 =

−1 0 00 −1 00 0 −10 0 00 0 10 1 01 0 0

1/√

2 0 0

0 1/√

2 0

0 0 1/√

2

t. (70)



In Figure 2.11 48 extreme points are visible just like it was for the six-dimensional case. This is expected since the transformation correspondsto choosing x4 = 0 which restricts us to a six-dimensional subspace of R7

which can then be visualized in the same way as the six-dimensional case.The remaining extreme points can be found using arguments analogous thefive-dimensional case.

By Theorem 2.3 the extreme points on the unit sphere for v4 is describedby the roots of this polynomial

P7(x) = x7 − 1

2x5 +

5

84x3 − 5

3528x.

101


Seven dimensions

As for v6(x6) we use symmetry to visualize v7(x7). We select a subspace ofR7 that contains all symmetrical points (x1, x2, x3, 0,−x3,−x2,−x1) on thesphere.


x7 =

−1 0 00 −1 00 0 −10 0 00 0 10 1 01 0 0

1/√

2 0 0

0 1/√

2 0

0 0 1/√

2

t. (70)



In Figure 2.11 48 extreme points are visible just like it was for the six-dimensional case. This is expected since the transformation correspondsto choosing x4 = 0 which restricts us to a six-dimensional subspace of R7

which can then be visualized in the same way as the six-dimensional case.The remaining extreme points can be found using arguments analogous thefive-dimensional case.

By Theorem 2.3 the extreme points on the unit sphere for v4 is describedby the roots of this polynomial

P7(x) = x7 − 1

2x5 +

5

84x3 − 5

3528x.

101

103



x71 =− x77, x72 = −x76, x73 = −x75, x74 = 0,

x75 =(−1)

34

2√

21

(14i− 3

√14

(z6w

136 + z6w

136

)) 12

=1

2√

21

√14− 2

√14(√

3l6 − k6

), (71)

x76 =(−1)

14

2√

21

(−14i− 3

√14

(z6w

136 + z6w

136

)) 12

=1

2√

21

√14− 2

√14(√

3l7 + k7

), (72)

x77 =

√1

42

(3√

14

(w

136 + w

136

)+ 5

) 12

=

√1

42

(2√

14k7 + 5), (73)

z6 =√

3 + i, w6 = 2 + i√

10

k7 = cos

(1

3arctan

(√5

2

)),

l7 = sin

(1

3arctan

(√5

2

)).

102



x71 =− x77, x72 = −x76, x73 = −x75, x74 = 0,

x75 =(−1)

34

2√

21

(14i− 3

√14

(z6w

136 + z6w

136

)) 12

=1

2√

21

√14− 2

√14(√

3l6 − k6

), (71)

x76 =(−1)

14

2√

21

(−14i− 3

√14

(z6w

136 + z6w

136

)) 12

=1

2√

21

√14− 2

√14(√

3l7 + k7

), (72)

x77 =

√1

42

(3√

14

(w

136 + w

136

)+ 5

) 12

=

√1

42

(2√

14k7 + 5), (73)

z6 =√

3 + i, w6 = 2 + i√

10

k7 = cos

(1

3arctan

(√5

2

)),

l7 = sin

(1

3arctan

(√5

2

)).

102

104

2.3 OPTIMIZATION OF THE VANDERMONDEDETERMINANT ON SURFACES DEFINED BY A

UNIVARIATE POLYNOMIAL

2.3 Extreme points of the Vandermonde determi-nant on some surfaces implicitly defined by aunivariate polynomial

This section is based on Paper C

In this section the objective is to find the extreme points of the Vander-monde determinant on a surface implicitly defined by

gR(x) =n∑i=1

R(xi) = 0, where R(x) =m∑i=0

rixi, ri ∈ R. (74)

Lemma 2.8. The problem of finding the extreme points of the Vandermondedeterminant on the surface defined by gR(x) = 0 can be rewritten as anordinary differential equation of the form

f ′′(x)− 2ρR′(x)f ′(x)− P (x)f(x) = 0 (75)

that has a unique (up to a multiplicative constant) polynomial solution, f ,and any permutation of the roots of f will give the coordinates of a criticalpoint of the Vandermonde determinant.

Proof. Using the method of Lagrange multipliers we get

∂vn∂xj

= λ∂gR∂xj

⇔n∑i=1i 6=j

vn(x)

xj − xi= λR′(xj)

for some λ ∈ R.If we only consider this expression in a single point we can consider vn(x)

as a constant value and then the expression can be rewritten as

n∑i=1i 6=j

1

xj − xi= ρR′(xj) (76)

where ρ is some unknown constant.Consider the polynomial

f(x) =

n∏i=1

(x− xi)

and note that1

2

f ′′(xj)

f ′(xj)=

n∑i=1i 6=j

1

xj − xi. (77)

103



2.3 Extreme points of the Vandermonde determi-nant on some surfaces implicitly defined by aunivariate polynomial

This section is based on Paper C

In this section the objective is to find the extreme points of the Vander-monde determinant on a surface implicitly defined by

gR(x) =n∑i=1

R(xi) = 0, where R(x) =m∑i=0

rixi, ri ∈ R. (74)

Lemma 2.8. The problem of finding the extreme points of the Vandermondedeterminant on the surface defined by gR(x) = 0 can be rewritten as anordinary differential equation of the form

f ′′(x)− 2ρR′(x)f ′(x)− P (x)f(x) = 0 (75)

that has a unique (up to a multiplicative constant) polynomial solution, f ,and any permutation of the roots of f will give the coordinates of a criticalpoint of the Vandermonde determinant.

Proof. Using the method of Lagrange multipliers we get

∂vn∂xj

= λ∂gR∂xj

⇔n∑i=1i 6=j

vn(x)

xj − xi= λR′(xj)

for some λ ∈ R.If we only consider this expression in a single point we can consider vn(x)

as a constant value and then the expression can be rewritten as

n∑i=1i 6=j

1

xj − xi= ρR′(xj) (76)

where ρ is some unknown constant.Consider the polynomial

f(x) =

n∏i=1

(x− xi)

and note that1

2

f ′′(xj)

f ′(xj)=

n∑i=1i 6=j

1

xj − xi. (77)

103

105


In each critical point we can combine (76) and (77) thus in each of theextreme points we will have the relation

f ′′(xj)− 2ρR′(xj)f′(xj) = 0, j = 1, 2, . . . , n

for some ρ ∈ R. Since each xj is a root of f(x) we see that the left handside in the differential equation must be a polynomial with the same rootsas f(x), thus we can conclude that for any x ∈ R

f ′′(x)− 2ρR′(x)f ′(x)− P (x)f(x) = 0 (78)

where P (x) is a polynomial of degree m− 2.

Using this technique it is also easy to find the coordinates on a spheretranslated in the (1, . . . , 1) direction.

Corollary 2.1. If x = (x1, x2, . . . , xn) is a critical point of the Vander-monde determinant on a surface S ⊂ Cn then (x1 + a, x2 + a, . . . , xn + a) isa critical point of the Vandermonde determinant on the surface x + a1 ∈Cn|x ∈ S.Proof. Follows immediately from

vn (x1 + a, x2 + a, . . . , xn + a) =∏

1≤i<j≤n(xj + a− xi − a)

=∏

1≤i<j≤n(xj − xi) = vn(x1, . . . , xn).

In several cases it is possible to find the extreme points by identifying theunknown parameters, ρ and the coefficients of P (x), by comparing the termsin (75) with different degrees and solving the resulting equation system. Wewill discuss the cases in the upcoming sections.

2.3.1 Critical points on surfaces given by a first degree uni-variate polynomial

When R(x) = r1x + r0 the surface defined by

n∑i=1

R(xi) = 0 will always be

a plane with normal (1, 1, . . . , 1) through the point(r0r1, r0r1 , . . . ,

r0r1

).

Since

vn

(x1 +

r0

r1, x2 +

r0

r1, . . . , xn +

r0

r1

)=

∏1≤i<j≤n

(xj +

r0

r1− xi −

r0

r1

)=

∏1≤i<j≤n

(xj − xi) = vn(x1, . . . , xn).

So the Vandermonde determinant will have no extreme point unless afurther constraint is added.

104


In each critical point we can combine (76) and (77) thus in each of theextreme points we will have the relation

f ′′(xj)− 2ρR′(xj)f′(xj) = 0, j = 1, 2, . . . , n

for some ρ ∈ R. Since each xj is a root of f(x) we see that the left handside in the differential equation must be a polynomial with the same rootsas f(x), thus we can conclude that for any x ∈ R

f ′′(x)− 2ρR′(x)f ′(x)− P (x)f(x) = 0 (78)

where P (x) is a polynomial of degree m− 2.

Using this technique it is also easy to find the coordinates on a spheretranslated in the (1, . . . , 1) direction.

Corollary 2.1. If x = (x1, x2, . . . , xn) is a critical point of the Vander-monde determinant on a surface S ⊂ Cn then (x1 + a, x2 + a, . . . , xn + a) isa critical point of the Vandermonde determinant on the surface x + a1 ∈Cn|x ∈ S.Proof. Follows immediately from

vn (x1 + a, x2 + a, . . . , xn + a) =∏

1≤i<j≤n(xj + a− xi − a)

=∏

1≤i<j≤n(xj − xi) = vn(x1, . . . , xn).

In several cases it is possible to find the extreme points by identifying theunknown parameters, ρ and the coefficients of P (x), by comparing the termsin (75) with different degrees and solving the resulting equation system. Wewill discuss the cases in the upcoming sections.

2.3.1 Critical points on surfaces given by a first degree uni-variate polynomial

When R(x) = r1x + r0 the surface defined by

n∑i=1

R(xi) = 0 will always be

a plane with normal (1, 1, . . . , 1) through the point(r0r1, r0r1 , . . . ,

r0r1

).

Since

vn

(x1 +

r0

r1, x2 +

r0

r1, . . . , xn +

r0

r1

)=

∏1≤i<j≤n

(xj +

r0

r1− xi −

r0

r1

)=

∏1≤i<j≤n

(xj − xi) = vn(x1, . . . , xn).

So the Vandermonde determinant will have no extreme point unless afurther constraint is added.

104

106



2.3.2 Critical points on surfaces given by a second degreeunivariate polynomial

Surfaces defined by letting R(x) = 12x

2 +r1x+r0 = 12

((x+ r1)2 − r2

1 + 2r0

)will all be spheres around (−r1,−r1, . . . ,−r1) with radius

√n(r212 − r0

).

Thus the critical points can be found by a small modification of thetechnique used on the unit sphere described in Section 2.2.

Theorem 2.5. On the surface defined by g(x) =

n∑i=1

1

2x2i + r1xi + r0 the

coordinates of the critical points of the Vandermonde determinant are givenby the roots of

f(x) = Hn

((n− 1

2(r21 − 2r0)

) 12 (x+ r1)

2

)

= n!

bn2 c∑i=0

(−1)i

i!

(n− 1

2(r21 − 2r0)

)n−2i2 (x+ r1)n−2i

(n− 2i)!

where Hn denotes the nth (physicist) Hermite polynomial.

Proof. Since R(x) = 12x

2 + r1x+ r0 the differential equation (75) will be ofthe form

f ′′(x)− 2ρ(x+ r1)f ′(x)− p0f(x) = 0.

By considering the terms with degree n it is easy to see that p0 = −2ρn andthus we get

f ′′(x)− 2ρ(x+ r1)f ′(x) + 2ρnf(x) = 0.

Setting y = ρ12 (x + r1) gives x = y

ρ12− r1 and by considering the function

g(y) = f

(y

ρ12− r1

)we can rewrite the differential equation as follows

d2g

dx2− 2ρ

(√y

ρ− r1 + r1

)dg

dx+ 2 ρn g(y) = 0

⇔ ρ g′′(y)− 2 ρy

ρ12

ρ12 g′(x) + 2 ρn g(y) = 0

⇔ g′′(y)− 2 y g′(x) + 2n g(y) = 0. (79)

Equation (79) defines a class of orthogonal polynomials called the Hermite

polynomials [2], Hn(y). Thus f(x) = cHn(ρ12 (x + r1)) for some arbitrary

constant c. To find the value of ρ we can exploit some properties of theroots of the Hermite polynomials.

105



2.3.2 Critical points on surfaces given by a second degreeunivariate polynomial

Surfaces defined by letting R(x) = 12x

2 +r1x+r0 = 12

((x+ r1)2 − r2

1 + 2r0

)will all be spheres around (−r1,−r1, . . . ,−r1) with radius

√n(r212 − r0

).

Thus the critical points can be found by a small modification of thetechnique used on the unit sphere described in Section 2.2.

Theorem 2.5. On the surface defined by g(x) =

n∑i=1

1

2x2i + r1xi + r0 the

coordinates of the critical points of the Vandermonde determinant are givenby the roots of

f(x) = Hn

((n− 1

2(r21 − 2r0)

) 12 (x+ r1)

2

)

= n!

bn2 c∑i=0

(−1)i

i!

(n− 1

2(r21 − 2r0)

)n−2i2 (x+ r1)n−2i

(n− 2i)!

where Hn denotes the nth (physicist) Hermite polynomial.

Proof. Since R(x) = 12x

2 + r1x+ r0 the differential equation (75) will be ofthe form

f ′′(x)− 2ρ(x+ r1)f ′(x)− p0f(x) = 0.

By considering the terms with degree n it is easy to see that p0 = −2ρn andthus we get

f ′′(x)− 2ρ(x+ r1)f ′(x) + 2ρnf(x) = 0.

Setting y = ρ12 (x + r1) gives x = y

ρ12− r1 and by considering the function

g(y) = f

(y

ρ12− r1

)we can rewrite the differential equation as follows

d2g

dx2− 2ρ

(√y

ρ− r1 + r1

)dg

dx+ 2 ρn g(y) = 0

⇔ ρ g′′(y)− 2 ρy

ρ12

ρ12 g′(x) + 2 ρn g(y) = 0

⇔ g′′(y)− 2 y g′(x) + 2n g(y) = 0. (79)

Equation (79) defines a class of orthogonal polynomials called the Hermite

polynomials [2], Hn(y). Thus f(x) = cHn(ρ12 (x + r1)) for some arbitrary

constant c. To find the value of ρ we can exploit some properties of theroots of the Hermite polynomials.

105

107


If we let yi, i = 1, . . . , n be the roots of Hn(y). On page 95 we show thatthese roots have the following properties

n∑i=1

yi = 0, (80)

n∑i=1

y2i =

n(n− 1)

2. (81)

We now take the change of variables x = y

ρ12− r1 into consideration and

get

n∑i=1

xi =n∑i=1

(yi

ρ12

− r1

)2

=1

ρ12

(n∑i=1

yi

)− nr1,

n∑i=1

x2i =

n∑i=1

(yi

ρ12

− r1

)2

=n∑i=1

1

ρ

(n∑i=1

y2i

)− 2r1

ρ12

(n∑i=1

yi

)+ nr2

1.

Using (80) and (81) we can simplify these expression

n∑i=1

xi = −nr1,

n∑i=1

x2i =

n(n− 1)

2ρ+ nr2

1.

This allow us to rephrase the constraint g(x) = 0 as follows

g(x) =n∑i=1

1

2x2i + r1xi + r0 =

n(n− 1)

4ρ− nr2

1

2+ nr0 = 0

and from this it is easy to find an expression for ρ

ρ =n− 1

8(r21 − 2r0)

.

Thus the coordinates of the extreme points are the roots of the polynomialgiven in Theorem 2.5.

Remark 2.3. Note that Remarks 2.1 and 2.2 apply in this case as well.

For more details and demonstrations of how to visualize this result seeSection 2.2.2.

106


If we let yi, i = 1, . . . , n be the roots of Hn(y). On page 95 we show thatthese roots have the following properties

n∑i=1

yi = 0, (80)

n∑i=1

y2i =

n(n− 1)

2. (81)

We now take the change of variables x = y

ρ12− r1 into consideration and

get

n∑i=1

xi =n∑i=1

(yi

ρ12

− r1

)2

=1

ρ12

(n∑i=1

yi

)− nr1,

n∑i=1

x2i =

n∑i=1

(yi

ρ12

− r1

)2

=n∑i=1

1

ρ

(n∑i=1

y2i

)− 2r1

ρ12

(n∑i=1

yi

)+ nr2

1.

Using (80) and (81) we can simplify these expression

n∑i=1

xi = −nr1,

n∑i=1

x2i =

n(n− 1)

2ρ+ nr2

1.

This allow us to rephrase the constraint g(x) = 0 as follows

g(x) =n∑i=1

1

2x2i + r1xi + r0 =

n(n− 1)

4ρ− nr2

1

2+ nr0 = 0

and from this it is easy to find an expression for ρ

ρ =n− 1

8(r21 − 2r0)

.

Thus the coordinates of the extreme points are the roots of the polynomialgiven in Theorem 2.5.

Remark 2.3. Note that Remarks 2.1 and 2.2 apply in this case as well.

For more details and demonstrations of how to visualize this result seeSection 2.2.2.

106

108



2.3.3 Critical points on the sphere defined by a p-norm

Definition 2.2. The p−norm of x ∈ Rn denoted ‖x‖p is defined as

‖x‖p =

(n∑i=1

|xi|p) 1

p

, for p > 0. (82)

Definition 2.3. The infinity norm of x ∈ Rn denoted ‖x‖∞ is defined as

‖x‖∞ = sup|xi| : 1 ≤ i ≤ n. (83)

Definition 2.4. The sphere defined by the p-norm, denoted Sn−1p (r), for

positive integer p, is the set of all x ∈ Rn such that

n∑i=1

|xi|p = ‖x‖pp = rp. (84)

When r = 1 this is the unit sphere defined by the p-norm, denoted simplySn−1p . When p increases the points on Sn−1

p approach the points on the cubeso for convenience we define Sn−1

∞ as the cube defined by the boundary of[−1, 1]n.

Spheres defined by p-norms describes many well-known geometric shapes.For instance when n = 2, p = 2, then S1

2(r) = (x1, x2) ∈ R2 : x21 +x2

2 = r2is a circle and when n = 3, p = 2, then

S13(r) = (x1, x2, x3) ∈ R2 : x2

1 + x22 + x2

3 = r2

is the standard 2-sphere with radius r. In the previous section we discussedhow the extreme points of the Vandermonde determinant are distributed forthe case p = 2 and n ≥ 2. In this section we will examine how the extremepoints of the Vandermonde determinant are distributed on the sphere definedby the p-norm for the cases p ∈ 4, 6, 8 for a few different values of n. InFigure 2.12 Sn−1

p for p = 2, p = 4, p = 6, p = 8, and p = ∞ with a sectioncut out are illustrated.

Similarly to the previous section we will construct a polynomial whoseroots give the coordinates of the extreme points of the Vandermonde deter-minant. First we will consider the case p = 4, n = 4.

2.3.4 The case p = 4 and n = 4

We will illustrate the construction of a polynomial that has the coordinatesof the points as roots with the case p = 4, n = 4. If we denote the poly-nomial whose roots give the coordinates with P 4

4 (x) and use the same typeof argument that was used to get equation (75). Taking P (x) to be of theform:

P (x) = xn + cn−2xn−2 + cn−4x

n−4 + · · · (85)

107



2.3.3 Critical points on the sphere defined by a p-norm

Definition 2.2. The p−norm of x ∈ Rn denoted ‖x‖p is defined as

‖x‖p =

(n∑i=1

|xi|p) 1

p

, for p > 0. (82)

Definition 2.3. The infinity norm of x ∈ Rn denoted ‖x‖∞ is defined as

‖x‖∞ = sup|xi| : 1 ≤ i ≤ n. (83)

Definition 2.4. The sphere defined by the p-norm, denoted Sn−1p (r), for

positive integer p, is the set of all x ∈ Rn such that

n∑i=1

|xi|p = ‖x‖pp = rp. (84)

When r = 1 this is the unit sphere defined by the p-norm, denoted simplySn−1p . When p increases the points on Sn−1

p approach the points on the cubeso for convenience we define Sn−1

∞ as the cube defined by the boundary of[−1, 1]n.

Spheres defined by p-norms describes many well-known geometric shapes.For instance when n = 2, p = 2, then S1

2(r) = (x1, x2) ∈ R2 : x21 +x2

2 = r2is a circle and when n = 3, p = 2, then

S13(r) = (x1, x2, x3) ∈ R2 : x2

1 + x22 + x2

3 = r2

is the standard 2-sphere with radius r. In the previous section we discussedhow the extreme points of the Vandermonde determinant are distributed forthe case p = 2 and n ≥ 2. In this section we will examine how the extremepoints of the Vandermonde determinant are distributed on the sphere definedby the p-norm for the cases p ∈ 4, 6, 8 for a few different values of n. InFigure 2.12 Sn−1

p for p = 2, p = 4, p = 6, p = 8, and p = ∞ with a sectioncut out are illustrated.

Similarly to the previous section we will construct a polynomial whoseroots give the coordinates of the extreme points of the Vandermonde deter-minant. First we will consider the case p = 4, n = 4.

2.3.4 The case p = 4 and n = 4

We will illustrate the construction of a polynomial that has the coordinatesof the points as roots with the case p = 4, n = 4. If we denote the poly-nomial whose roots give the coordinates with P 4

4 (x) and use the same typeof argument that was used to get equation (75). Taking P (x) to be of theform:

P (x) = xn + cn−2xn−2 + cn−4x

n−4 + · · · (85)

107

109


Figure 2.12: Illustration of S2p for p = 2, p = 4, p = 6, p = 8, and p = ∞

with a section cut out. The outer cube corresponds to p = 0 andp = 2 corresponds to the sphere in the middle.

with every other coefficient zero, when n is even of we have even powersand when n is odd we have odd powers. By identifying the powers in thedifferential equation (75) for the case p = 4:

P′′(x) + ρnx

3P′(x) + (σnx

2 + τnx+ νn)P (x) = 0, (86)

we obtain that τnxP (x) does not share any powers with any other part of theequation and thus τn = 0. Similarly, identifying the coefficients we obtainpρn + σn = 0. This leads us to the differential equation

P′′(x) + ρnx

3P′(x) + (−pρnx2 + νn)P (x) = 0. (87)

Basing on (85) and (87), and setting n = 4, p = 4 we get to generate thesystem of

Sn−1p =

n∑i=1

xpi = 1,

P 44 (x) = xn + cn−2x

n−2 + cn−4xn−4 + · · · ,

P 44

′′(x) + ρnx

3P 44

′(x) + (−pρnx2 + νn)P 4

4 (x) = 0.

It follows that

S44 =

4∑i=1

x4i = x4

1 + x42 + x4

3 + x44 = 1,

P 44 (x) = x4 + c2x

2 + c0 ⇒ P 44

′(x) = 4x3 + 2c2x ⇒ P 4

4

′′(x) = 12x2 + 2c2,

and by substitution into the differential equation

(12x2 + 2c2) + ρnx3(4x3 + 2c2x) + (−pρnx2 + νn)(x4 + c2x

2 + c0) = 0,

(ν − 2ρc2)x4 + (νc2 − 4ρc0 + 12)x2 + (2c2 + c0ν) = 0.

108


Figure 2.12: Illustration of S2p for p = 2, p = 4, p = 6, p = 8, and p = ∞

with a section cut out. The outer cube corresponds to p = 0 andp = 2 corresponds to the sphere in the middle.

with every other coefficient zero, when n is even of we have even powersand when n is odd we have odd powers. By identifying the powers in thedifferential equation (75) for the case p = 4:

P′′(x) + ρnx

3P′(x) + (σnx

2 + τnx+ νn)P (x) = 0, (86)

we obtain that τnxP (x) does not share any powers with any other part of theequation and thus τn = 0. Similarly, identifying the coefficients we obtainpρn + σn = 0. This leads us to the differential equation

P′′(x) + ρnx

3P′(x) + (−pρnx2 + νn)P (x) = 0. (87)

Basing on (85) and (87), and setting n = 4, p = 4 we get to generate thesystem of

Sn−1p =

n∑i=1

xpi = 1,

P 44 (x) = xn + cn−2x

n−2 + cn−4xn−4 + · · · ,

P 44

′′(x) + ρnx

3P 44

′(x) + (−pρnx2 + νn)P 4

4 (x) = 0.

It follows that

S44 =

4∑i=1

x4i = x4

1 + x42 + x4

3 + x44 = 1,

P 44 (x) = x4 + c2x

2 + c0 ⇒ P 44

′(x) = 4x3 + 2c2x ⇒ P 4

4

′′(x) = 12x2 + 2c2,

and by substitution into the differential equation

(12x2 + 2c2) + ρnx3(4x3 + 2c2x) + (−pρnx2 + νn)(x4 + c2x

2 + c0) = 0,

(ν − 2ρc2)x4 + (νc2 − 4ρc0 + 12)x2 + (2c2 + c0ν) = 0.

108

110



Equating corresponding coefficients as in P (x) we get:

ν − 2ρc2 = 1,

νc2 − 4ρc0 + 12 = c2,

2c2 + c0ν = c0.

Setting t = x2 we can express S43 and P (x) as follows:

S43 = 2t21 + 2t22 = 2

4∑i=1

t2i = 1

P 44 (x) = x4 + c2x

2 + c0 = t2 − (t0 + t1)t+ t0t1 = 0.

Equating coefficient in P 44 (x) gives

t0 + t1 = c2, t0t1 = c0

⇒t0t1 + t21 = c2t1 ⇒ c0 + t21 = c2t1 ⇒ t21 = c2t1 − c0

⇒t20 + t0t1 = c2t0 ⇒ t20 + c0 = c2t0 ⇒ t20 = c2t0 − c0

⇒t20 + t21 = c2(t0 + t1)− 2c0 = c22 − 2c0 ⇒ 2

4∑i=1

t2i = 2(c22 − 2c0) = 1

This now gives a fourth equation so as to solve the system:

ν − 2ρc2 = 1, (88)

νc2 − 4ρc0 + 12 = c2, (89)

2c2 + c0ν = c0, (90)

2(c22 − 2c0) = 1. (91)

From (88) we obtain ν = 1 + 2ρc2 and substituting this into (89) gives

c2(1 + 2ρc2)− 4ρc0 + 12 = c2 ⇒ ρ(2(c2

2 − 2c0))

= −12⇒ ρ = −12.

To get the last equality use (91) and the fact that c2 6= 0.Using this value in the expression for ν we obtain ν = 1 − 24c2 and

substituting this value into (90) gives

2c2 + c0(1− 24c2) = c0 ⇒ 2c2(1− 12c0) = 0⇒ 1− 12c0 = 0⇒ c0 =1

12,

where the last equality follows from c2 6= 0.Now with ρ = −12, c0 = 1/12, using (91) we obtain

2(c22 − 2c0) = 1⇒ c2 =

1

2+

2

12=

8

12⇒ c2 =

2√6

Therefore we obtain P 44 (x) = x4 − 2√

6x2 + 1

12 .

In Section 2.3.5 we will generalise this technique somewhat.

109



Equating corresponding coefficients as in P (x) we get:

ν − 2ρc2 = 1,

νc2 − 4ρc0 + 12 = c2,

2c2 + c0ν = c0.

Setting t = x2 we can express S43 and P (x) as follows:

S43 = 2t21 + 2t22 = 2

4∑i=1

t2i = 1

P 44 (x) = x4 + c2x

2 + c0 = t2 − (t0 + t1)t+ t0t1 = 0.

Equating coefficient in P 44 (x) gives

t0 + t1 = c2, t0t1 = c0

⇒t0t1 + t21 = c2t1 ⇒ c0 + t21 = c2t1 ⇒ t21 = c2t1 − c0

⇒t20 + t0t1 = c2t0 ⇒ t20 + c0 = c2t0 ⇒ t20 = c2t0 − c0

⇒t20 + t21 = c2(t0 + t1)− 2c0 = c22 − 2c0 ⇒ 2

4∑i=1

t2i = 2(c22 − 2c0) = 1

This now gives a fourth equation so as to solve the system:

ν − 2ρc2 = 1, (88)

νc2 − 4ρc0 + 12 = c2, (89)

2c2 + c0ν = c0, (90)

2(c22 − 2c0) = 1. (91)

From (88) we obtain ν = 1 + 2ρc2 and substituting this into (89) gives

c2(1 + 2ρc2)− 4ρc0 + 12 = c2 ⇒ ρ(2(c2

2 − 2c0))

= −12⇒ ρ = −12.

To get the last equality use (91) and the fact that c2 6= 0.Using this value in the expression for ν we obtain ν = 1 − 24c2 and

substituting this value into (90) gives

2c2 + c0(1− 24c2) = c0 ⇒ 2c2(1− 12c0) = 0⇒ 1− 12c0 = 0⇒ c0 =1

12,

where the last equality follows from c2 6= 0.Now with ρ = −12, c0 = 1/12, using (91) we obtain

2(c22 − 2c0) = 1⇒ c2 =

1

2+

2

12=

8

12⇒ c2 =

2√6

Therefore we obtain P 44 (x) = x4 − 2√

6x2 + 1

12 .

In Section 2.3.5 we will generalise this technique somewhat.

109

111


2.3.5 Some results for even n and p

In this section we will discuss the case when n and p are positive and evenintegers, and n > p. We will discuss a method that can give the coordinatesextreme points of the Vandermonde determinant constrained to Sn−1

p , asdefined in (84), as the roots of a polynomial.

First we will examine how this optimisation problem can be rewrittenas a differential equation similar to (86).

Lemma 2.9. Let n and p be even positive integers. Consider the unit spheregiven by the p - norm, in other words the surface given by

Spn =

(x1, . . . , xn) ∈ Rn

∣∣∣∣∣n∑i=1

xpi = 1

.

There exists a second order differential equation

P pn′′(x)− ap−2

nxp−1P pn

′(x) +Qpn(x)P pn(x) = 0, (92)

where P pn(x) and Qpn(x) are of the forms

P pn(x) = x2n +

12n−1∑i=0

c2ix2i

and

Qpn(x) = −ap−2xp−2 +

12p−2∑i=0

(−1)ia2ix2i.

There is also a relation between the coefficients of P pn and Qpn given by

2j(2j − 1)c2j +

(j−1∑k=0

a2k c2(j−k−1)

)+n+ p− 2j

nap−2 c2j−p = 0 (93)

for 1 ≤ j ≤ n+p−22 where cn = 1, ck = 0 for k 6∈ 0, 2, 4, . . . , n and ak = 0

for k 6∈ 0, 2, 4, . . . , p− 2.

Proof. This result is proved analogously to how (75) is found. Define

P pn(x) =n∏i=1

(x− xi)

and note that1

2

P pn′′(x)

P pn′(x)

=n∑i=1

1

x− xi.

110


2.3.5 Some results for even n and p

In this section we will discuss the case when n and p are positive and evenintegers, and n > p. We will discuss a method that can give the coordinatesextreme points of the Vandermonde determinant constrained to Sn−1

p , asdefined in (84), as the roots of a polynomial.

First we will examine how this optimisation problem can be rewrittenas a differential equation similar to (86).

Lemma 2.9. Let n and p be even positive integers. Consider the unit spheregiven by the p - norm, in other words the surface given by

Spn =

(x1, . . . , xn) ∈ Rn

∣∣∣∣∣n∑i=1

xpi = 1

.

There exists a second order differential equation

P pn′′(x)− ap−2

nxp−1P pn

′(x) +Qpn(x)P pn(x) = 0, (92)

where P pn(x) and Qpn(x) are of the forms

P pn(x) = x2n +

12n−1∑i=0

c2ix2i

and

Qpn(x) = −ap−2xp−2 +

12p−2∑i=0

(−1)ia2ix2i.

There is also a relation between the coefficients of P pn and Qpn given by

2j(2j − 1)c2j +

(j−1∑k=0

a2k c2(j−k−1)

)+n+ p− 2j

nap−2 c2j−p = 0 (93)

for 1 ≤ j ≤ n+p−22 where cn = 1, ck = 0 for k 6∈ 0, 2, 4, . . . , n and ak = 0

for k 6∈ 0, 2, 4, . . . , p− 2.

Proof. This result is proved analogously to how (75) is found. Define

P pn(x) =n∏i=1

(x− xi)

and note that1

2

P pn′′(x)

P pn′(x)

=n∑i=1

1

x− xi.

110

112



Now apply the method of Lagrange multipliers and see that in the criticalpoints

n∑i=1i 6=j

1

xj − xi= ρR′(xj)

where ρ is some unknown constant.In each critical point we can combine the two expressions and get

P pn′′(xj)− 2ρR′(xj)P

pn′(xj) = 0, j = 1, 2, . . . , n

for some ρ ∈ R. Since each xj is a root of f(x) we see that the left handside in the differential equation must be a polynomial with the same rootsas P pn(x), thus we can conclude that for any x ∈ R

P pn′′(x)− 2ρR′(x)P pn

′(x)−Q(x)f(x) = 0

where Q(x) is a polynomial of degree p− 2.By applying the principles of polynomial solutions to linear second order

differential equation [10, 50], expanding the expression and matching thecoefficients of the terms with different powers of x you can see that thecoefficients of P (x) and Q(x) must obey the relation given in (93).

Noting that the relations between the two sets of coefficients are lin-ear we will consider the equations given by (93) corresponding to j ∈n−2

2 , n2 , . . . ,n+p−2

2

, the corresponding system of equations in matrix form

becomes

cn−2 cn−4 cn−6 · · · c4pn cn−p−2

1 cn−2 cn−4 · · · c6p−2n cn−p

0 1 cn−2 · · · c8p−4n cn−p+2

......

.... . .

......

0 0 0 · · · cn−24n cn−4

0 0 0 · · · 1 2n cn−2

a0

a2

a4...

ap−4

ap−2

=

−n(n− 1)00...00

.

(94)By solving this system we can reduce the n+p−2

2 equations given bymatching the terms to n−2

2 equations that together with the condition givenby (84) gives a system of polynomial equations that determines all the un-known coefficients of P (x).

To describe how we can express the solution to (94) we will use a few well-known relations between elementary symmetric polynomials and power sumsoften referred to as the Newton–Girard formulae (Theorem 2.7), and Vieta’sformula (Theorem 2.6) that describes the relation between the coefficientsof a polynomial and its roots.

Here we will give some useful properties of elementary symmetric poly-nomials and power sums and relations between them.

111



Now apply the method of Lagrange multipliers and see that in the criticalpoints

n∑i=1i 6=j

1

xj − xi= ρR′(xj)

where ρ is some unknown constant.In each critical point we can combine the two expressions and get

P pn′′(xj)− 2ρR′(xj)P

pn′(xj) = 0, j = 1, 2, . . . , n

for some ρ ∈ R. Since each xj is a root of f(x) we see that the left handside in the differential equation must be a polynomial with the same rootsas P pn(x), thus we can conclude that for any x ∈ R

P pn′′(x)− 2ρR′(x)P pn

′(x)−Q(x)f(x) = 0

where Q(x) is a polynomial of degree p− 2.By applying the principles of polynomial solutions to linear second order

differential equation [10, 50], expanding the expression and matching thecoefficients of the terms with different powers of x you can see that thecoefficients of P (x) and Q(x) must obey the relation given in (93).

Noting that the relations between the two sets of coefficients are lin-ear we will consider the equations given by (93) corresponding to j ∈n−2

2 , n2 , . . . ,n+p−2

2

, the corresponding system of equations in matrix form

becomes

cn−2 cn−4 cn−6 · · · c4pn cn−p−2

1 cn−2 cn−4 · · · c6p−2n cn−p

0 1 cn−2 · · · c8p−4n cn−p+2

......

.... . .

......

0 0 0 · · · cn−24n cn−4

0 0 0 · · · 1 2n cn−2

a0

a2

a4...

ap−4

ap−2

=

−n(n− 1)00...00

.

(94)By solving this system we can reduce the n+p−2

2 equations given bymatching the terms to n−2

2 equations that together with the condition givenby (84) gives a system of polynomial equations that determines all the un-known coefficients of P (x).

To describe how we can express the solution to (94) we will use a few well-known relations between elementary symmetric polynomials and power sumsoften referred to as the Newton–Girard formulae (Theorem 2.7), and Vieta’sformula (Theorem 2.6) that describes the relation between the coefficientsof a polynomial and its roots.

Here we will give some useful properties of elementary symmetric poly-nomials and power sums and relations between them.

111

113


Definition 2.5. The elementary symmetric polynomials are defined by

e1(x1, . . . , xn) =

n∑i=1

xi,

e2(x1, . . . , xn) =∑

1≤i1<i2<nxi1xi2 ,

e3(x1, . . . , xn) =∑

1≤i1<i2<i3<nxi1xi2xi3 ,

...

em(x1, . . . , xn) =∑

1≤i1<...<im<nxi1xi2xi3 · · ·xim ,

...

en(x1, . . . , xn) =x1x2 · · ·xn.

The elementary symmetric polynomials can be used to describe a wellknown relation between the roots of a polynomial and its coefficients oftenreferred to as Vieta’s formula.

Theorem 2.6 (Vieta’s formula). Suppose x1, . . . , xn are the n roots of apolynomial

xn + c1xn−1 + . . .+ cn.

Then ck = (−1)kek(x1, . . . , xn).

Definition 2.6. A power sum is an expression of the form

pk(x1, . . . , xn) =

n∑i=1

xki .

Theorem 2.7 (Newton–Girard formulae). The Newton–Girard formulaecan be expressed in many ways. For us the most useful version is the de-terminantal expressions. Let ek = ek(x1, . . . , xn) and pk = pk(x1, . . . , xn)denote the elementary symmetric polynomials and the power sums as inDefinitions 2.5 and 2.6. Then the power sum can be expressed in terms ofelementary symmetric polynomials in this way

pk =

∣∣∣∣∣∣∣∣∣∣∣∣∣

e1 1 0 · · · 0 02e2 e1 1 · · · 0 03e3 e2 e1 · · · 0 0

......

.... . .

......

(p− 1)en−1 en−2 en−3 · · · e1 1pen en−1 en−2 · · · e2 e1

∣∣∣∣∣∣∣∣∣∣∣∣∣.

Proof. See for example [198].

112


Definition 2.5. The elementary symmetric polynomials are defined by

e1(x1, . . . , xn) =

n∑i=1

xi,

e2(x1, . . . , xn) =∑

1≤i1<i2<nxi1xi2 ,

e3(x1, . . . , xn) =∑

1≤i1<i2<i3<nxi1xi2xi3 ,

...

em(x1, . . . , xn) =∑

1≤i1<...<im<nxi1xi2xi3 · · ·xim ,

...

en(x1, . . . , xn) =x1x2 · · ·xn.

The elementary symmetric polynomials can be used to describe a wellknown relation between the roots of a polynomial and its coefficients oftenreferred to as Vieta’s formula.

Theorem 2.6 (Vieta’s formula). Suppose x1, . . . , xn are the n roots of apolynomial

xn + c1xn−1 + . . .+ cn.

Then ck = (−1)kek(x1, . . . , xn).

Definition 2.6. A power sum is an expression of the form

pk(x1, . . . , xn) =

n∑i=1

xki .

Theorem 2.7 (Newton–Girard formulae). The Newton–Girard formulaecan be expressed in many ways. For us the most useful version is the de-terminantal expressions. Let ek = ek(x1, . . . , xn) and pk = pk(x1, . . . , xn)denote the elementary symmetric polynomials and the power sums as inDefinitions 2.5 and 2.6. Then the power sum can be expressed in terms ofelementary symmetric polynomials in this way

pk =

∣∣∣∣∣∣∣∣∣∣∣∣∣

e1 1 0 · · · 0 02e2 e1 1 · · · 0 03e3 e2 e1 · · · 0 0

......

.... . .

......


∣∣∣∣∣∣∣∣∣∣∣∣∣.

Proof. See for example [198].

112

114



Lemma 2.10. Using the following notation

tn(c1, c2, . . . , cm) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

cm cm−1 cm−2 · · · c22mn c1

1 cm cm−1 · · · c32m−2n c2

0 1 cm · · · c42m−4n c3

......

.... . .

......

0 0 0 · · · cm4n cm−1

0 0 0 · · · 1 2n cm

∣∣∣∣∣∣∣∣∣∣∣∣∣(95)

and tn(c) = 2cn then tn can be written

tn(c1, . . . , cp) =∑

r1+2r2+3r3+···+nrn=n

r1≥0, ..., rn≥0

n(r1 + r2 + · · ·+ rn − 1)

r1!r2! · · · rn!

p−1∏i=0

crip−i

and it obeys the recursive relation

tn(c1, . . . , cp) =2p

nc1 −

p∑i=2

ci+1 tn(cp−i+2, . . . , cp).

Proof. Comparing the expression for tn with the relations given in Theo-rem 2.7 it is clear that these relations are equivalent to the Newton-Girardformulae with some minor modifications.

Lemma 2.11. For even n and p the condition (84) can be rewritten as

−n tn(cn−p−2, cn−p, . . . , cn−2) = 1

where tn is defined by (95).

Proof. Note that the expression gp(x1, . . . , xn) =

n∑1

xpi = 1 is a power sum.

By Theorem 2.7 the following relation holds:

gp(x) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

e1 1 0 · · · 0 02e2 e1 1 · · · 0 03e3 e2 e1 · · · 0 0

......

.... . .

......


∣∣∣∣∣∣∣∣∣∣∣∣∣where ek is the k:th elementary symmetric polynomial of x1, . . . , xn. UsingVieta’s formula we can relate the elementary symmetric polynomials to thecoefficients of P (x) by noting that

P (x) = x2n +

n2−1∑j=1

c2jx2j =

n∑k=1

(−1)kekxn−k

or more compactly e2k = cn−2k.

113



Lemma 2.10. Using the following notation

tn(c1, c2, . . . , cm) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

cm cm−1 cm−2 · · · c22mn c1

1 cm cm−1 · · · c32m−2n c2

0 1 cm · · · c42m−4n c3

......

.... . .

......

0 0 0 · · · cm4n cm−1

0 0 0 · · · 1 2n cm

∣∣∣∣∣∣∣∣∣∣∣∣∣(95)

and tn(c) = 2cn then tn can be written

tn(c1, . . . , cp) =∑

r1+2r2+3r3+···+nrn=n

r1≥0, ..., rn≥0

n(r1 + r2 + · · ·+ rn − 1)

r1!r2! · · · rn!

p−1∏i=0

crip−i

and it obeys the recursive relation

tn(c1, . . . , cp) =2p

nc1 −

p∑i=2

ci+1 tn(cp−i+2, . . . , cp).

Proof. Comparing the expression for tn with the relations given in Theo-rem 2.7 it is clear that these relations are equivalent to the Newton-Girardformulae with some minor modifications.

Lemma 2.11. For even n and p the condition (84) can be rewritten as

−n tn(cn−p−2, cn−p, . . . , cn−2) = 1

where tn is defined by (95).

Proof. Note that the expression gp(x1, . . . , xn) =

n∑1

xpi = 1 is a power sum.

By Theorem 2.7 the following relation holds:

gp(x) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

e1 1 0 · · · 0 02e2 e1 1 · · · 0 03e3 e2 e1 · · · 0 0

......

.... . .

......


∣∣∣∣∣∣∣∣∣∣∣∣∣where ek is the k:th elementary symmetric polynomial of x1, . . . , xn. UsingVieta’s formula we can relate the elementary symmetric polynomials to thecoefficients of P (x) by noting that

P (x) = x2n +

n2−1∑j=1

c2jx2j =

n∑k=1

(−1)kekxn−k

or more compactly e2k = cn−2k.

113

115


With e2k = cn−2k and e2k+1 = 0 we get

gp(x) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 0 · · · 0 02cn−2 0 1 · · · 0 0

0 cn−2 0 · · · 0 04cn−4 0 cn−2 · · · 0 0

......

.... . .

......

0 cn−p−2 0 · · · 0 1pcn−p 0 cn−p−2 · · · cn−2 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣.

Using Laplace expansion on every other row gives

gp(x) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 0 0 · · · 0 02cn−2 0 1 0 · · · 0 0

0 cn−2 0 1 · · · 0 04cn−4 0 cn−2 0 · · · 0 0

......

......

. . ....

...0 c2 0 c4 · · · 0 1

pcn−p 0 c2 0 · · · cn−2 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= −

∣∣∣∣∣∣∣∣∣∣∣∣∣

2cn−2 1 0 · · · 0 00 0 1 · · · 0 0

4cn−4 cn−2 0 · · · 0 0...

......

. . ....

...0 0 c4 · · · 0 1pc0 c2 0 · · · cn−2 0

∣∣∣∣∣∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣∣∣∣∣∣

2cn−2 1 0 0 · · · 0 04cn−4 cn−2 1 0 · · · 0 0

0 0 0 1 · · · 0 0...

......

.... . .

......

0 0 0 0 · · · 0 1pc0 c2 c4 c6 · · · cn−2 0

∣∣∣∣∣∣∣∣∣∣∣∣∣= −

∣∣∣∣∣∣∣∣∣∣∣∣∣

2cn−2 1 0 · · · 0 04cn−4 cn−2 1 · · · 0 06cn−6 cn−4 cn−2 · · · 0 0

......

.... . .

......

(p− 2)c2 c4 c6 · · · cn−2 1pc0 c2 c4 · · · en−4 cn−2

∣∣∣∣∣∣∣∣∣∣∣∣∣

= −n

∣∣∣∣∣∣∣∣∣∣∣∣∣

cp cp−1 cp−2 · · · c2pn c1

1 cp cp−1 · · · c3p−2n c2

0 1 cp · · · c4p−4n c3

......

.... . .

......

0 0 0 · · · cp 4n cp−1

0 0 0 · · · 1 2n cp

∣∣∣∣∣∣∣∣∣∣∣∣∣= (−1)

p2n tn(c2, c4, . . . , cp)

Thus gp(x1, . . . , xn) = 1 is equivalent to −ntn(c2, c4, . . . , cp) = 1.

Lemma 2.12. The coefficients of the polynomial Q(x) in (92) can be ex-pressed using the coefficients of P (x) as follows

a2k−2 = (−1)k+1n2(n− 1)tn(cn−p+2k+2, . . . , cn−2), k = 1, 2, . . . ,p

2. (96)

114


With e2k = cn−2k and e2k+1 = 0 we get

gp(x) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 0 · · · 0 02cn−2 0 1 · · · 0 0

0 cn−2 0 · · · 0 04cn−4 0 cn−2 · · · 0 0

......

.... . .

......

0 cn−p−2 0 · · · 0 1pcn−p 0 cn−p−2 · · · cn−2 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣.

Using Laplace expansion on every other row gives

gp(x) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 0 0 · · · 0 02cn−2 0 1 0 · · · 0 0

0 cn−2 0 1 · · · 0 04cn−4 0 cn−2 0 · · · 0 0

......

......

. . ....

...0 c2 0 c4 · · · 0 1

pcn−p 0 c2 0 · · · cn−2 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= −

∣∣∣∣∣∣∣∣∣∣∣∣∣

2cn−2 1 0 · · · 0 00 0 1 · · · 0 0

4cn−4 cn−2 0 · · · 0 0...

......

. . ....

...0 0 c4 · · · 0 1pc0 c2 0 · · · cn−2 0

∣∣∣∣∣∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣∣∣∣∣∣

2cn−2 1 0 0 · · · 0 04cn−4 cn−2 1 0 · · · 0 0

0 0 0 1 · · · 0 0...

......

.... . .

......

0 0 0 0 · · · 0 1pc0 c2 c4 c6 · · · cn−2 0

∣∣∣∣∣∣∣∣∣∣∣∣∣= −

∣∣∣∣∣∣∣∣∣∣∣∣∣

2cn−2 1 0 · · · 0 04cn−4 cn−2 1 · · · 0 06cn−6 cn−4 cn−2 · · · 0 0

......

.... . .

......

(p− 2)c2 c4 c6 · · · cn−2 1pc0 c2 c4 · · · en−4 cn−2

∣∣∣∣∣∣∣∣∣∣∣∣∣

= −n

∣∣∣∣∣∣∣∣∣∣∣∣∣

cp cp−1 cp−2 · · · c2pn c1

1 cp cp−1 · · · c3p−2n c2

0 1 cp · · · c4p−4n c3

......

.... . .

......

0 0 0 · · · cp 4n cp−1

0 0 0 · · · 1 2n cp

∣∣∣∣∣∣∣∣∣∣∣∣∣= (−1)

p2n tn(c2, c4, . . . , cp)

Thus gp(x1, . . . , xn) = 1 is equivalent to −ntn(c2, c4, . . . , cp) = 1.

Lemma 2.12. The coefficients of the polynomial Q(x) in (92) can be ex-pressed using the coefficients of P (x) as follows

a2k−2 = (−1)k+1n2(n− 1)tn(cn−p+2k+2, . . . , cn−2), k = 1, 2, . . . ,p

2. (96)

114

116



Proof. By (94) we can write

a0

a2

a4...

ap−4

ap−2

=

cn−2 cn−4 cn−6 · · · cn−p−4pn cn−p−2

1 cn−2 cn−4 · · · cn−p−6p−2n cn−p

0 1 cn−2 · · · cn−p−8p−4n cn−p+2

......

.... . .

......

0 0 0 · · · cn−24n cn−4

0 0 0 · · · 1 2n cn−2

−1

−n(n− 1)00...00

.

and using Cramer’s rule we get ap−2k =det(Tn,p,k)

tn(cn−p−2, . . . , cn−2)where

Tn,p,k =

cn−2 cn−4 · · · cn−2k+2 −n(n− 1) cn−2k−2 · · · pn cn−p−2

1 cn−2 · · · cn−2k 0 cn−2k−4 · · · p−2n cn−p

0 1 · · · cn−2k−2 0 cn−2k−6 · · · p−4n cn−p+2

......

. . ....

...... · · ·

...0 0 · · · 0 0 0 · · · 4

n cn−4

0 0 · · · 0 0 0 · · · 2n cn−2

.

︸︷︷︸M

By moving the kth column to the first column and using Laplace expansiondet(Tk) can be rewritten on the form

det(Tn,p,k) =(−1)kn(n− 1)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 cn−2 · · · cn−2k

0 1 · · · cn−2k−2...

.... . .

...0 0 · · · 10 0 · · · 0 M0 0 · · · 0...

.... . .

...0 0 · · · 00 0 · · · 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= −n(n− 1)|M |

=− n(n− 1)

∣∣∣∣∣∣∣∣∣∣∣

cn−2 · · · cn−p+2kp−2kn cn−p+2k+2

1 · · · cn−p+2k+2p−2k−2

n cn−p+2k+4...

. . ....

...0 · · · cn−2

4n cn−4

0 · · · 1 2n cn−2

∣∣∣∣∣∣∣∣∣∣∣=(−1)kn(n− 1)tn(cn−p+2k+2, . . . , cn−2)

We can use Lemma 2.11 to see that tn(cn−p−2, . . . , cn−2) =−1

nand thus

ap−2k =det(Tn,p,k)

tn(cn−p−2, . . . , cn−2)= (−1)k+1n2(n− 1)tn(cn−p+2k+2, . . . , cn−2)

115



Proof. By (94) we can write

a0

a2

a4...

ap−4

ap−2

=

cn−2 cn−4 cn−6 · · · cn−p−4pn cn−p−2

1 cn−2 cn−4 · · · cn−p−6p−2n cn−p

0 1 cn−2 · · · cn−p−8p−4n cn−p+2

......

.... . .

......

0 0 0 · · · cn−24n cn−4

0 0 0 · · · 1 2n cn−2

−1

−n(n− 1)00...00

.

and using Cramer’s rule we get ap−2k =det(Tn,p,k)

tn(cn−p−2, . . . , cn−2)where

Tn,p,k =

cn−2 cn−4 · · · cn−2k+2 −n(n− 1) cn−2k−2 · · · pn cn−p−2

1 cn−2 · · · cn−2k 0 cn−2k−4 · · · p−2n cn−p

0 1 · · · cn−2k−2 0 cn−2k−6 · · · p−4n cn−p+2

......

. . ....

...... · · ·

...0 0 · · · 0 0 0 · · · 4

n cn−4

0 0 · · · 0 0 0 · · · 2n cn−2

.

︸︷︷︸M

By moving the kth column to the first column and using Laplace expansiondet(Tk) can be rewritten on the form

det(Tn,p,k) =(−1)kn(n− 1)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 cn−2 · · · cn−2k

0 1 · · · cn−2k−2...

.... . .

...0 0 · · · 10 0 · · · 0 M0 0 · · · 0...

.... . .

...0 0 · · · 00 0 · · · 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= −n(n− 1)|M |

=− n(n− 1)

∣∣∣∣∣∣∣∣∣∣∣

cn−2 · · · cn−p+2kp−2kn cn−p+2k+2

1 · · · cn−p+2k+2p−2k−2

n cn−p+2k+4...

. . ....

...0 · · · cn−2

4n cn−4

0 · · · 1 2n cn−2

∣∣∣∣∣∣∣∣∣∣∣=(−1)kn(n− 1)tn(cn−p+2k+2, . . . , cn−2)

We can use Lemma 2.11 to see that tn(cn−p−2, . . . , cn−2) =−1

nand thus

ap−2k =det(Tn,p,k)

tn(cn−p−2, . . . , cn−2)= (−1)k+1n2(n− 1)tn(cn−p+2k+2, . . . , cn−2)

115

117


Theorem 2.8. The non-zero coefficients, c2k, in P pn that solves (92) can befound by solving the polynomial equation system given by

2j(2j − 1)c2j +

(j−1∑k=0

(−1)p−2k+1n2(n− 1)tn(cn−2k+2, . . . , cn−2)

)+ n(n− 1)(n+ p− 2j)tn(cn−p+4, . . . , cn−2) = 0,

for j = 0, . . . , n2 − 1.

Proof. The equation system is found by using (96) to substitute ak in (93).

Using Lagrange multipliers directly gives a polynomial equation systemwith n equations while Theorem 2.8 gives n

2 equations.As an example we can consider the case n = 8, p = 4. Matching the

coefficients for (92) gives the system

a0c0 + 2c0 = 0,

a0c2 + a2c0 + 12c4 = 0,

30c6 + a0c4 +3

4a2c2 = 0,

56 + a0c6 +1

2a2c4 = 0,

a0 +1

4a2c6 = 0,

and rewriting the constraint that the points lie on S74 gives 2c2

6 − 4c4 = 0.In this case the expressions for a0 and a2 becomes quite simple

a0 = −112c6,

a2 = 448.

By resubstituting the expressions into the system, or using Theorem 2.8directly an equation systems for the c0, c2, c4 and c6 is given by

112c0c6 + 2c0 = 0,

−112c2c6 + 448c0 + 12c4 = 0,

−112c4c6 + 332c2 + 30c6 = 0,

−2c26 + 4c4 + 1 = 0.

The authors are not aware of any method that can be used to easily andreliably solve the system given by Theorem 2.8. In Table 2.2 results fora number of systems, both with even and odd n and various values for pare given. These were found by manually experimentation combined withcomputer aided symbolic computations.

116


Theorem 2.8. The non-zero coefficients, c2k, in P pn that solves (92) can befound by solving the polynomial equation system given by

2j(2j − 1)c2j +

(j−1∑k=0

(−1)p−2k+1n2(n− 1)tn(cn−2k+2, . . . , cn−2)

)+ n(n− 1)(n+ p− 2j)tn(cn−p+4, . . . , cn−2) = 0,

for j = 0, . . . , n2 − 1.

Proof. The equation system is found by using (96) to substitute ak in (93).

Using Lagrange multipliers directly gives a polynomial equation systemwith n equations while Theorem 2.8 gives n

2 equations.As an example we can consider the case n = 8, p = 4. Matching the

coefficients for (92) gives the system

a0c0 + 2c0 = 0,

a0c2 + a2c0 + 12c4 = 0,

30c6 + a0c4 +3

4a2c2 = 0,

56 + a0c6 +1

2a2c4 = 0,

a0 +1

4a2c6 = 0,

and rewriting the constraint that the points lie on S74 gives 2c2

6 − 4c4 = 0.In this case the expressions for a0 and a2 becomes quite simple

a0 = −112c6,

a2 = 448.

By resubstituting the expressions into the system, or using Theorem 2.8directly an equation systems for the c0, c2, c4 and c6 is given by

112c0c6 + 2c0 = 0,

−112c2c6 + 448c0 + 12c4 = 0,

−112c4c6 + 332c2 + 30c6 = 0,

−2c26 + 4c4 + 1 = 0.

The authors are not aware of any method that can be used to easily andreliably solve the system given by Theorem 2.8. In Table 2.2 results fora number of systems, both with even and odd n and various values for pare given. These were found by manually experimentation combined withcomputer aided symbolic computations.

116

118



n = 2

P 22 (x) = x2 − 1

2 , P 24 (x) = x2 − 1

2

√2, P 2

6 (x) = x2 − 223

2 , P 28 (x) = x2 − 2

34

2

n = 3

P 32 (x) = x3 − 1

2x, P 43 (x) = x3 − 1

2

√2x, P 3

6 (x) = x3 − 223

2 x, P 38 (x) = x2 − 2

34

2

n = 4

P 42 (x) = x4 − 1

2x2 + 1

48 , P 44 (x) = x4 −

√6

3 x2 + 1

12 ,

P 46 (x) = x4 − 1

4(√

33 + 1)13x2 + 1

96

(9−√

33)

(√

33 + 1)23

P 48 (x) = x4 −

√3

6 (30√

5− 30)14x2 + 1

120

(√5− 5

)√30√

5− 30

n = 5

P 52 (x) = x5 − 1

4x, P 54 (x) = x5 − 2

√5

5 x3 + 320x, P 5

6 (x) = x5 − 1013

2 x3 + 1023

20 x

P 58 (x) = x5 −

√10

10 (50√

13 + 10)14x3 + 1

1800

(5√

13− 55)√

50√

13 + 10

n = 6

P 62 (x) = x6 − 1

2x4 + 1

20x2 − 1

1800

P 64 (x) = x6 −

√50+20

√5

10 x4 +√

510 x

2 − (−4+2√

5)√

50+20√

5600

n = 7

P 72 (x) = x7 − 1

2x5 + 5

84x3 − 5

3528

P 74 (x) = x7 −

√1050+84

√109

42 x5 +(

121 +

√10942

)x3 − (−16+2

√109)√

105+84√

10910584

n = 8

P 82 (x) = x8 − 1

2x6 + 15

224x4 − 15

6272x2 + 15

1404928 ,

P 84 (x) = x8 −

√140+42

√6

14 x6 +(

328 + 3

√6

28

)x4

−(−(140+42

√6)

32

16464 + 29√

140+42√

62352

)x2 − 3

3136 +√

61568

Table 2.2: Polynomials, Pnp , whose roots give the coordinates of the extremepoints of the Vandermonde determinant on the sphere defined bythe p-norm in n dimensions.

117



n = 2

P 22 (x) = x2 − 1

2 , P 24 (x) = x2 − 1

2

√2, P 2

6 (x) = x2 − 223

2 , P 28 (x) = x2 − 2

34

2

n = 3

P 32 (x) = x3 − 1

2x, P 43 (x) = x3 − 1

2

√2x, P 3

6 (x) = x3 − 223

2 x, P 38 (x) = x2 − 2

34

2

n = 4

P 42 (x) = x4 − 1

2x2 + 1

48 , P 44 (x) = x4 −

√6

3 x2 + 1

12 ,

P 46 (x) = x4 − 1

4(√

33 + 1)13x2 + 1

96

(9−√

33)

(√

33 + 1)23

P 48 (x) = x4 −

√3

6 (30√

5− 30)14x2 + 1

120

(√5− 5

)√30√

5− 30

n = 5

P 52 (x) = x5 − 1

4x, P 54 (x) = x5 − 2

√5

5 x3 + 320x, P 5

6 (x) = x5 − 1013

2 x3 + 1023

20 x

P 58 (x) = x5 −

√10

10 (50√

13 + 10)14x3 + 1

1800

(5√

13− 55)√

50√

13 + 10

n = 6

P 62 (x) = x6 − 1

2x4 + 1

20x2 − 1

1800

P 64 (x) = x6 −

√50+20

√5

10 x4 +√

510 x

2 − (−4+2√

5)√

50+20√

5600

n = 7

P 72 (x) = x7 − 1

2x5 + 5

84x3 − 5

3528

P 74 (x) = x7 −

√1050+84

√109

42 x5 +(

121 +

√10942

)x3 − (−16+2

√109)√

105+84√

10910584

n = 8

P 82 (x) = x8 − 1

2x6 + 15

224x4 − 15

6272x2 + 15

1404928 ,

P 84 (x) = x8 −

√140+42

√6

14 x6 +(

328 + 3

√6

28

)x4

−(−(140+42

√6)

32

16464 + 29√

140+42√

62352

)x2 − 3

3136 +√

61568

Table 2.2: Polynomials, Pnp , whose roots give the coordinates of the extremepoints of the Vandermonde determinant on the sphere defined bythe p-norm in n dimensions.

117

119


2.3.6 Some results for cubes and intersections of planes

It can be noted that when p → ∞ then Sn−1p as defined in the previous

section will converge towards the cube.A similar technique to the described technique for surfaces implicitly

defined by a univariate polynomial can be employed on the cube. Themaximum value for the Vandermonde determinant on the cube [−1, 1]n hasbeen known for a long time (at least since [90]). Here we will show a shortderivation.

Theorem 2.9. The coordinates of the critical points of vn(x) on the cubexn ∈ [−1, 1]n are given by x1 = −1, xn = 1 and xi equal to the ith root ofPn−2(x) where Pn are the Legendre polynomials

Pn(x) = 2nn∑k=0

xk(n

k

)(n+k−12

n

)or some permutation of them.

Proof. It is easy to show that the coordinates −1 and +1 must be present inthe maxima points, if they were not then we could rescale the point so thatthe value of vn(x) is increased, which is not allowed. We may thus assumethe ordered sequence of coordinates

−1 = x1 < · · · < xn = +1.

The Vandermonde determinant then becomes

vn(x) = 2n−1∏i=2

(1 + xi)(1− xi)∏

1<i<j<n

(xj − xi).

and the partial derivatives become

∂vnxk

= vn(x)

1

xk + 1+

1

xk − 1+n−1∑i=2i 6=k

1

xk − xi

, 1 < k < n.

Using Lagrange multipliers the resulting equations system becomes

∂vnxk

= 0, k = 2, . . . , n− 1

and choosing f(x) =n−1∏k=2

(x − xk) gives that in each coordinate of a critical

point

1

xk + 1+

1

xk − 1+

1

2

f ′′(xk)

f ′(xk)= 0, 1 < k < n,

⇔ (1− x2)f ′′(xk) + 2xkf′(xk) = 0, 1 < k < n

118


2.3.6 Some results for cubes and intersections of planes

It can be noted that when p → ∞ then Sn−1p as defined in the previous

section will converge towards the cube.A similar technique to the described technique for surfaces implicitly

defined by a univariate polynomial can be employed on the cube. Themaximum value for the Vandermonde determinant on the cube [−1, 1]n hasbeen known for a long time (at least since [90]). Here we will show a shortderivation.

Theorem 2.9. The coordinates of the critical points of vn(x) on the cubexn ∈ [−1, 1]n are given by x1 = −1, xn = 1 and xi equal to the ith root ofPn−2(x) where Pn are the Legendre polynomials

Pn(x) = 2nn∑k=0

xk(n

k

)(n+k−12

n


Proof. It is easy to show that the coordinates −1 and +1 must be present inthe maxima points, if they were not then we could rescale the point so thatthe value of vn(x) is increased, which is not allowed. We may thus assumethe ordered sequence of coordinates

−1 = x1 < · · · < xn = +1.


vn(x) = 2n−1∏i=2

(1 + xi)(1− xi)∏

1<i<j<n

(xj − xi).


∂vnxk

= vn(x)

1

xk + 1+

1

xk − 1+n−1∑i=2i 6=k

1

xk − xi

, 1 < k < n.


∂vnxk

= 0, k = 2, . . . , n− 1



point

1

xk + 1+

1

xk − 1+

1

2

f ′′(xk)

f ′(xk)= 0, 1 < k < n,

⇔ (1− x2)f ′′(xk) + 2xkf′(xk) = 0, 1 < k < n

118

120



and thus the left hand side of the expression must form a polynomial thatcan be expressed as some multiple of f(x)

(1− x2)f ′′(x)− 2xf ′(x)− σf(x) = 0. (97)

The constant σ is found by considering the coefficient for xn−2:

(n− 2)(n− 3) + 2(n− 2)− σ = 0 ⇔ σ = (n− 2)(n− 1).

This gives us the differential equation that defines the Legendre polynomialPn−2(x) [2].

The technique above can also easily be used to find critical points on theintersection of two planes given by x1 = a and xn = b, b > a.

Theorem 2.10. The coordinates of the critical points of vn(x) on the in-tersection of two planes given by x1 = a and xn = b are given by xn−1 = a,

xn = b and xi is the ith root of Pn−2

(x−ab−a

)where Pn are the Legendre

polynomials

Pn(x) = 2nn∑k=0

xk(n

k

)(n+k−12

n


Proof. We assume the ordered sequence of coordinates

−1 = x1 < · · · < xn = +1.


vn(x) = (b− a)

n−1∏i=2

(a− xi)(b− xi)∏

1<i<j<n

(xj − xi).


∂vnxk

= vn(x)

1

xk − a+

1

xk − b+n−1∑i=2i 6=k

1

xk − xi

, 1 < k < n.


∂vnxk

= 0, k = 2, . . . , n− 1



point

1

xk − a+

1

xk − b+

1

2

f ′′(xk)

f ′(xk)= 0, 1 < k < n,

⇔ (1− x2)f ′′(xk) + 2xkf′(xk) = 0, 1 < k < n,

119




(1− x2)f ′′(x)− 2xf ′(x)− σf(x) = 0. (97)


(n− 2)(n− 3) + 2(n− 2)− σ = 0 ⇔ σ = (n− 2)(n− 1).

This gives us the differential equation that defines the Legendre polynomialPn−2(x) [2].

The technique above can also easily be used to find critical points on theintersection of two planes given by x1 = a and xn = b, b > a.

Theorem 2.10. The coordinates of the critical points of vn(x) on the in-tersection of two planes given by x1 = a and xn = b are given by xn−1 = a,

xn = b and xi is the ith root of Pn−2

(x−ab−a

)where Pn are the Legendre

polynomials

Pn(x) = 2nn∑k=0

xk(n

k

)(n+k−12

n


Proof. We assume the ordered sequence of coordinates

−1 = x1 < · · · < xn = +1.


vn(x) = (b− a)

n−1∏i=2

(a− xi)(b− xi)∏

1<i<j<n

(xj − xi).


∂vnxk

= vn(x)

1

xk − a+

1

xk − b+n−1∑i=2i 6=k

1

xk − xi

, 1 < k < n.


∂vnxk

= 0, k = 2, . . . , n− 1



point

1

xk − a+

1

xk − b+

1

2

f ′′(xk)

f ′(xk)= 0, 1 < k < n,

⇔ (1− x2)f ′′(xk) + 2xkf′(xk) = 0, 1 < k < n,

119

121



(x− a)(x− b)f ′′(x) + (2x− a− b)f ′(x)− σf(x) = 0.


(n− 2)(n− 3) + 2(n− 2)− σ = 0 ⇔ σ = (n− 2)(n− 1).

The resulting differential equation is

(x− a)(x− b)f ′′(x) + (2x− a− b)f ′(x)− (n− 2)(n− 1)f(x) = 0.

If we change variables according to y = x−ab−a and let g(y) = f(y(b− a) + a)

then the differential equation becomes

y(y − 1)g′′(y) + (2y − 1)g′(y)− (n− 1)(n− 2)g(y) = 0

which we can recognize as a special case of Euler’s hypergeometric differen-tial equation whose solution can be expressed as

g(y) = c ·2F1(1− n, n+ 2; 1; y), for some arbitrary c ∈ R,

where 2F1 is the hypergeometric function [2]. In this case the hypergeometricfunction is a polynomial and relates to the Legendre polynomials as follows

2F1(1− n, n+ 2; 1; y) = n!Pn−2(y)

thus it is sufficient to consider the roots of Pn−2

(x−ab−a

).

2.3.7 Optimising the probability density function of theeigenvalues of the Wishart matrix

This section is based on Section 5 of Paper D

Here we will show an example of how the results in Section 2.2 canbe applied to find the extreme points of the eigenvalue distribution of theensembles discussed in Section 1.1.7.

Lemma 2.13. Suppose we have a Wishart distributed matrix W with theprobability density function of its eigenvalues given by

P(λ) = Cnvn(λ)m exp

(−β

2

n∑k=1

P (λk)

)(98)

where Cn is a normalising constant, m is a positive integer, β > 1 and Pis a polynomial with real coefficients. Then the vector of eigenvalues of Wwill lie on the surface defined by

n∑k=1

P (λk) = Tr(P (W)). (99)

120



(x− a)(x− b)f ′′(x) + (2x− a− b)f ′(x)− σf(x) = 0.


(n− 2)(n− 3) + 2(n− 2)− σ = 0 ⇔ σ = (n− 2)(n− 1).

The resulting differential equation is

(x− a)(x− b)f ′′(x) + (2x− a− b)f ′(x)− (n− 2)(n− 1)f(x) = 0.

If we change variables according to y = x−ab−a and let g(y) = f(y(b− a) + a)

then the differential equation becomes

y(y − 1)g′′(y) + (2y − 1)g′(y)− (n− 1)(n− 2)g(y) = 0

which we can recognize as a special case of Euler’s hypergeometric differen-tial equation whose solution can be expressed as

g(y) = c ·2F1(1− n, n+ 2; 1; y), for some arbitrary c ∈ R,

where 2F1 is the hypergeometric function [2]. In this case the hypergeometricfunction is a polynomial and relates to the Legendre polynomials as follows

2F1(1− n, n+ 2; 1; y) = n!Pn−2(y)

thus it is sufficient to consider the roots of Pn−2

(x−ab−a

).

2.3.7 Optimising the probability density function of theeigenvalues of the Wishart matrix

This section is based on Section 5 of Paper D

Here we will show an example of how the results in Section 2.2 canbe applied to find the extreme points of the eigenvalue distribution of theensembles discussed in Section 1.1.7.

Lemma 2.13. Suppose we have a Wishart distributed matrix W with theprobability density function of its eigenvalues given by

P(λ) = Cnvn(λ)m exp

(−β

2

n∑k=1

P (λk)

)(98)

where Cn is a normalising constant, m is a positive integer, β > 1 and Pis a polynomial with real coefficients. Then the vector of eigenvalues of Wwill lie on the surface defined by

n∑k=1

P (λk) = Tr(P (W)). (99)

120

122



Proof. Since W is symmetric by Lemma 1.2 then it will also have real eigen-values. By Lemma 1.1

n∑k=1

P (λk) = Tr(P (W))

and thus the point given by λ = (λ1, λ2, . . . , λn) will be on the surfacedefined by

n∑k=1

P (λk) = Tr(P (W)).

To find the maximum values we can use the method of Lagrange multi-pliers and find eigenvectors such that

∂P∂λk

= η∂

∂λk

(Tr(P (W))−

n∑k=1

P (λk)

)= −ηdP (λk)

dλk, k = 1, . . . , n,

where η is some real-valued constant. Computing the left-hand side gives

∂P(β)

∂λk= P(λ)

−β2

dP (λk)

dλk+

n∑i=1i 6=k

m

λk − λi

.

Thus the stationary points of (98) on the surface given by (99) are thesolution to the equation system

P(λ)

−β2

dP (λk)

dλk+

n∑i=1i 6=k

m

λk − λi

= −ηdP (λk)

dλk, k = 1, . . . , n.

If we denote the value of P in a stationary point with Ps then the systemabove can be rewritten as

n∑i=1i 6=k

1

λk − λi=

1

m

(β

2− η

Ps

)dP (λk)

dλk= ρ

dP (λk)

dλk, k = 1, . . . , n. (100)

The equation system described by (100) appears when one tries to op-timize the Vandermonde determinant on a surface defined by a univariatepolynomial. This equation system can be rewritten as an ordinary differen-tial equation. For more details see Section 2.3

Consider the polynomial

f(λ) =n∏i=1

(λ− λi)

121



Proof. Since W is symmetric by Lemma 1.2 then it will also have real eigen-values. By Lemma 1.1

n∑k=1

P (λk) = Tr(P (W))

and thus the point given by λ = (λ1, λ2, . . . , λn) will be on the surfacedefined by

n∑k=1

P (λk) = Tr(P (W)).

To find the maximum values we can use the method of Lagrange multi-pliers and find eigenvectors such that

∂P∂λk

= η∂

∂λk

(Tr(P (W))−

n∑k=1

P (λk)

)= −ηdP (λk)

dλk, k = 1, . . . , n,

where η is some real-valued constant. Computing the left-hand side gives

∂P(β)

∂λk= P(λ)

−β2

dP (λk)

dλk+

n∑i=1i 6=k

m

λk − λi

.

Thus the stationary points of (98) on the surface given by (99) are thesolution to the equation system

P(λ)

−β2

dP (λk)

dλk+

n∑i=1i 6=k

m

λk − λi

= −ηdP (λk)

dλk, k = 1, . . . , n.

If we denote the value of P in a stationary point with Ps then the systemabove can be rewritten as

n∑i=1i 6=k

1

λk − λi=

1

m

(β

2− η

Ps

)dP (λk)

dλk= ρ

dP (λk)

dλk, k = 1, . . . , n. (100)

The equation system described by (100) appears when one tries to op-timize the Vandermonde determinant on a surface defined by a univariatepolynomial. This equation system can be rewritten as an ordinary differen-tial equation. For more details see Section 2.3

Consider the polynomial

f(λ) =n∏i=1

(λ− λi)

121

123


and note that1

2

f ′′(λj)

f ′(λj)=

n∑i=1i 6=j

1

λj − λi.

Thus in each of the extreme points we will have the relation

d2f

dλ2

∣∣∣∣λ=λj

− 2ρdP

dλ

∣∣∣∣λ=λj

df

dλ

∣∣∣∣λ=λj

= 0, j = 1, 2, . . . , n

for some ρ ∈ R. Since each λj is a root of f(λ) we see that the left handside in the differential equation must be a polynomial with the same rootsas f(λ), thus we can conclude that for any λ ∈ R

d2f

dλ− 2ρ

dP

dλ

df

dλ−Q(λ)f(λ) = 0 (101)

where Q is a polynomial of degree (deg(p)− 2).Consider the β ensemble described by (16). For this ensemble the poly-

nomial that defines the surface that the eigenvalues will be on is p(λ) = λ2.Thus by Lemma 2.13 the surface becomes a sphere with radius

√Tr(W2).

The solution to the equation system given by (100) was found in Section2.2. The solution is given as the roots of a polynomial, in this case thesolution can be written as the roots of the rescaled Hermite polynomials,the explicit expression for the polynomial whose roots give the maximumpoints is

f(x) = Hn

((n− 1

2(r21 − 2r0)

) 12 (x+ r1)

2

)

= n!

bn2 c∑i=0

(−1)i

i!

(n− 1

2(r21 − 2r0)

)n−2i2 (x+ r1)n−2i

(n− 2i)!(102)

where Hn denotes the nth (physicist) Hermite polynomial [2].The solution on the unit sphere can then be used to find the vector of

eigenvalues that maximizes the probability density function P(λ) given by(16). Since rescaling the vector of eigenvalues affects the probability densitydepending on the length of the original vector in the following way

P(cλ) = cn(n−1)m

2 exp

(β

2(1− c2)|λ|2

)P(λ)

the unit sphere solution can be rescaled so that it ends up on the appropriatesphere.

122


and note that1

2

f ′′(λj)

f ′(λj)=

n∑i=1i 6=j

1

λj − λi.

Thus in each of the extreme points we will have the relation

d2f

dλ2

∣∣∣∣λ=λj

− 2ρdP

dλ

∣∣∣∣λ=λj

df

dλ

∣∣∣∣λ=λj

= 0, j = 1, 2, . . . , n

for some ρ ∈ R. Since each λj is a root of f(λ) we see that the left handside in the differential equation must be a polynomial with the same rootsas f(λ), thus we can conclude that for any λ ∈ R

d2f

dλ− 2ρ

dP

dλ

df

dλ−Q(λ)f(λ) = 0 (101)

where Q is a polynomial of degree (deg(p)− 2).Consider the β ensemble described by (16). For this ensemble the poly-

nomial that defines the surface that the eigenvalues will be on is p(λ) = λ2.Thus by Lemma 2.13 the surface becomes a sphere with radius

√Tr(W2).

The solution to the equation system given by (100) was found in Section2.2. The solution is given as the roots of a polynomial, in this case thesolution can be written as the roots of the rescaled Hermite polynomials,the explicit expression for the polynomial whose roots give the maximumpoints is

f(x) = Hn

((n− 1

2(r21 − 2r0)

) 12 (x+ r1)

2

)

= n!

bn2 c∑i=0

(−1)i

i!

(n− 1

2(r21 − 2r0)

)n−2i2 (x+ r1)n−2i

(n− 2i)!(102)

where Hn denotes the nth (physicist) Hermite polynomial [2].The solution on the unit sphere can then be used to find the vector of

eigenvalues that maximizes the probability density function P(λ) given by(16). Since rescaling the vector of eigenvalues affects the probability densitydepending on the length of the original vector in the following way

P(cλ) = cn(n−1)m

2 exp

(β

2(1− c2)|λ|2

)P(λ)

the unit sphere solution can be rescaled so that it ends up on the appropriatesphere.

122

124

Chapter 3

Approximation ofelectrostatic dischargecurrents using theanalytically extendedfunction

This chapter is based on Papers E, F and G




Chapter 3

Approximation ofelectrostatic dischargecurrents using theanalytically extendedfunction

This chapter is based on Papers E, F and G




125

126

3.1. THE ANALYTICALLY EXTENDED FUNCTION (AEF)

3.1 The analytically extended function (AEF)

In this section we consider least square approximation using a particularfunction we call the power-exponential function, as a basic component.

Definition 3.1. Here we will refer to the function defined by (103) as thepower-exponential function,

x(β; t) =(te1−t)β , 0 ≤ t. (103)

For non-negative values of t and β the power-exponential function hasa steeply rising initial part followed by a more slowly decaying part, seeFigure 3.1. This makes it qualitatively similar to several functions thatare popular for approximation of important phenomena in different fieldssuch as approximation of lightning discharge currents and pharmacokinetics.Examples include the biexponential function [38], [256], the Heidler function[117] and the Pulse function [299].

Figure 3.1: An illustration of how the steepness of the power exponential func-tion varies with β.

The power-exponential function has been used in other applications, forexample to model attack rate of predatory fish, see [232,233].

Here we examine linear combinations of piecewise power exponentialfunctions that will be used in later sections to approximate electrostaticdischarge current functions.

125


3.1 The analytically extended function (AEF)

In this section we consider least square approximation using a particularfunction we call the power-exponential function, as a basic component.

Definition 3.1. Here we will refer to the function defined by (103) as thepower-exponential function,

x(β; t) =(te1−t)β , 0 ≤ t. (103)

For non-negative values of t and β the power-exponential function hasa steeply rising initial part followed by a more slowly decaying part, seeFigure 3.1. This makes it qualitatively similar to several functions thatare popular for approximation of important phenomena in different fieldssuch as approximation of lightning discharge currents and pharmacokinetics.Examples include the biexponential function [38], [256], the Heidler function[117] and the Pulse function [299].

Figure 3.1: An illustration of how the steepness of the power exponential func-tion varies with β.

The power-exponential function has been used in other applications, forexample to model attack rate of predatory fish, see [232,233].

Here we examine linear combinations of piecewise power exponentialfunctions that will be used in later sections to approximate electrostaticdischarge current functions.

125

127


3.1.1 The p-peak analytically extended function

This section is based on Section 2 of Paper E

The p -peaked AEF is constructed using the power exponential functiongiven in Definition 3.1. In order to get a function with multiple peaks andwhere the steepness of the rise between each peak as well as the slope ofthe decaying part is not dependent on each other, we define the analyti-cally extended function (AEF) as a function that consist of piecewise linearcombinations of the power exponential function that has been scaled andtranslated so that the resulting function is continuous. With a given differ-ence in height between subsequent peaks Im1 , Im2 , . . . , Imp , correspondingtimes tm1 , tm2 , . . . , tmp , integers nq > 0, real values βq,k, ηq,k, 1 ≤ q ≤ p+1,1 ≤ k ≤ nq such that the sum over k of ηq,k is equal to one, the p -peakedAEF i(t) is given by (104).

Definition 3.2. Given Imq ∈ R and tmq ∈ R, q = 1, 2, . . . , p such thattm0 = 0 < tm1 < tm2 < . . . < tmp along with ηq,k, βq,k ∈ R and 0 < nq ∈ Z

for q = 1, 2, . . . , p+ 1, k = 1, 2, . . . , nq such that

nq∑k=1

ηq,k = 1.

The analytically extended function (AEF), i(t), with p peaks is defined as

i(t)=

(q−1∑k=1

Imk

)+Imq

nq∑k=1

ηq,kxq(t)β2q,k+1, tmq−1≤ t ≤ tmq , 1≤q≤p,(

p∑k=1

Imk

) np+1∑k=1

ηp+1,kxp+1(t)β2p+1,k , tmp ≤ t,

(104)

where

xq(t) =

t− tmq−1

∆tmqexp

(tmq − t∆tmq

), 1 ≤ q ≤ p,

t

tmqexp

(1− t

tmq

), q = p+ 1,

and ∆tmq = tmq − tmq−1 .

Sometimes the notation i(t;β,η) with

β =[β1,1 β1,2 . . . βq,k . . . βp+1,np+1

],

η =[η1,1 η1,2 . . . ηq,k . . . ηp+1,np+1

],

will be used to clarify what the particular parameters for a certain AEF are.

Remark 3.1. The p -peak AEF can be written more compactly if we intro-

126


3.1.1 The p-peak analytically extended function

This section is based on Section 2 of Paper E

The p -peaked AEF is constructed using the power exponential functiongiven in Definition 3.1. In order to get a function with multiple peaks andwhere the steepness of the rise between each peak as well as the slope ofthe decaying part is not dependent on each other, we define the analyti-cally extended function (AEF) as a function that consist of piecewise linearcombinations of the power exponential function that has been scaled andtranslated so that the resulting function is continuous. With a given differ-ence in height between subsequent peaks Im1 , Im2 , . . . , Imp , correspondingtimes tm1 , tm2 , . . . , tmp , integers nq > 0, real values βq,k, ηq,k, 1 ≤ q ≤ p+1,1 ≤ k ≤ nq such that the sum over k of ηq,k is equal to one, the p -peakedAEF i(t) is given by (104).

Definition 3.2. Given Imq ∈ R and tmq ∈ R, q = 1, 2, . . . , p such thattm0 = 0 < tm1 < tm2 < . . . < tmp along with ηq,k, βq,k ∈ R and 0 < nq ∈ Z

for q = 1, 2, . . . , p+ 1, k = 1, 2, . . . , nq such that

nq∑k=1

ηq,k = 1.

The analytically extended function (AEF), i(t), with p peaks is defined as

i(t)=

(q−1∑k=1

Imk

)+Imq

nq∑k=1

ηq,kxq(t)β2q,k+1, tmq−1≤ t ≤ tmq , 1≤q≤p,(

p∑k=1

Imk

) np+1∑k=1

ηp+1,kxp+1(t)β2p+1,k , tmp ≤ t,

(104)

where

xq(t) =

t− tmq−1

∆tmqexp

(tmq − t∆tmq

), 1 ≤ q ≤ p,

t

tmqexp

(1− t

tmq

), q = p+ 1,

and ∆tmq = tmq − tmq−1 .

Sometimes the notation i(t;β,η) with

β =[β1,1 β1,2 . . . βq,k . . . βp+1,np+1

],

η =[η1,1 η1,2 . . . ηq,k . . . ηp+1,np+1

],

will be used to clarify what the particular parameters for a certain AEF are.

Remark 3.1. The p -peak AEF can be written more compactly if we intro-

126

128


duce the vectors

ηq = [ηq,1 ηq,2 . . . ηq,nq ]>, (105)

xq(t) =

[xq(t)

β2q,1+1 xq(t)

β2q,2+1 . . . xq(t)

β2q,nq

+1]>, 1 ≤ q ≤ p,[

xq(t)β2q,1 xq(t)

β2q,2 . . . xq(t)

β2q,nq

]>, q = p+ 1.

(106)

The more compact form is

i(t) =

(q−1∑k=1

Imk

)+ Imq · η>q xq(t), tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,(

q∑k=1

Imk

)· η>q xq(t), tmq ≤ t, q = p+ 1.

(107)

If the AEF is used to model an electrical current, than the derivativeof the AEF determines the induced electrical voltage in conductive loops inthe lightning field. For this reason it is desirable to guarantee that the firstderivative of the AEF is continuous.

Since the AEF is a linear function of elementary functions its derivativecan be found using standard methods.

Theorem 3.1. The derivative of the p -peak AEF is

di(t)

dt=

Imq

tmq − tt− tmq−1

xq(t)

∆tmqη>q Bq xq(t), tmq−1≤ t ≤ tmq , 1 ≤ q ≤ p,

Imqxq(t)

t

tmq − ttmq

η>q Bq xq(t), tmq ≤ t, q = p+ 1,

(108)where

Bq =

β2q,1 + 1 0 . . . 0

0 β2q,2 + 1 . . . 0

......

. . ....

0 0 . . . β2q,nq + 1

,

Bp+1 =

β2p+1,1 0 . . . 0

0 β2p+1,2 . . . 0

......

. . ....

0 0 . . . β2p+1,np+1

,for 1 ≤ q ≤ p.

Proof. From the definition of the AEF (see (104)) and the derivative of thepower exponential function (103) given by

d

dtx(β; t) = β(1− t)tβ−1eβ(1−t),

127


duce the vectors

ηq = [ηq,1 ηq,2 . . . ηq,nq ]>, (105)

xq(t) =

[xq(t)

β2q,1+1 xq(t)

β2q,2+1 . . . xq(t)

β2q,nq

+1]>, 1 ≤ q ≤ p,[

xq(t)β2q,1 xq(t)

β2q,2 . . . xq(t)

β2q,nq

]>, q = p+ 1.

(106)

The more compact form is

i(t) =

(q−1∑k=1

Imk

)+ Imq · η>q xq(t), tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,(

q∑k=1

Imk

)· η>q xq(t), tmq ≤ t, q = p+ 1.

(107)

If the AEF is used to model an electrical current, than the derivativeof the AEF determines the induced electrical voltage in conductive loops inthe lightning field. For this reason it is desirable to guarantee that the firstderivative of the AEF is continuous.

Since the AEF is a linear function of elementary functions its derivativecan be found using standard methods.

Theorem 3.1. The derivative of the p -peak AEF is

di(t)

dt=

Imq

tmq − tt− tmq−1

xq(t)

∆tmqη>q Bq xq(t), tmq−1≤ t ≤ tmq , 1 ≤ q ≤ p,

Imqxq(t)

t

tmq − ttmq

η>q Bq xq(t), tmq ≤ t, q = p+ 1,

(108)where

Bq =

β2q,1 + 1 0 . . . 0

0 β2q,2 + 1 . . . 0

......

. . ....

0 0 . . . β2q,nq + 1

,

Bp+1 =

β2p+1,1 0 . . . 0

0 β2p+1,2 . . . 0

......

. . ....

0 0 . . . β2p+1,np+1

,for 1 ≤ q ≤ p.

Proof. From the definition of the AEF (see (104)) and the derivative of thepower exponential function (103) given by

d

dtx(β; t) = β(1− t)tβ−1eβ(1−t),

127

129


expression (108) can easily be derived since differentiation is a linear oper-ation and the result can be rewritten in the compact form analogously to(107).

Illustration of the AEF function and its derivative for various values ofβq,k-parameters is shown in Figure 3.2.

Figure 3.2: Illustration of the AEF (solid line) and its derivative (dashedline) with different βq,k-parameters but the same Imq andtmq . (a) 0 < βq,k < 1, (b) 4 < βq,k < 5,(c) 12 < βq,k < 13, (d) a mixture of large and small βq,k-parameters.

Lemma 3.1. The AEF is continuous and at each tmq the derivative is equalto zero.

Proof. Within each interval tmq−1 ≤ t ≤ tmq the AEF is a linear combinationof continuous functions and at each tmq the function will approach the samevalue from both directions unless all ηq,k ≤ 0, but if all ηq,k ≤ 0 thennq∑k=1

ηq,k 6= 1.

Noting that for any diagonal matrix B the expression

η>q B xq(t) =

nq∑k=1

ηq,kBkkxq(t)β2q,k+1, 1 ≤ q ≤ p,

128


expression (108) can easily be derived since differentiation is a linear oper-ation and the result can be rewritten in the compact form analogously to(107).

Illustration of the AEF function and its derivative for various values ofβq,k-parameters is shown in Figure 3.2.

Figure 3.2: Illustration of the AEF (solid line) and its derivative (dashedline) with different βq,k-parameters but the same Imq andtmq . (a) 0 < βq,k < 1, (b) 4 < βq,k < 5,(c) 12 < βq,k < 13, (d) a mixture of large and small βq,k-parameters.

Lemma 3.1. The AEF is continuous and at each tmq the derivative is equalto zero.

Proof. Within each interval tmq−1 ≤ t ≤ tmq the AEF is a linear combinationof continuous functions and at each tmq the function will approach the samevalue from both directions unless all ηq,k ≤ 0, but if all ηq,k ≤ 0 thennq∑k=1

ηq,k 6= 1.

Noting that for any diagonal matrix B the expression

η>q B xq(t) =

nq∑k=1

ηq,kBkkxq(t)β2q,k+1, 1 ≤ q ≤ p,

128

130


is well-defined and that the equivalent statement holds for q = p and consid-ering (108) it is easy to see that the factor (tmq − t) in the derivative ensuresthat the derivative is zero every time t = tmq .

When interpolating a waveform with p peaks it is natural to require thatthere will not appear new peaks between the chosen peaks. This correspondsto requiring monotonicity in each interval. One way to achieve this is givenin Lemma 3.2.

Lemma 3.2. If ηq,k ≥ 0, k = 1, . . . , nq the AEF, i(t), is strictly monotonicon the interval tmq−1 < t < tmq .

Proof. The AEF will be strictly monotonic in an interval if the derivativehas the same sign everywhere in the interval. That this is the case followsfrom (108) since every term in η>q Bq xq(t) is non-negative if ηq,k ≥ 0, k =1, . . . , nq, so the sign of the derivative it determined by Imq .

If we allow some of the ηq,k-parameters to be negative, the derivativecan change sign and the function might get an extra peak between twoother peaks, see Figure 3.3.

Figure 3.3: An example of a two-peaked AEF where some of the ηq,k-parameters are negative, so that it has points where the first deriva-tive changes sign between two peaks. The solid line is the AEFand the dashed lines is the derivative of the AEF.

The integral of the electric current represents the charge flow. Unlikethe Heidler function the integral of the AEF is relatively straightforward tofind. How to do this is detailed in Lemma 3.3, Lemma 3.4, Theorem 3.2,and Theorem 3.3.

Lemma 3.3. For any tmq−1 ≤ t0 ≤ t1 ≤ tmq , 1 ≤ q ≤ p,∫ t1

t0

xq(t)β dt =

eβ

ββ+1∆γ

(β + 1,

t1 − tmqβ∆tmq

,t0 − tmqβ∆tmq

)(109)

129


is well-defined and that the equivalent statement holds for q = p and consid-ering (108) it is easy to see that the factor (tmq − t) in the derivative ensuresthat the derivative is zero every time t = tmq .

When interpolating a waveform with p peaks it is natural to require thatthere will not appear new peaks between the chosen peaks. This correspondsto requiring monotonicity in each interval. One way to achieve this is givenin Lemma 3.2.

Lemma 3.2. If ηq,k ≥ 0, k = 1, . . . , nq the AEF, i(t), is strictly monotonicon the interval tmq−1 < t < tmq .

Proof. The AEF will be strictly monotonic in an interval if the derivativehas the same sign everywhere in the interval. That this is the case followsfrom (108) since every term in η>q Bq xq(t) is non-negative if ηq,k ≥ 0, k =1, . . . , nq, so the sign of the derivative it determined by Imq .

If we allow some of the ηq,k-parameters to be negative, the derivativecan change sign and the function might get an extra peak between twoother peaks, see Figure 3.3.

Figure 3.3: An example of a two-peaked AEF where some of the ηq,k-parameters are negative, so that it has points where the first deriva-tive changes sign between two peaks. The solid line is the AEFand the dashed lines is the derivative of the AEF.

The integral of the electric current represents the charge flow. Unlikethe Heidler function the integral of the AEF is relatively straightforward tofind. How to do this is detailed in Lemma 3.3, Lemma 3.4, Theorem 3.2,and Theorem 3.3.

Lemma 3.3. For any tmq−1 ≤ t0 ≤ t1 ≤ tmq , 1 ≤ q ≤ p,∫ t1

t0

xq(t)β dt =

eβ

ββ+1∆γ

(β + 1,

t1 − tmqβ∆tmq

,t0 − tmqβ∆tmq

)(109)

129

131


with ∆tmq = tmq − tmq−1 and

∆γ(β, t0, t1) = γ (β + 1, βt1)− γ (β + 1, βt0) ,

where

γ(β, t) =

∫ t

0τβ−1e−τ dτ

is the lower incomplete Gamma function [2].

If t0 = tmq−1 and t1 = tmq then∫ tmq

tmq−1

xq(t)β dt =

eβ

ββ+1γ (β + 1, β) . (110)

Proof.∫ t1

t0

xq(t)β dt =

∫ t1

t0

(t− tmq∆tmq

exp

(1−

t− tmq∆tmq

))βdt

=eβ−1

ββ

∫ t1

t0

(βt− tmq∆tmq

)βexp

(1− β

t− tmq∆tmq

)dt.

Changing variables according to τ = βt−tmq∆tmq

gives

∫ t1

t0

xq(t)β dt =

eβ

ββ+1

∫ τ1

τ0

τβe−τ dt =

=eβ

ββ+1(γ(β + 1, τ1)− γ(β + 1, τ0))

=eβ

ββ+1∆γ(β + 1, τ1, τ0)

=eβ

ββ+1∆γ

(β + 1, β

t1 − tmq∆tmq

, βt0 − tmq

∆tmq

).

When t0 = tmq−1 and t1 = tmq then∫ t1

t0

xq(t)β dt =

eβ

ββ+1∆γ (β + 1, β)

and with γ(β + 1, 0) = 0 we get (110).

Lemma 3.4. For any tmq−1 ≤ t0 ≤ t1 ≤ tmq , 1 ≤ q ≤ p,

∫ t1

t0

i(t) dt = (t1 − t0)

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k gq(t1, t0), (111)

130


with ∆tmq = tmq − tmq−1 and

∆γ(β, t0, t1) = γ (β + 1, βt1)− γ (β + 1, βt0) ,

where

γ(β, t) =

∫ t

0τβ−1e−τ dτ

is the lower incomplete Gamma function [2].

If t0 = tmq−1 and t1 = tmq then∫ tmq

tmq−1

xq(t)β dt =

eβ

ββ+1γ (β + 1, β) . (110)

Proof.∫ t1

t0

xq(t)β dt =

∫ t1

t0

(t− tmq∆tmq

exp

(1−

t− tmq∆tmq

))βdt

=eβ−1

ββ

∫ t1

t0

(βt− tmq∆tmq

)βexp

(1− β

t− tmq∆tmq

)dt.

Changing variables according to τ = βt−tmq∆tmq

gives

∫ t1

t0

xq(t)β dt =

eβ

ββ+1

∫ τ1

τ0

τβe−τ dt =

=eβ

ββ+1(γ(β + 1, τ1)− γ(β + 1, τ0))

=eβ

ββ+1∆γ(β + 1, τ1, τ0)

=eβ

ββ+1∆γ

(β + 1, β

t1 − tmq∆tmq

, βt0 − tmq

∆tmq

).

When t0 = tmq−1 and t1 = tmq then∫ t1

t0

xq(t)β dt =

eβ

ββ+1∆γ (β + 1, β)

and with γ(β + 1, 0) = 0 we get (110).

Lemma 3.4. For any tmq−1 ≤ t0 ≤ t1 ≤ tmq , 1 ≤ q ≤ p,

∫ t1

t0

i(t) dt = (t1 − t0)

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k gq(t1, t0), (111)

130

132


where

gq(t1, t0) =eβ

2q,k(

β2q,k + 1

)β2q,k+1

∆γ

(β2q,k + 2,

t1 − tmq−1

∆tmq,t0 − tmq−1

∆tmq

)

with ∆γ(β, t0, t1) defined as in (109).

Proof.∫ t1

t0

i(t) dt =

∫ t1

t0

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,kxq(t)β2q,k+1 dt

= (t1 − t0)

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k

∫ t1

t0

xq(t)β2q,k+1 dt

= (t1 − t0)

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k gq(t0, t1).

Theorem 3.2. If tma−1 ≤ ta ≤ tma, tmb−1≤ tb ≤ tmb and 0 ≤ ta ≤ tb ≤ tmp

then∫ tb

ta

i(t) dt = (tma − ta)

(a−1∑k=1

Imk

)+ Ima

na∑k=1

ηa,k ga(ta, tma)

+b−1∑

q=a+1

(∆tmq

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k g(β2q,k + 1

))

+ (tb − tmb)

(b−1∑k=1

Imk

)+ Imb

nb∑k=1

ηb,k gb(tmb , tb), (112)

where gq(t0, t1) is defined as in Lemma 3.4 and

g(β) =eβ

ββ+1γ (β + 1, β) .

Proof. This theorem follows from integration being linear and Lemma 3.4.

Theorem 3.3. For tmp ≤ t0 < t1 <∞ the integral of the AEF is∫ t1

t0

i(t) dt =

(p∑

k=1

Imk

) np+1∑k=1

ηp+1,k gp+1(t1, t0), (113)

where gq(t0, t1) is defined as in Lemma 3.4.

131


where

gq(t1, t0) =eβ

2q,k(

β2q,k + 1

)β2q,k+1

∆γ

(β2q,k + 2,

t1 − tmq−1

∆tmq,t0 − tmq−1

∆tmq

)

with ∆γ(β, t0, t1) defined as in (109).

Proof.∫ t1

t0

i(t) dt =

∫ t1

t0

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,kxq(t)β2q,k+1 dt

= (t1 − t0)

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k

∫ t1

t0

xq(t)β2q,k+1 dt

= (t1 − t0)

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k gq(t0, t1).

Theorem 3.2. If tma−1 ≤ ta ≤ tma, tmb−1≤ tb ≤ tmb and 0 ≤ ta ≤ tb ≤ tmp

then∫ tb

ta

i(t) dt = (tma − ta)

(a−1∑k=1

Imk

)+ Ima

na∑k=1

ηa,k ga(ta, tma)

+b−1∑

q=a+1

(∆tmq

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k g(β2q,k + 1

))

+ (tb − tmb)

(b−1∑k=1

Imk

)+ Imb

nb∑k=1

ηb,k gb(tmb , tb), (112)

where gq(t0, t1) is defined as in Lemma 3.4 and

g(β) =eβ

ββ+1γ (β + 1, β) .


Theorem 3.3. For tmp ≤ t0 < t1 <∞ the integral of the AEF is∫ t1

t0

i(t) dt =

(p∑

k=1

Imk

) np+1∑k=1

ηp+1,k gp+1(t1, t0), (113)

where gq(t0, t1) is defined as in Lemma 3.4.

131

133


When t0 = tmp and t1 →∞ the integral becomes

∫ ∞tmp

i(t) dt =

(p∑

k=1

Imk

) np+1∑k=1

ηp+1,k g(β2p+1,k

), (114)

where

g(β) =eβ

ββ+1(Γ(β + 1)− γ (β + 1, β))

with

Γ(β) =

∫ ∞0

tβ−1e−t dt

is the Gamma function [2].


In the next section we will estimate the parameters of the AEF that givesthe best fit with respect to some data and for this the partial derivativeswith respect to the βmq parameters will be useful. Since the AEF is a linearfunction of elementary functions these partial derivatives can easily be foundusing standard methods.

Theorem 3.4. The partial derivatives of the p-peak AEF with respect tothe β parameters are

∂i

∂βq,k=

0, 0 ≤ t ≤ tmq−1 ,

2 Imqηq,k βq,k hq(t)xq(t)β2q,k+1, tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,

0, tmq ≤ t,(115)

∂i

∂βp+1,k=

0, 0 ≤ t ≤ tmp ,2 Imp+1ηp+1,k βp+1,k hp+1(t)xp+1(t)β

2p+1,k , tmp ≤ t,

(116)where

hq(t) =

ln

(t− tmq−1

∆tmq

)−t− tmq−1

∆tmq+ 1, 1 ≤ q ≤ p,

ln

(t

tmq

)− t

tmq+ 1, q = p+ 1.

Proof. Since the βq,k parameters are independent, differentiation with re-spect to βq,k will annihilate all terms but one in each linear combination.The expressions (115) and (116) then follow from the standard rules fordifferentiation of composite functions and products of functions.

132


When t0 = tmp and t1 →∞ the integral becomes

∫ ∞tmp

i(t) dt =

(p∑

k=1

Imk

) np+1∑k=1

ηp+1,k g(β2p+1,k

), (114)

where

g(β) =eβ

ββ+1(Γ(β + 1)− γ (β + 1, β))

with

Γ(β) =

∫ ∞0

tβ−1e−t dt

is the Gamma function [2].


In the next section we will estimate the parameters of the AEF that givesthe best fit with respect to some data and for this the partial derivativeswith respect to the βmq parameters will be useful. Since the AEF is a linearfunction of elementary functions these partial derivatives can easily be foundusing standard methods.

Theorem 3.4. The partial derivatives of the p-peak AEF with respect tothe β parameters are

∂i

∂βq,k=

0, 0 ≤ t ≤ tmq−1 ,

2 Imqηq,k βq,k hq(t)xq(t)β2q,k+1, tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p,

0, tmq ≤ t,(115)

∂i

∂βp+1,k=

0, 0 ≤ t ≤ tmp ,2 Imp+1ηp+1,k βp+1,k hp+1(t)xp+1(t)β

2p+1,k , tmp ≤ t,

(116)where

hq(t) =

ln

(t− tmq−1

∆tmq

)−t− tmq−1

∆tmq+ 1, 1 ≤ q ≤ p,

ln

(t

tmq

)− t

tmq+ 1, q = p+ 1.

Proof. Since the βq,k parameters are independent, differentiation with re-spect to βq,k will annihilate all terms but one in each linear combination.The expressions (115) and (116) then follow from the standard rules fordifferentiation of composite functions and products of functions.

132

134

3.2. APPROXIMATION OF LIGHTNING DISCHARGECURRENT FUNCTIONS

3.2 Approximation of lightning discharge currentfunctions

This section is based on Section 3 of Paper F

Many different types of systems, objects and equipment are susceptibleto damage from lightning discharges. Lightning effects are usually anal-ysed using lightning discharge models. Most of the engineering and electro-magnetic models imply channel-base current functions. Various single andmulti-peaked functions are proposed in the literature for modelling lightningchannel-base currents, examples include [117, 118, 140, 141, 146]. For engi-neering and electromagnetic models, a general function that would be ableto reproduce desired waveshapes is needed, such that analytical solutionsfor its derivatives, integrals, and integral transformations, exist. A multi-peaked channel-base current function has been proposed by Javor [140] asa generalization of the so-called TRF (two-rise front) function from [141],which possesses such properties.

In this paper we analyse a modification of such multi-peaked function,a so-called p -peak analytically extended function (AEF). The possibilityof application of the AEF to approximation of various multi-peaked wave-shapes is investigated. Estimation of its parameters has been performedusing the Marquardt least squares method (MLSM), an efficient method forthe estimation of non-linear function parameters, see Section 1.2.6. It hasbeen applied in many fields, including lightning research, e.g. for optimizingparameters of the Heidler function [178], or the Pulse function [181,182].

Some numerical results are presented, including those for the StandardIEC 62305 [133] current of the first-positive strokes, and an example of a fast-decaying lightning current waveform. Fitting a p-peaked AEF to recordedcurrent data (from [257]) is also illustrated.

3.2.1 Fitting the AEF

Suppose that we have kq points (tq,k, iq,k) ordered with respect to tq,k,tmq−1 < tq,1 < tq,2 < . . . < tq,kq < tmq , and we wish to choose parame-ters ηq,k and βq,k such that the sum of the squares of the residuals,

Sq =

kq∑k=1

(i(tq,k)− iq,k)2 , (117)

is minimized. One way to estimate these parameters is to use the Marquardtleast squares method described in Section 1.2.6.

In order to fit the AEF it is sufficient that kq ≥ nq. Suppose we have someestimate of the β-parameters which is collected in the vector b. It is thenfairly simple to calculate an estimate for the η-parameters, see Section 3.2.4,

133


3.2 Approximation of lightning discharge currentfunctions


Many different types of systems, objects and equipment are susceptibleto damage from lightning discharges. Lightning effects are usually anal-ysed using lightning discharge models. Most of the engineering and electro-magnetic models imply channel-base current functions. Various single andmulti-peaked functions are proposed in the literature for modelling lightningchannel-base currents, examples include [117, 118, 140, 141, 146]. For engi-neering and electromagnetic models, a general function that would be ableto reproduce desired waveshapes is needed, such that analytical solutionsfor its derivatives, integrals, and integral transformations, exist. A multi-peaked channel-base current function has been proposed by Javor [140] asa generalization of the so-called TRF (two-rise front) function from [141],which possesses such properties.

In this paper we analyse a modification of such multi-peaked function,a so-called p -peak analytically extended function (AEF). The possibilityof application of the AEF to approximation of various multi-peaked wave-shapes is investigated. Estimation of its parameters has been performedusing the Marquardt least squares method (MLSM), an efficient method forthe estimation of non-linear function parameters, see Section 1.2.6. It hasbeen applied in many fields, including lightning research, e.g. for optimizingparameters of the Heidler function [178], or the Pulse function [181,182].

Some numerical results are presented, including those for the StandardIEC 62305 [133] current of the first-positive strokes, and an example of a fast-decaying lightning current waveform. Fitting a p-peaked AEF to recordedcurrent data (from [257]) is also illustrated.

3.2.1 Fitting the AEF

Suppose that we have kq points (tq,k, iq,k) ordered with respect to tq,k,tmq−1 < tq,1 < tq,2 < . . . < tq,kq < tmq , and we wish to choose parame-ters ηq,k and βq,k such that the sum of the squares of the residuals,

Sq =

kq∑k=1

(i(tq,k)− iq,k)2 , (117)

is minimized. One way to estimate these parameters is to use the Marquardtleast squares method described in Section 1.2.6.

In order to fit the AEF it is sufficient that kq ≥ nq. Suppose we have someestimate of the β-parameters which is collected in the vector b. It is thenfairly simple to calculate an estimate for the η-parameters, see Section 3.2.4,

133

135


which we collect in h. We define a residual vector by (e)k = i(tq,k; b,h)−iq,kwhere i(t; b,h) is the AEF with the estimated parameters.

The Jacobian matrix, J, can in this case be described as

J =

∂i∂βq,1

∣∣∣t=tq,1

∂i∂βq,2

∣∣∣t=tq,1

. . . ∂i∂βq,nq

∣∣∣t=tq,1

∂i∂βq,1

∣∣∣t=tq,2

∂i∂βq,2

∣∣∣t=tq,2

. . . ∂i∂βq,nq

∣∣∣t=tq,2

......

. . ....

∂i∂βq,1

∣∣∣t=tq,kq

∂i∂βq,2

∣∣∣t=tq,kq

. . . ∂i∂βq,nq

∣∣∣t=tq,kq

(118)

where the partial derivatives are given by (115) and (116).

3.2.2 Estimating parameters for underdetermined systems


For the Marquardt least squares method to work at least one data point perunknown parameter is needed, m ≥ k. It can still be possible to estimateall unknown parameters if there is insufficient data, m < k if we know somefurther relations between the parameters.

Suppose that k − m = p and let γj = βm+j , j = 1, 2, · · · , p. If thereare at least p known relations between the unknown parameters such thatγj = γj(β1, · · · , βm) for j = 1, 2, · · · , p then the Marquardt least squaresmethod can be used to give estimates on β1, · · · , βm and the still unknownparameters can be estimated from these. Denoting the estimated parametersb = (b1, · · · , bm) and c = (c1, · · · , cp) the following algorithm can be used:

• Input: v > 1 and initial values b(0), λ(0).

• r = 0

/ Find c(r) using b(r) together with extra relations.

• Find b(r+1) and δ(r) using MLSM.

• Check chosen termination condition for MLSM, if it is not satisfied goto /.

• Output: b, c.

The algorithm is illustrated in Figure 3.4.

134


which we collect in h. We define a residual vector by (e)k = i(tq,k; b,h)−iq,kwhere i(t; b,h) is the AEF with the estimated parameters.

The Jacobian matrix, J, can in this case be described as

J =

∂i∂βq,1

∣∣∣t=tq,1

∂i∂βq,2

∣∣∣t=tq,1

. . . ∂i∂βq,nq

∣∣∣t=tq,1

∂i∂βq,1

∣∣∣t=tq,2

∂i∂βq,2

∣∣∣t=tq,2

. . . ∂i∂βq,nq

∣∣∣t=tq,2

......

. . ....

∂i∂βq,1

∣∣∣t=tq,kq

∂i∂βq,2

∣∣∣t=tq,kq

. . . ∂i∂βq,nq

∣∣∣t=tq,kq

(118)

where the partial derivatives are given by (115) and (116).

3.2.2 Estimating parameters for underdetermined systems


For the Marquardt least squares method to work at least one data point perunknown parameter is needed, m ≥ k. It can still be possible to estimateall unknown parameters if there is insufficient data, m < k if we know somefurther relations between the parameters.

Suppose that k − m = p and let γj = βm+j , j = 1, 2, · · · , p. If thereare at least p known relations between the unknown parameters such thatγj = γj(β1, · · · , βm) for j = 1, 2, · · · , p then the Marquardt least squaresmethod can be used to give estimates on β1, · · · , βm and the still unknownparameters can be estimated from these. Denoting the estimated parametersb = (b1, · · · , bm) and c = (c1, · · · , cp) the following algorithm can be used:

• Input: v > 1 and initial values b(0), λ(0).

• r = 0

/ Find c(r) using b(r) together with extra relations.

• Find b(r+1) and δ(r) using MLSM.

• Check chosen termination condition for MLSM, if it is not satisfied goto /.

• Output: b, c.

The algorithm is illustrated in Figure 3.4.

134

136


Input: choose v andinitial values for b(0) and λ(0) r = 0

Find b(r+1) and δ(r)

using MLSMFind h(r) using b(r)

together with extra relations

Termination conditionsatisfied

r = r + 1

Output: b, h

YES

NO

Figure 3.4: Schematic description of the parameter estimation algorithm.

3.2.3 Fitting with data points as well as charge flow andspecific energy conditions

By considering the charge flow at the striking point, Q0, unitary resistanceR and the specific energy, W0, we get two further conditions:

Q0 =

∫ ∞0

i(t) dt, (119)

W0 =

∫ ∞0

i(t)2 dt. (120)

First we will define

Q(b,h) =

∫ ∞0

i(t; b,h) dt

W (b,h) =

∫ ∞0

i(t; b,h)2 dt.

These two quantities can be calculated as follows.

Theorem 3.5.

Q(b,h) =

p∑q=1

(∆tmq

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k g(β2q,k + 1)

)

+

(p∑

k=1

Imk

) np+1∑k=1

ηp+1,k g(β2p+1,k), (121)

135


Input: choose v andinitial values for b(0) and λ(0) r = 0

Find b(r+1) and δ(r)

using MLSMFind h(r) using b(r)

together with extra relations

Termination conditionsatisfied

r = r + 1

Output: b, h

YES

NO

Figure 3.4: Schematic description of the parameter estimation algorithm.

3.2.3 Fitting with data points as well as charge flow andspecific energy conditions

By considering the charge flow at the striking point, Q0, unitary resistanceR and the specific energy, W0, we get two further conditions:

Q0 =

∫ ∞0

i(t) dt, (119)

W0 =

∫ ∞0

i(t)2 dt. (120)

First we will define

Q(b,h) =

∫ ∞0

i(t; b,h) dt

W (b,h) =

∫ ∞0

i(t; b,h)2 dt.

These two quantities can be calculated as follows.

Theorem 3.5.

Q(b,h) =

p∑q=1

(∆tmq

(q−1∑k=1

Imk

)+ Imq

nq∑k=1

ηq,k g(β2q,k + 1)

)

+

(p∑

k=1

Imk

) np+1∑k=1

ηp+1,k g(β2p+1,k), (121)

135

137


W (b,h) =

p∑q=1

(q−1∑k=1

Imk

)2

+

(q−1∑k=1

Imk

)Imq

nq∑k=1

ηq,k g(β2q,k + 1)

+ I2mq

nq∑k=1

η2q,k g

(2β2

q,k + 2)

+ 2 I2mq

nq−1∑r=1

nq∑s=r+1

ηq,r ηq,s g(β2q,r + β2

q,s + 2)

+

(p∑

k=1

Imk

)2( np∑k=1

η2p,k g

(2β2

p,k

)+ 2

np+1−1∑r=1

np+1∑s=r+1

ηp+1,r ηp+1,s g(β2p+1,r + β2

p+1,s

)(122)

where g(β) and g(β) are defined in Theorems 3.2 and 3.3.

Proof. Formula (121) is found by combining (112) and (113).Formula (122) is found by noting that

(n∑k=1

ak

)2

=

n∑k=1

a2k +

n−1∑r=1

n∑s=r+1

ar as

and then reasoning analogously to the proofs for (112) and (113).

We can calculate the charge flow and specific energy given by the AEFwith formulas (121) and (122), respectively, and get two additional residualterms EQ0 = Q(b,h)−Q0 and EW0 = W (b,h)−W0. Since these are globalconditions this means that the parameters η and β no longer can be fittedseparately in each interval. This means that we need to consider all datapoints simultaneously. The resulting J-matrix is

J =

J1 . . . 0...

. . ....

0 . . . Jp+1∂EQ0∂β1,1

. . .∂EQ0∂β1,n1

. . .∂EQ0∂βp+1,1

. . .∂EQ0

∂βp+1,np+1∂EW0∂β1,1

. . .∂EW0∂β1,n1

. . .∂EW0∂βp+1,1

. . .∂EW0

∂βp+1,np+1

(123)

136


W (b,h) =

p∑q=1

(q−1∑k=1

Imk

)2

+

(q−1∑k=1

Imk

)Imq

nq∑k=1

ηq,k g(β2q,k + 1)

+ I2mq

nq∑k=1

η2q,k g

(2β2

q,k + 2)

+ 2 I2mq

nq−1∑r=1

nq∑s=r+1

ηq,r ηq,s g(β2q,r + β2

q,s + 2)

+

(p∑

k=1

Imk

)2( np∑k=1

η2p,k g

(2β2

p,k

)+ 2

np+1−1∑r=1

np+1∑s=r+1

ηp+1,r ηp+1,s g(β2p+1,r + β2

p+1,s

)(122)

where g(β) and g(β) are defined in Theorems 3.2 and 3.3.

Proof. Formula (121) is found by combining (112) and (113).Formula (122) is found by noting that

(n∑k=1

ak

)2

=

n∑k=1

a2k +

n−1∑r=1

n∑s=r+1

ar as

and then reasoning analogously to the proofs for (112) and (113).

We can calculate the charge flow and specific energy given by the AEFwith formulas (121) and (122), respectively, and get two additional residualterms EQ0 = Q(b,h)−Q0 and EW0 = W (b,h)−W0. Since these are globalconditions this means that the parameters η and β no longer can be fittedseparately in each interval. This means that we need to consider all datapoints simultaneously. The resulting J-matrix is

J =

J1 . . . 0...

. . ....

0 . . . Jp+1∂EQ0∂β1,1

. . .∂EQ0∂β1,n1

. . .∂EQ0∂βp+1,1

. . .∂EQ0

∂βp+1,np+1∂EW0∂β1,1

. . .∂EW0∂β1,n1

. . .∂EW0∂βp+1,1

. . .∂EW0

∂βp+1,np+1

(123)

136

138


where

Jq =

∂i∂βq,1

∣∣∣t=tq,1

∂i∂βq,2

∣∣∣t=tq,1

. . . ∂i∂βq,nq

∣∣∣t=tq,1

∂i∂βq,1

∣∣∣t=tq,2

∂i∂βq,2

∣∣∣t=tq,2

. . . ∂i∂βq,nq

∣∣∣t=tq,2

......

. . ....

∂i∂βq,1

∣∣∣t=tq,kq

∂i∂βq,2

∣∣∣t=tq,kq

. . . ∂i∂βq,nq

∣∣∣t=tq,kq

and the partial derivatives in the last two rows are given by

∂Q

∂βq,s=

2 Imqηq,s βq,s

dg

dβ

∣∣∣∣β=β2

q,s+1

, 1 ≤ q ≤ p,

2 Impηp+1,s βp+1,sdg

dβ

∣∣∣∣β=β2

p+1,s

, q = p+ 1.

For 1 ≤ q ≤ p

∂W

∂βq,s= 2

(q−1∑k=1

Imk

)Imqηq,s βq,s

dg

dβ

∣∣∣∣β=β2

q,s+1

+ 4 I2mqηq,sβq,s

ηq,s dg

dβ

∣∣∣∣β=2β2

q,s+2

+

nq∑k=1k 6=s

ηq,kdg

dβ

∣∣∣∣β=β2

q,s+β2q,k+2

and

∂W

∂βp+1,s= 4

(p∑

k=1

Imk

)ηp+1,sβp+1,sηp+1,s

dg

dβ

∣∣∣∣β=2β2

p+1,s

+

nq∑k=1k 6=s

ηp+1,kdg

dβ

∣∣∣∣β=β2

p+1,s+β2p+1,k

.

The derivatives of g(β) and g(β) are

dg

dβ=

1

e

(1 +

eβ

ββ(Γ(β + 1)

(Ψ(β)− ln(β)

)−G(β)

)), (124)

dg

dβ=

1

e

(eβ

ββG(β)− 1

), (125)

where Γ(β) is the Gamma function, Ψ(β) is the digamma function, see forexample [2], and G(β) is a special case of the Meijer G-function and can bedefined as

G(β) = G3,02,3

(β

∣∣∣∣ 1, 10, 0, β + 1

)137


where

Jq =

∂i∂βq,1

∣∣∣t=tq,1

∂i∂βq,2

∣∣∣t=tq,1

. . . ∂i∂βq,nq

∣∣∣t=tq,1

∂i∂βq,1

∣∣∣t=tq,2

∂i∂βq,2

∣∣∣t=tq,2

. . . ∂i∂βq,nq

∣∣∣t=tq,2

......

. . ....

∂i∂βq,1

∣∣∣t=tq,kq

∂i∂βq,2

∣∣∣t=tq,kq

. . . ∂i∂βq,nq

∣∣∣t=tq,kq

and the partial derivatives in the last two rows are given by

∂Q

∂βq,s=

2 Imqηq,s βq,s

dg

dβ

∣∣∣∣β=β2

q,s+1

, 1 ≤ q ≤ p,

2 Impηp+1,s βp+1,sdg

dβ

∣∣∣∣β=β2

p+1,s

, q = p+ 1.

For 1 ≤ q ≤ p

∂W

∂βq,s= 2

(q−1∑k=1

Imk

)Imqηq,s βq,s

dg

dβ

∣∣∣∣β=β2

q,s+1

+ 4 I2mqηq,sβq,s

ηq,s dg

dβ

∣∣∣∣β=2β2

q,s+2

+

nq∑k=1k 6=s

ηq,kdg

dβ

∣∣∣∣β=β2

q,s+β2q,k+2

and

∂W

∂βp+1,s= 4

(p∑

k=1

Imk

)ηp+1,sβp+1,sηp+1,s

dg

dβ

∣∣∣∣β=2β2

p+1,s

+

nq∑k=1k 6=s

ηp+1,kdg

dβ

∣∣∣∣β=β2

p+1,s+β2p+1,k

.

The derivatives of g(β) and g(β) are

dg

dβ=

1

e

(1 +

eβ

ββ(Γ(β + 1)

(Ψ(β)− ln(β)

)−G(β)

)), (124)

dg

dβ=

1

e

(eβ

ββG(β)− 1

), (125)

where Γ(β) is the Gamma function, Ψ(β) is the digamma function, see forexample [2], and G(β) is a special case of the Meijer G-function and can bedefined as

G(β) = G3,02,3

(β

∣∣∣∣ 1, 10, 0, β + 1

)137

139


using the notation from [236]. When evaluating this function it might bemore practical to rewrite G using other special functions

G(β) = G3,02,3

(β

∣∣∣∣ 1, 10, 0, β + 1

)=

ββ+1

(β + 1)2 2F2(β + 1, β + 1; β + 2, β + 2; −β)

−(Ψ(β) + π cot(πβ) + ln(β)

)π csc (πβ)

Γ (−β)

where

2F2(β + 1, β + 1; β + 2, β + 2; −β) =∞∑k=0

(−1)kβk(β + 1)2

(β + k + 1)2

is a special case of the hypergeometric function. These partial derivativeswere found using software for symbolic computation [200].

Note that all η-parameters must be recalculated for each step and howthis is done is detailed in Section 3.2.4.

3.2.4 Calculating the η-parameters from the β-parameters

Suppose that we have nq − 1 points (tq,k, iq,k) such that

tmq−1 < tq,1 < tq,2 < . . . < tq,nq−1 < tmq .

For an AEF that interpolates these points it must be true that

q−1∑k=1

Imk + Imq

nq∑s=1

ηq,sxq(tq,k)βq,s = iq,k, k = 1, 2, . . . , nq − 1. (126)

Since ηq,1 + ηq,2 + . . .+ ηq,nq = 1 equation (126) can be rewritten as

Imq

nq−1∑s=1

ηq,s

(xq(tq,k)

βq,s − xq(tq,k)βq,nq)

= iq,k − xq(tq,k)βq,nq −q−1∑s=1

Ims

(127)for k = 1, 2, . . . , nq − 1. This can be written as a matrix equation

ImqXqηq = iq, (128)

ηq =[ηq,1 ηq,2 . . . ηq,nq−1

]>,(iq

)k


Ims ,(Xq

)k,s

= xq(k, s) = xq(tq,k)βq,s − xq(tq,k)βq,nq ,

and xq(t) given by (105).When all βq,k, k = 1, 2, . . . , nq are known then ηq,k, k = 1, 2, . . . , nq − 1 can

be found by solving equation (128) and ηq,nq = 1−nq−1∑k=1

ηq,k.

138


using the notation from [236]. When evaluating this function it might bemore practical to rewrite G using other special functions

G(β) = G3,02,3

(β

∣∣∣∣ 1, 10, 0, β + 1

)=

ββ+1

(β + 1)2 2F2(β + 1, β + 1; β + 2, β + 2; −β)

−(Ψ(β) + π cot(πβ) + ln(β)

)π csc (πβ)

Γ (−β)

where

2F2(β + 1, β + 1; β + 2, β + 2; −β) =∞∑k=0

(−1)kβk(β + 1)2

(β + k + 1)2

is a special case of the hypergeometric function. These partial derivativeswere found using software for symbolic computation [200].

Note that all η-parameters must be recalculated for each step and howthis is done is detailed in Section 3.2.4.

3.2.4 Calculating the η-parameters from the β-parameters

Suppose that we have nq − 1 points (tq,k, iq,k) such that

tmq−1 < tq,1 < tq,2 < . . . < tq,nq−1 < tmq .

For an AEF that interpolates these points it must be true that

q−1∑k=1

Imk + Imq

nq∑s=1

ηq,sxq(tq,k)βq,s = iq,k, k = 1, 2, . . . , nq − 1. (126)

Since ηq,1 + ηq,2 + . . .+ ηq,nq = 1 equation (126) can be rewritten as

Imq

nq−1∑s=1

ηq,s

(xq(tq,k)

βq,s − xq(tq,k)βq,nq)


Ims

(127)for k = 1, 2, . . . , nq − 1. This can be written as a matrix equation

ImqXqηq = iq, (128)

ηq =[ηq,1 ηq,2 . . . ηq,nq−1

]>,(iq

)k


Ims ,(Xq

)k,s

= xq(k, s) = xq(tq,k)βq,s − xq(tq,k)βq,nq ,

and xq(t) given by (105).When all βq,k, k = 1, 2, . . . , nq are known then ηq,k, k = 1, 2, . . . , nq − 1 can

be found by solving equation (128) and ηq,nq = 1−nq−1∑k=1

ηq,k.

138

140


If we have kq > nq−1 data points then the parameters can be estimatedwith the least squares solution to (128), more specifically the solution to

I2mqX

>q Xqηq = X>q iq.

3.2.5 Explicit formulas for a single-peak AEF

Consider the case where p = 1, n1 = n2 = 2 and τ = ttm1

. Then the explicit

formula for the AEF is

i(τ)

Im1

=

η1,1 τ

β21,1+1e(β2

1,1+1)(1−τ)+ η1,2 τβ21,2+1e(β2

1,2+1)(1−τ), 0≤τ≤1,

η2,1 τβ22,1 eβ

22,1(1−τ)+ η2,2 τ

β22,2 eβ

22,2(1−τ) , 1≤τ.

(129)

Assume that four datapoints, (ik, τk), k = 1, 2, 3, 4, as well as the chargeflow Q0 and specific energy W0, are known.

If we want to fit the AEF to this data using MLSM equation (123) gives

J =

f1(τ1) f2(τ1) 0 0f1(τ2) f2(τ2) 0 0

0 0 g1(τ3) g2(τ3)0 0 g1(τ4) g2(τ4)

∂

∂β1,1Q(β,η)

∂

∂β1,2Q(β,η)

∂

∂β2,1Q(β,η)

∂

∂β2,2Q(β,η)

∂

∂β1,1W (β,η)

∂

∂β1,2W (β,η)

∂

∂β2,1W (β,η)

∂

∂β2,2W (β,η)

,

fk(τ) = 2 η1,k β1,kτβ21,k+1e(β2

1,k+1)(1−τ)( ln(τ) + 1− τ),

η1,1 =i1Im1

− τβ21,2

1 e(β21,2+1)(1−τ1), η1,2 = 1− η1,1,

gk(τ) = 2 η2,k β2,kτβ22,keβ

22,k(1−τ)( ln(τ) + 1− τ

),

η2,1 =i3Im1

− τβ22,2

3 eβ21,2(1−τ3), η2,2 = 1− η2,1,

β =[(β2

1,1 + 1) (

β21,2 + 1

)β2

2,1 β22,2

],

η =[η1,1 η1,2 η2,1 η2,2

],

Q(β,η)

Im1

=2∑s=1

η1,seβ

21,s(

β21,s + 1

)β21,s+1

γ(β2

1,s + 2, β22,s + 1

)

+2∑s=1

η2,seβ

22,s

β2β2

2,s+1

2,s

(Γ(β2

2,s + 1)− γ

(β2

2,s + 1, β22,s

)),

∂Q

∂βq,s=

2 Im1η1,s β1,s

dg

dβ

∣∣∣∣β=β2

1,s+1

, q = 1,

2 Imqηp,s β2,sdg

dβ

∣∣∣∣β=β2

2,s

, q = 2,

139


If we have kq > nq−1 data points then the parameters can be estimatedwith the least squares solution to (128), more specifically the solution to

I2mqX

>q Xqηq = X>q iq.

3.2.5 Explicit formulas for a single-peak AEF

Consider the case where p = 1, n1 = n2 = 2 and τ = ttm1

. Then the explicit

formula for the AEF is

i(τ)

Im1

=

η1,1 τ

β21,1+1e(β2

1,1+1)(1−τ)+ η1,2 τβ21,2+1e(β2

1,2+1)(1−τ), 0≤τ≤1,

η2,1 τβ22,1 eβ

22,1(1−τ)+ η2,2 τ

β22,2 eβ

22,2(1−τ) , 1≤τ.

(129)

Assume that four datapoints, (ik, τk), k = 1, 2, 3, 4, as well as the chargeflow Q0 and specific energy W0, are known.

If we want to fit the AEF to this data using MLSM equation (123) gives

J =

f1(τ1) f2(τ1) 0 0f1(τ2) f2(τ2) 0 0

0 0 g1(τ3) g2(τ3)0 0 g1(τ4) g2(τ4)

∂

∂β1,1Q(β,η)

∂

∂β1,2Q(β,η)

∂

∂β2,1Q(β,η)

∂

∂β2,2Q(β,η)

∂

∂β1,1W (β,η)

∂

∂β1,2W (β,η)

∂

∂β2,1W (β,η)

∂

∂β2,2W (β,η)

,

fk(τ) = 2 η1,k β1,kτβ21,k+1e(β2

1,k+1)(1−τ)( ln(τ) + 1− τ),

η1,1 =i1Im1

− τβ21,2

1 e(β21,2+1)(1−τ1), η1,2 = 1− η1,1,

gk(τ) = 2 η2,k β2,kτβ22,keβ

22,k(1−τ)( ln(τ) + 1− τ

),

η2,1 =i3Im1

− τβ22,2

3 eβ21,2(1−τ3), η2,2 = 1− η2,1,

β =[(β2

1,1 + 1) (

β21,2 + 1

)β2

2,1 β22,2

],

η =[η1,1 η1,2 η2,1 η2,2

],

Q(β,η)

Im1

=2∑s=1

η1,seβ

21,s(

β21,s + 1

)β21,s+1

γ(β2

1,s + 2, β22,s + 1

)

+2∑s=1

η2,seβ

22,s

β2β2

2,s+1

2,s

(Γ(β2

2,s + 1)− γ

(β2

2,s + 1, β22,s

)),

∂Q

∂βq,s=

2 Im1η1,s β1,s

dg

dβ

∣∣∣∣β=β2

1,s+1

, q = 1,

2 Imqηp,s β2,sdg

dβ

∣∣∣∣β=β2

2,s

, q = 2,

139

141


with derivatives of g(β) and g(β) given by (124) and (125),

β =[(β2

1,1 + β21,2 + 2

) (β2

1,1 + β21,2 + 2

)(β2

2,1 + β22,2) (β2

2,1 + β22,2)],

η =[η2

1,1 η21,2 η2

2,1 η22,2

],

η =[(η1,1η1,2) (η1,1η1,2)(η2,1η2,2) (η2,1η2,2)

],

∂

∂βq,sW (β,η) = 2βq,s

∂

∂βq,sQ (2β, η) + β

q,(

(s−1 mod 2)+1) ∂

∂βq,sQ(β, η

).

3.2.6 Fitting to lightning discharge currents


In this section some results of fitting the AEF to a few different waveformswill be demonstrated. Some single-peak waveforms given by Heidler func-tions in the IEC 62305-1 standard [133] will be approximated using theAEF, and furthermore, fitting the multi-peaked waveform to experimentaldata will be presented.

Single-peak waveforms

In this section some numerical results of fitting the AEF function to single-peak waveshapes are presented and compared with the corresponding fittingof the Heidler function. The AEF given by (129) is used to model few light-ning current waveshapes whose parameters (rise/decay time ratio, T1/T2,peak current value, Im1, time to peak current, tm1, charge flow at the strik-ing point, Q0, specific energy, W0, and time to 0.1Im1, t1) are given in Table3.1. Data points were chosen as follows:

(i1, τ1) = (0.1 Im1 , t1),

(i2, τ2) = (0.9 Im1 , t2 = t1 + 0.8T1),

(i3, τ3) = (0.5 Im1 , th = t1 − 0.1T1 + T2),

(i4, τ4) = (i(1.5 th), 1.5 th).

The AEF representation of the waveshape denoted as the first positivestroke current in IEC 62305 standard [133], is shown in Figure 3.5. Ris-ing and decaying parts of the first negative stroke current from IEC 62305standard [133] are shown in Figure 3.6, left and right, respectively. β andη parameters of both waveshapes optimized by the MLSM are given in Ta-ble 3.1.

We have also observed a so-called fast-decaying waveshape whose pa-rameters are given in Table 3.1. It’s representation using the AEF functionis shown in Figure 3.7, and corresponding β and η parameter values in Ta-ble 3.1.

140


with derivatives of g(β) and g(β) given by (124) and (125),

β =[(β2

1,1 + β21,2 + 2

) (β2

1,1 + β21,2 + 2

)(β2

2,1 + β22,2) (β2

2,1 + β22,2)],

η =[η2

1,1 η21,2 η2

2,1 η22,2

],

η =[(η1,1η1,2) (η1,1η1,2)(η2,1η2,2) (η2,1η2,2)

],

∂

∂βq,sW (β,η) = 2βq,s

∂

∂βq,sQ (2β, η) + β

q,(

(s−1 mod 2)+1) ∂

∂βq,sQ(β, η

).

3.2.6 Fitting to lightning discharge currents


In this section some results of fitting the AEF to a few different waveformswill be demonstrated. Some single-peak waveforms given by Heidler func-tions in the IEC 62305-1 standard [133] will be approximated using theAEF, and furthermore, fitting the multi-peaked waveform to experimentaldata will be presented.

Single-peak waveforms

In this section some numerical results of fitting the AEF function to single-peak waveshapes are presented and compared with the corresponding fittingof the Heidler function. The AEF given by (129) is used to model few light-ning current waveshapes whose parameters (rise/decay time ratio, T1/T2,peak current value, Im1, time to peak current, tm1, charge flow at the strik-ing point, Q0, specific energy, W0, and time to 0.1Im1, t1) are given in Table3.1. Data points were chosen as follows:

(i1, τ1) = (0.1 Im1 , t1),

(i2, τ2) = (0.9 Im1 , t2 = t1 + 0.8T1),

(i3, τ3) = (0.5 Im1 , th = t1 − 0.1T1 + T2),

(i4, τ4) = (i(1.5 th), 1.5 th).

The AEF representation of the waveshape denoted as the first positivestroke current in IEC 62305 standard [133], is shown in Figure 3.5. Ris-ing and decaying parts of the first negative stroke current from IEC 62305standard [133] are shown in Figure 3.6, left and right, respectively. β andη parameters of both waveshapes optimized by the MLSM are given in Ta-ble 3.1.

We have also observed a so-called fast-decaying waveshape whose pa-rameters are given in Table 3.1. It’s representation using the AEF functionis shown in Figure 3.7, and corresponding β and η parameter values in Ta-ble 3.1.

140

142


Figure 3.5: First-positive stroke represented by the AEF function. Here it isfitted with respect to both the data points as well as Q0 and W0.

Figure 3.6: First-negative stroke represented by the AEF function. Here it isfitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters.

Apart from the AEF function (solid line), the Heidler function represen-tation of the same waveshapes (dashed line), and used data points (red solidcircles) are also shown in the figures.

Multi-peaked AEF waveforms for measured data

In this section the AEF will be constructed by fitting to measured datarather than approximation of the Heidler function. We will use data basedon the measurements of flash number 23 in [257]. Two AEFs have beenconstructed, one by choosing peaks corresponding to local maxima, see Fig-ure 3.8, and one by choosing peaks corresponding to local maxima and localminima, see Figure 3.9. For both AEFs there are two terms in each intervalwhich means that for each peak there are two parameters that are chosenmanually (time and current for each peak) and for each interval there aretwo parameters that are fitted using the MLSM.

The AEF in Figure 3.8 demonstrates that the AEF can handle caseswhere the function is not constant or monotonically increasing/decreasingbetween peaks. This is only possible if the AEF has more than one term inthe interval.

141


Figure 3.5: First-positive stroke represented by the AEF function. Here it isfitted with respect to both the data points as well as Q0 and W0.

Figure 3.6: First-negative stroke represented by the AEF function. Here it isfitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters.

Apart from the AEF function (solid line), the Heidler function represen-tation of the same waveshapes (dashed line), and used data points (red solidcircles) are also shown in the figures.

Multi-peaked AEF waveforms for measured data

In this section the AEF will be constructed by fitting to measured datarather than approximation of the Heidler function. We will use data basedon the measurements of flash number 23 in [257]. Two AEFs have beenconstructed, one by choosing peaks corresponding to local maxima, see Fig-ure 3.8, and one by choosing peaks corresponding to local maxima and localminima, see Figure 3.9. For both AEFs there are two terms in each intervalwhich means that for each peak there are two parameters that are chosenmanually (time and current for each peak) and for each interval there aretwo parameters that are fitted using the MLSM.

The AEF in Figure 3.8 demonstrates that the AEF can handle caseswhere the function is not constant or monotonically increasing/decreasingbetween peaks. This is only possible if the AEF has more than one term inthe interval.

141

143


Figure 3.7: Fast-decaying waveshape represented by the AEF function. Here itis fitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters.

Figure 3.8: AEF fitted to measurements from [257]. Here the peaks have beenchosen to correspond to local maxima in the measured data.

Conclusions

This section investigated the possibility to approximate, in general, multi-peaked lightning currents using an AEF function. Furthermore, existence ofthe analytical solution for the derivative and the integral of such function hasbeen proven, which is needed in order to perform lightning electromagneticfield (LEMF) calculations based on it.

Two single-peak Standard IEC 62305-1 waveforms, and a fast-decayingone, have been represented using a variation of the proposed AEF function(129). The estimation of their parameters has been performed applying theMLS method using two pairs of data points for each function part (risingand decaying). The results show that there are several factors that need tobe taken into consideration to get the best possible approximation of a givenwaveform. The accuracy of the approximation varies with the chosen datapoints and the number of terms in the AEF. In several cases the two-termsum converged towards a single term sum. This can probably be improvedby choosing the number of terms and the number and placement of datapoints in other ways which the authors intend to examine further. Further

142


Figure 3.7: Fast-decaying waveshape represented by the AEF function. Here itis fitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters.

Figure 3.8: AEF fitted to measurements from [257]. Here the peaks have beenchosen to correspond to local maxima in the measured data.

Conclusions

This section investigated the possibility to approximate, in general, multi-peaked lightning currents using an AEF function. Furthermore, existence ofthe analytical solution for the derivative and the integral of such function hasbeen proven, which is needed in order to perform lightning electromagneticfield (LEMF) calculations based on it.

Two single-peak Standard IEC 62305-1 waveforms, and a fast-decayingone, have been represented using a variation of the proposed AEF function(129). The estimation of their parameters has been performed applying theMLS method using two pairs of data points for each function part (risingand decaying). The results show that there are several factors that need tobe taken into consideration to get the best possible approximation of a givenwaveform. The accuracy of the approximation varies with the chosen datapoints and the number of terms in the AEF. In several cases the two-termsum converged towards a single term sum. This can probably be improvedby choosing the number of terms and the number and placement of datapoints in other ways which the authors intend to examine further. Further

142

144

3.3. APPROXIMATION OF ELECTROSTATIC DISCHARGECURRENTS USING THE AEF BY INTERPOLATION ON A

D-OPTIMAL DESIGN

First-positive First-negative Fast-decayingstroke stroke

T1/T2 10/350 1/200 8/20

tm1 [µs] 31.428 3.552 15.141

Im1 [kA] 200 100 0.001

Q0 [C] 100 / /

W0 [MJ/Ω] 10 / /

t1 [µs] 14.5 1.47 6.34

β1,1 0.114 1.84 7.666

β1,2 2.17 9.99 2.626

β2,1 0.284 0.099 0.925

β2,2 0 0.127 2.420

η1,1 −0.197 1 0

η1,2 1.197 0 1

η2,1 1 0.401 0.2227

η2,2 0 0.599 0.7773

Table 3.1: AEF function’s parameters for some current waveshapes

examples of fitted (single- and multi-peaked) waveforms can be found in [189]and [143].

3.3 Approximation of electrostatic discharge cur-rents using the AEF by interpolation on a D-optimal design

This section is based on Paper G

In this section we analyse the applicability of the AEF with p peaks torepresentation of ESD currents by interpolation of data points chosen ac-cording to a D-optimal design. This is illustrated through examples fromtwo applications. The first application is modelling of ESDs from IEC stan-dards commonly used in electrostatic discharge immunity testing, and thesecond modelling of lightning discharges.

For the ESD immunity testing application we model the IEC Standard61000-4-2 waveshape, [131, 132] and an experimentally measured ESD cur-rent from [151].

For the lightning discharge application we model the IEC 61312-1 stan-dard waveshape [117,134] and a more complex measured lightning dischargecurrent from [69]. We also use the same method to approximate a measuredderivative of a lightning discharge current derivative from [130].

143


D-OPTIMAL DESIGN

First-positive First-negative Fast-decayingstroke stroke

T1/T2 10/350 1/200 8/20

tm1 [µs] 31.428 3.552 15.141

Im1 [kA] 200 100 0.001

Q0 [C] 100 / /

W0 [MJ/Ω] 10 / /

t1 [µs] 14.5 1.47 6.34

β1,1 0.114 1.84 7.666

β1,2 2.17 9.99 2.626

β2,1 0.284 0.099 0.925

β2,2 0 0.127 2.420

η1,1 −0.197 1 0

η1,2 1.197 0 1

η2,1 1 0.401 0.2227

η2,2 0 0.599 0.7773

Table 3.1: AEF function’s parameters for some current waveshapes

examples of fitted (single- and multi-peaked) waveforms can be found in [189]and [143].

3.3 Approximation of electrostatic discharge cur-rents using the AEF by interpolation on a D-optimal design

This section is based on Paper G

In this section we analyse the applicability of the AEF with p peaks torepresentation of ESD currents by interpolation of data points chosen ac-cording to a D-optimal design. This is illustrated through examples fromtwo applications. The first application is modelling of ESDs from IEC stan-dards commonly used in electrostatic discharge immunity testing, and thesecond modelling of lightning discharges.

For the ESD immunity testing application we model the IEC Standard61000-4-2 waveshape, [131, 132] and an experimentally measured ESD cur-rent from [151].

For the lightning discharge application we model the IEC 61312-1 stan-dard waveshape [117,134] and a more complex measured lightning dischargecurrent from [69]. We also use the same method to approximate a measuredderivative of a lightning discharge current derivative from [130].

143

145


Figure 3.9: AEF fitted to measurements from [257]. Here the peaks have beenchosen to correspond to local maxima and minima in the measureddata.

In both applications the basic properties of the current (or current deriva-tive) are the same, these properties and how they are modelled with the AEFis discussed in the next section.

Multi-peaked analytically extended function

A so-called multi-peaked analytically extended function (AEF) has beenproposed and applied to lightning discharge current modelling in Section 3.1and [183]. Initial considerations on applying the function to ESD currentshave also been made in [189].

The AEF consists of scaled and translated power-exponential functions,

that is functions of the form x(β; t) =(te1−t)β, see Definition 3.1.

Here we define the AEF with p peaks as

i(t) =

q−1∑k=1

Imk + Imq

nq∑k=1

ηq,kxq,k(t), (130)

for tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p, and

p∑k=1

Imk

np+1∑k=1

ηp+1,kxp+1,k(t), (131)

for tmp ≤ t.The current value of the first peak is denoted by Im1 , the difference

between each pair of subsequent peaks by Im2 , Im3 , . . . , Imp , and their cor-responding times by tm1 , tm2 , . . . , tmp . In each time interval q, with 1 ≤ q ≤p+ 1, the number of terms is given by nq, 0 < nq ∈ Z. Parameters ηq,k are

such that ηq,k ∈ R for q = 1, 2, . . . , p + 1, k = 1, 2, . . . , nq and

nq∑k=1

ηq,k = 1.

144


Figure 3.9: AEF fitted to measurements from [257]. Here the peaks have beenchosen to correspond to local maxima and minima in the measureddata.

In both applications the basic properties of the current (or current deriva-tive) are the same, these properties and how they are modelled with the AEFis discussed in the next section.

Multi-peaked analytically extended function

A so-called multi-peaked analytically extended function (AEF) has beenproposed and applied to lightning discharge current modelling in Section 3.1and [183]. Initial considerations on applying the function to ESD currentshave also been made in [189].

The AEF consists of scaled and translated power-exponential functions,

that is functions of the form x(β; t) =(te1−t)β, see Definition 3.1.

Here we define the AEF with p peaks as

i(t) =

q−1∑k=1

Imk + Imq

nq∑k=1

ηq,kxq,k(t), (130)

for tmq−1 ≤ t ≤ tmq , 1 ≤ q ≤ p, and

p∑k=1

Imk

np+1∑k=1

ηp+1,kxp+1,k(t), (131)

for tmp ≤ t.The current value of the first peak is denoted by Im1 , the difference

between each pair of subsequent peaks by Im2 , Im3 , . . . , Imp , and their cor-responding times by tm1 , tm2 , . . . , tmp . In each time interval q, with 1 ≤ q ≤p+ 1, the number of terms is given by nq, 0 < nq ∈ Z. Parameters ηq,k are

such that ηq,k ∈ R for q = 1, 2, . . . , p + 1, k = 1, 2, . . . , nq and

nq∑k=1

ηq,k = 1.

144

146


D-OPTIMAL DESIGN

Furthermore xq,k(t), 1 ≤ q ≤ p+ 1 is given by

xq,k(t) =

x(βq,k;

t−tmq−1

tmq−tmq−1

), 1 ≤ q ≤ p,

x(βq,k;

ttmq

), q = p+ 1.

(132)

Remark 3.2. When previously applying the AEF, see Section 3.1.1, allexponents (β-parameters) of the AEF were set to β2+1 in order to guaranteethat the derivative of the AEF is continuous. Here this condition will besatisfied in a different manner.

Since the AEF is a linear function of elementary functions its derivativeand integral can be found using standard methods. For explicit formulaeplease refer to Theorems 3.1–3.3.

Previously, the authors have fitted AEF functions to lightning dischargecurrents and ESD currents using the Marquardt least square method buthave noticed that the obtained result varies greatly depending on how thewaveforms are sampled. This is problematic, especially since the methodol-ogy becomes computationally demanding when applied to large amounts ofdata. Here we will try one way to minimize the data needed but still enoughto get an as good approximation as possible.

The method examined here will be based on D-optimality of a regressionmodel. A D-optimal design is found by choosing sample points such thatthe determinant of the Fisher information matrix of the model is minimized.For a standard linear regression model this is also equivalent, by the so-called Kiefer-Wolfowitz equivalence criterion, to G-optimality which meansthat the maximum of the prediction variance will be minimized. These arestandard results in the theory of optimal experiment design and a summarycan be found in for example [208].

Minimizing the prediction variance will in our case mean maximizing therobustness of the model. This does not guarantee a good approximation butit will increase the chances of the method working well when working withlimited precision and noisy data and thus improve the chances of finding agood approximation when it is possible.

3.3.1 D-optimal approximation for exponents given by aclass of arithmetic sequences

It can be desirable to minimize the number of points used when construct-ing the approximation. One way of doing this is choosing the D-optimalsampling points.

In this section we will only consider the case where in each interval then exponents, β1, . . . , βn, are chosen according to

βm =k +m− 1

c, m = 1, 2, . . . , n

145


D-OPTIMAL DESIGN

Furthermore xq,k(t), 1 ≤ q ≤ p+ 1 is given by

xq,k(t) =

x(βq,k;

t−tmq−1

tmq−tmq−1

), 1 ≤ q ≤ p,

x(βq,k;

ttmq

), q = p+ 1.

(132)

Remark 3.2. When previously applying the AEF, see Section 3.1.1, allexponents (β-parameters) of the AEF were set to β2+1 in order to guaranteethat the derivative of the AEF is continuous. Here this condition will besatisfied in a different manner.

Since the AEF is a linear function of elementary functions its derivativeand integral can be found using standard methods. For explicit formulaeplease refer to Theorems 3.1–3.3.

Previously, the authors have fitted AEF functions to lightning dischargecurrents and ESD currents using the Marquardt least square method buthave noticed that the obtained result varies greatly depending on how thewaveforms are sampled. This is problematic, especially since the methodol-ogy becomes computationally demanding when applied to large amounts ofdata. Here we will try one way to minimize the data needed but still enoughto get an as good approximation as possible.

The method examined here will be based on D-optimality of a regressionmodel. A D-optimal design is found by choosing sample points such thatthe determinant of the Fisher information matrix of the model is minimized.For a standard linear regression model this is also equivalent, by the so-called Kiefer-Wolfowitz equivalence criterion, to G-optimality which meansthat the maximum of the prediction variance will be minimized. These arestandard results in the theory of optimal experiment design and a summarycan be found in for example [208].

Minimizing the prediction variance will in our case mean maximizing therobustness of the model. This does not guarantee a good approximation butit will increase the chances of the method working well when working withlimited precision and noisy data and thus improve the chances of finding agood approximation when it is possible.

3.3.1 D-optimal approximation for exponents given by aclass of arithmetic sequences

It can be desirable to minimize the number of points used when construct-ing the approximation. One way of doing this is choosing the D-optimalsampling points.

In this section we will only consider the case where in each interval then exponents, β1, . . . , βn, are chosen according to

βm =k +m− 1

c, m = 1, 2, . . . , n

145

147


where k is a non-negative integer and c a positive real number. Note thatin order to guarantee continuity of the AEF derivative the condition k > chas to be satisfied.

In each interval we want an approximation of the form

y(t) =

n∑i=1

ηitβieβi(1−t)

and by setting z(t) = (te1−t)lc we obtain

y(t) =n∑i=1

ηiz(t)k+i−1.

If we have n sample points, ti, i = 1, . . . , n, then the Fisher informationmatrix, M , of this system is M = U>U where

U =

z(t1)k z(t2)k . . . z(tn)k

z(t1)k+1 z(t2)k+1 . . . z(tn)k+1

......

. . ....

z(t1)k+n−1 z(t2)k+n−1 . . . z(tn)k+n−1

.Thus if we want to maximize det(M) = det(U)2 it is sufficient to maximize

or minimize the determinant det(U). Set z(ti) = (tie1−ti)

lc = xi then

un(t1, . . . , tn) = det(U) =

(n∏i=1

xki

) ∏1≤i<j≤n

(xj − xi)

. (133)

To find ti we will use the Lambert W function. Formally the LambertW function is the function W that satisfies t = W (tet). Using W we caninvert z(t) in the following way

te1−t = xc ⇔ −te−t = −e−1xc

⇔ t = −W (−e−1xc). (134)

The Lambert W is multivalued but since we are only interested in real-valued solutions we are restricted to the main branches W0 and W−1. SinceW0 ≥ −1 and W−1 ≤ −1 the two branches correspond to the rising anddecaying parts of the AEF respectively. We will deal with the details offinding the correct points for the two parts separately.

3.3.2 D-optimal interpolation on the rising part

The D-optimal points on the rising part can be found using Theorem 3.6.

146


where k is a non-negative integer and c a positive real number. Note thatin order to guarantee continuity of the AEF derivative the condition k > chas to be satisfied.

In each interval we want an approximation of the form

y(t) =

n∑i=1

ηitβieβi(1−t)

and by setting z(t) = (te1−t)lc we obtain

y(t) =n∑i=1

ηiz(t)k+i−1.

If we have n sample points, ti, i = 1, . . . , n, then the Fisher informationmatrix, M , of this system is M = U>U where

U =

z(t1)k z(t2)k . . . z(tn)k

z(t1)k+1 z(t2)k+1 . . . z(tn)k+1

......

. . ....

z(t1)k+n−1 z(t2)k+n−1 . . . z(tn)k+n−1

.Thus if we want to maximize det(M) = det(U)2 it is sufficient to maximize

or minimize the determinant det(U). Set z(ti) = (tie1−ti)

lc = xi then

un(t1, . . . , tn) = det(U) =

(n∏i=1

xki

) ∏1≤i<j≤n

(xj − xi)

. (133)

To find ti we will use the Lambert W function. Formally the LambertW function is the function W that satisfies t = W (tet). Using W we caninvert z(t) in the following way

te1−t = xc ⇔ −te−t = −e−1xc

⇔ t = −W (−e−1xc). (134)

The Lambert W is multivalued but since we are only interested in real-valued solutions we are restricted to the main branches W0 and W−1. SinceW0 ≥ −1 and W−1 ≤ −1 the two branches correspond to the rising anddecaying parts of the AEF respectively. We will deal with the details offinding the correct points for the two parts separately.

3.3.2 D-optimal interpolation on the rising part

The D-optimal points on the rising part can be found using Theorem 3.6.

146

148


D-OPTIMAL DESIGN

Theorem 3.6. The determinant

un(k;x1, . . . , xn) =

(n∏i=1

xki

) ∏1≤i<j≤n

(xj − xi)

where k ∈ R is maximized or minimized on the cube [0, 1]n when x1, . . . , xn−1

are roots of the Jacobi polynomial

P(2k−1,0)n−1 (1− 2x) =

(2k)n−1

(n− 1)!

n−1∑i=0

(−1)n(n− 1

i

)(2k + n)i

(2k)ixi

and xn = 1, or some permutation thereof.Here ab is the rising factorial ab = a(a+ 1) · · · (a+ b− 1).

Proof. Without loss of generality we can assume 0 < x1 < x2 < . . . <xn−1 < xn ≤ 1. Fix all xi except xn. When xn increases all factors of unthat contain xn will also increase, thus un will reach its maximum valueon the edge of the cube where xn = 1. Using the method of Lagrangemultipliers in the plane given by xn = 1 gives

∂un∂xj

= un(k;x1, . . . , xn)

k

xj+

n∑i=1i6=j

1

xj − xi

= 0,

for j = 1, . . . , n− 1. By setting f(x) =n∏i=1

(x− xi) we get

k

xj+

n∑i=1i6=j

1

xj − xi= 0⇔ k

xj+

1

2

f ′′(xj)

f ′(xj)= 0

⇔ xjf′′(xj) + 2kf ′(xj) = 0 (135)

for j = 1, . . . , n− 1. Since f(x) is a polynomial of degree n that has x = 1as a root then equation (135) implies

xf ′′(x) + 2kf ′(x) = cf(x)

x− 1

where c is some constant. Set f(x) = (x−1)g(x) and the resulting differentialequation is

x(x− 1)g′′(x) + ((2k + 2)x− 2k)g′(x) + (2k − c)g(x) = 0.

The constant c can be found by examining the terms with degree n− 1 andis given by c = 2k + (n− 1)(2k + n), thus

x(1− x)g′′(x) + (2k − (2k + 2)x)g′(x)

+(n− 1)(2k + n)g(x) = 0. (136)

147


D-OPTIMAL DESIGN

Theorem 3.6. The determinant

un(k;x1, . . . , xn) =

(n∏i=1

xki

) ∏1≤i<j≤n

(xj − xi)

where k ∈ R is maximized or minimized on the cube [0, 1]n when x1, . . . , xn−1

are roots of the Jacobi polynomial

P(2k−1,0)n−1 (1− 2x) =

(2k)n−1

(n− 1)!

n−1∑i=0

(−1)n(n− 1

i

)(2k + n)i

(2k)ixi

and xn = 1, or some permutation thereof.Here ab is the rising factorial ab = a(a+ 1) · · · (a+ b− 1).

Proof. Without loss of generality we can assume 0 < x1 < x2 < . . . <xn−1 < xn ≤ 1. Fix all xi except xn. When xn increases all factors of unthat contain xn will also increase, thus un will reach its maximum valueon the edge of the cube where xn = 1. Using the method of Lagrangemultipliers in the plane given by xn = 1 gives

∂un∂xj

= un(k;x1, . . . , xn)

k

xj+

n∑i=1i6=j

1

xj − xi

= 0,

for j = 1, . . . , n− 1. By setting f(x) =n∏i=1

(x− xi) we get

k

xj+

n∑i=1i6=j

1

xj − xi= 0⇔ k

xj+

1

2

f ′′(xj)

f ′(xj)= 0

⇔ xjf′′(xj) + 2kf ′(xj) = 0 (135)

for j = 1, . . . , n− 1. Since f(x) is a polynomial of degree n that has x = 1as a root then equation (135) implies

xf ′′(x) + 2kf ′(x) = cf(x)

x− 1

where c is some constant. Set f(x) = (x−1)g(x) and the resulting differentialequation is

x(x− 1)g′′(x) + ((2k + 2)x− 2k)g′(x) + (2k − c)g(x) = 0.

The constant c can be found by examining the terms with degree n− 1 andis given by c = 2k + (n− 1)(2k + n), thus

x(1− x)g′′(x) + (2k − (2k + 2)x)g′(x)

+(n− 1)(2k + n)g(x) = 0. (136)

147

149


Comparing (136) with the standard form of the hypergeometric function [2]

x(1− x)g′′(x) + (c− (a+ b+ 1)x)g′(x)− abg(x) = 0

shows that g(x) can be expressed as follows

g(x) = C · 2F1(1− n, 2k + n; 2k, x)

= C · (2k)n−1

(n− 1)!

n−1∑i=0

(−1)i(n− 1

i

)(2k + n)i

(2k)ixi

where C is an arbitrary constant and since we are only interested in theroots of the polynomial we can chose C so that it gives the desired form ofthe expression. The connection to the Jacobi polynomial is given by [2]

2F1(−m, 1 + α+ β + n;α+ 1;x) =m!

(α+ 1)mP (α,β)m (1− 2x),

and α = 2k − 1, β = 0, m = n − 1 gives the expression in Theorem 3.6.

Note that the Jacobi polynomials P(a,b)n (x) are orthogonal polynomials on

the interval [−1, 1] with respect to the weight function (1− x)a(1 + x)b andthus all of its zeros will be real, distinct and located in [−1, 1], see [48]. Thusall zeros of the polynomial given in Theorem 3.6 will be real, distinct andlocated in the interval [0, 1].

We can now find the D-optimal t-values using the upper branch of theLambert W function as described in equation (134),

ti = −W0(−e−1xci ),

where xi are the roots of the Jacobi polynomial given in Theorem 3.6. Since−1 ≤W0(x) ≤ 0 for −e−1 ≤ x ≤ 0 this will always give 0 ≤ ti ≤ 1.

Remark 3.3. Note that xn = 1 means that tn = tq and also is equivalent

to the condition

nq∑r=1

ηq,r = 1. In other words, we are interpolating the peak

and p− 1 points inside each interval.

3.3.3 D-optimal interpolation on the decaying part

Finding the D-optimal points for the decaying part is more difficult than itis for the rising part. Suppose we denote the largest value for time that canreasonably be used (for computational or experimental reasons) with tmax.

This corresponds to some value xmax = (tmax exp(1 − tmax))1c . Ideally we

would want a corresponding theorem to Theorem 3.6 over [1, xmax]n insteadof [0, 1]n. It is easy to see that if xi = 0 or xi = 1 for some 1 ≤ xi ≤ n − 1then wn(k;x1, . . . , xn) = 0 and thus there must exist some local extreme

148


Comparing (136) with the standard form of the hypergeometric function [2]

x(1− x)g′′(x) + (c− (a+ b+ 1)x)g′(x)− abg(x) = 0

shows that g(x) can be expressed as follows

g(x) = C · 2F1(1− n, 2k + n; 2k, x)

= C · (2k)n−1

(n− 1)!

n−1∑i=0

(−1)i(n− 1

i

)(2k + n)i

(2k)ixi

where C is an arbitrary constant and since we are only interested in theroots of the polynomial we can chose C so that it gives the desired form ofthe expression. The connection to the Jacobi polynomial is given by [2]

2F1(−m, 1 + α+ β + n;α+ 1;x) =m!

(α+ 1)mP (α,β)m (1− 2x),

and α = 2k − 1, β = 0, m = n − 1 gives the expression in Theorem 3.6.

Note that the Jacobi polynomials P(a,b)n (x) are orthogonal polynomials on

the interval [−1, 1] with respect to the weight function (1− x)a(1 + x)b andthus all of its zeros will be real, distinct and located in [−1, 1], see [48]. Thusall zeros of the polynomial given in Theorem 3.6 will be real, distinct andlocated in the interval [0, 1].

We can now find the D-optimal t-values using the upper branch of theLambert W function as described in equation (134),

ti = −W0(−e−1xci ),

where xi are the roots of the Jacobi polynomial given in Theorem 3.6. Since−1 ≤W0(x) ≤ 0 for −e−1 ≤ x ≤ 0 this will always give 0 ≤ ti ≤ 1.

Remark 3.3. Note that xn = 1 means that tn = tq and also is equivalent

to the condition

nq∑r=1

ηq,r = 1. In other words, we are interpolating the peak

and p− 1 points inside each interval.

3.3.3 D-optimal interpolation on the decaying part

Finding the D-optimal points for the decaying part is more difficult than itis for the rising part. Suppose we denote the largest value for time that canreasonably be used (for computational or experimental reasons) with tmax.

This corresponds to some value xmax = (tmax exp(1 − tmax))1c . Ideally we

would want a corresponding theorem to Theorem 3.6 over [1, xmax]n insteadof [0, 1]n. It is easy to see that if xi = 0 or xi = 1 for some 1 ≤ xi ≤ n − 1then wn(k;x1, . . . , xn) = 0 and thus there must exist some local extreme

148

150


D-OPTIMAL DESIGN

point such that 0 < x1 < x2 < . . . < xn−1 < 1. This is no longer guaranteedwhen considering the volume [1, xmax]n instead. Therefore we will insteadextend Theorem 3.6 to the volume [0, xmax]n and give an extra constrainton the parameter k that guarantees 1 < x1 < x2 < . . . < xn−1 < xn = xmax.

Theorem 3.7. Let y1 < y2 < . . . < yn−1 be the roots of the Jacobi poly-

nomial P(2k−1,0)n−1 (1 − 2y). If k is chosen such that 1 < xmax · y1 then the

determinant wn(k;x1, . . . , xn) given in Theorem 3.6 is maximized or min-imized on the cube [1, xmax]n (where xmax > 1) when xi = xmax · yi andxn = xmax, or some permutation thereof.

Proof. This theorem follows from Theorem 3.6 combined with the fact thatwn(k;x1, . . . , xn) is a homogeneous polynomial. Since wn(k; b · x1, . . . , c ·xn) = bk+

n(n−1)2 ·wn(k;x1, . . . , xn) if (x1, . . . , xn) is an extreme point in [0, 1]n

then (b·x1, . . . , b·xn) is an extreme point in [0, b]n. Thus by Theorem 3.6 thepoints given by xi = xmax · yi will maximize or minimize wn(k;x1, . . . , xn)on [0, xmax]n.

Remark 3.4. It is in many cases possible to ensure the condition 1 <

xmax · y1 without actually calculating the roots of P(2k−1,0)n−1 (1− 2y). In the

literature on orthogonal polynomials there are many expressions for upperand lower bounds of the roots of the Jacobi polynomials. For instance in [74]an upper bound on the largest root of a Jacobi polynomial is given and canbe, in our case, rewritten as

y1 > 1− 3

4k2 + 2kn+ n2 − k − 2n+ 1

and thus

1− 3

4k2 + 2kn+ n2 − k − 2n+ 1>

1

xmax

guarantees that 1 < xmax · y1. If a more precise condition is needed thereare expressions that give tighter bounds of the largest root of the Jacobipolynomials, see [179].

We can now find the D-optimal t-values using the lower branch of theLambert W function as in equation (134),

ti = −W−1(−e−1xci ),

where xi are the roots of the Jacobi polynomial given in Theorem 3.6. Since−1 ≤W−1(x) < −∞ for −e−1 ≤ x ≤ 0 this will always give 1 ≤ ti < tmax =−W−1(−e−1xmax) so xmax is given by the highest feasible t.

Remark 3.5. Note that here just like in the rising part tn = tp whichmeans that we will interpolate to the final peak as well as p − 1 points inthe decaying part.

149


D-OPTIMAL DESIGN

point such that 0 < x1 < x2 < . . . < xn−1 < 1. This is no longer guaranteedwhen considering the volume [1, xmax]n instead. Therefore we will insteadextend Theorem 3.6 to the volume [0, xmax]n and give an extra constrainton the parameter k that guarantees 1 < x1 < x2 < . . . < xn−1 < xn = xmax.

Theorem 3.7. Let y1 < y2 < . . . < yn−1 be the roots of the Jacobi poly-

nomial P(2k−1,0)n−1 (1 − 2y). If k is chosen such that 1 < xmax · y1 then the

determinant wn(k;x1, . . . , xn) given in Theorem 3.6 is maximized or min-imized on the cube [1, xmax]n (where xmax > 1) when xi = xmax · yi andxn = xmax, or some permutation thereof.

Proof. This theorem follows from Theorem 3.6 combined with the fact thatwn(k;x1, . . . , xn) is a homogeneous polynomial. Since wn(k; b · x1, . . . , c ·xn) = bk+

n(n−1)2 ·wn(k;x1, . . . , xn) if (x1, . . . , xn) is an extreme point in [0, 1]n

then (b·x1, . . . , b·xn) is an extreme point in [0, b]n. Thus by Theorem 3.6 thepoints given by xi = xmax · yi will maximize or minimize wn(k;x1, . . . , xn)on [0, xmax]n.

Remark 3.4. It is in many cases possible to ensure the condition 1 <

xmax · y1 without actually calculating the roots of P(2k−1,0)n−1 (1− 2y). In the

literature on orthogonal polynomials there are many expressions for upperand lower bounds of the roots of the Jacobi polynomials. For instance in [74]an upper bound on the largest root of a Jacobi polynomial is given and canbe, in our case, rewritten as

y1 > 1− 3

4k2 + 2kn+ n2 − k − 2n+ 1

and thus

1− 3

4k2 + 2kn+ n2 − k − 2n+ 1>

1

xmax

guarantees that 1 < xmax · y1. If a more precise condition is needed thereare expressions that give tighter bounds of the largest root of the Jacobipolynomials, see [179].

We can now find the D-optimal t-values using the lower branch of theLambert W function as in equation (134),

ti = −W−1(−e−1xci ),

where xi are the roots of the Jacobi polynomial given in Theorem 3.6. Since−1 ≤W−1(x) < −∞ for −e−1 ≤ x ≤ 0 this will always give 1 ≤ ti < tmax =−W−1(−e−1xmax) so xmax is given by the highest feasible t.

Remark 3.5. Note that here just like in the rising part tn = tp whichmeans that we will interpolate to the final peak as well as p − 1 points inthe decaying part.

149

151


3.3.4 Examples of models from applications and experiments

Here we will apply the described scheme to two different applications, mod-elling of ESD currents commonly used in electrostatic discharge immunitytesting and modelling of lightning discharge currents.

The values of n and peak-times have been chosen manually, and k andc have been chosen by first fixing k and then numerically finding a c thatgave a close approximation. For this purpose we used software for numericalcomputing [205], based on the interior reflective Newton method describedin [55, 56]. This is then repeated for k = 1, . . . , 10 and the best fittingset of parameters is chosen. Note that this methodology uses all availabledata points to evaluate fitting but could probably be simplified further. Forexample, by using a simpler method for choosing c given k, only use a subsetof available points to asses accuracy or, with sufficient experimentation findsome suitable heuristic for choosing the appropriate value of k. Since thewaveforms are given as data rather than explicit functions the D-optimalpoints have been calculated and then the closest available data points havebeen chosen. In these examples the coefficients in the linear sums can benegative.

3.3.5 Modelling of ESD currents

The IEC 61000-4-2 standard current waveshape

All ESD generators used in testing of equipment and devices must be ableto reproduce the same ESD current waveshape. The requirements for thiswaveshape are given in the IEC 61000-4-2 Standard, [132].

The IEC 61000-4-2 Standard gives a graphical representation of the typ-ical ESD current, Figure 3.10, and also defines, for a given test level voltage,required values of ESD current’s key parameters.

The values of the ESD currents key parameters are listed in Table 3.2for the case of the contact discharge, where Ipeak is the ESD current initialpeak, tr is the rising time defined as the difference between time momentscorresponding to 10% and 90% of the current peak Ipeak, I30 and I60 are theESD current values calculated for time periods of 30 and 60 ns, respectively,starting from the time point corresponding to 10% of Ipeak.

In this section we present the results of fitting a 2-peak AEF to theStandard ESD current given in IEC 61000-4-2. Data points which are usedin the optimization procedure are manually sampled from the graphicallygiven Standard [132] current function, Figure 3.10. The peak currents andcorresponding times are also extracted, and the results of D-optimal inter-polation with two peaks are illustrated, see Figure 3.11. The parameters arelisted in Table 3.3. In the illustrated examples a fairly good fit is found buttypically areas with steeply rising and decaying parts are somewhat moredifficult to fit with good accuracy than the other parts of the waveform.

150


3.3.4 Examples of models from applications and experiments

Here we will apply the described scheme to two different applications, mod-elling of ESD currents commonly used in electrostatic discharge immunitytesting and modelling of lightning discharge currents.

The values of n and peak-times have been chosen manually, and k andc have been chosen by first fixing k and then numerically finding a c thatgave a close approximation. For this purpose we used software for numericalcomputing [205], based on the interior reflective Newton method describedin [55, 56]. This is then repeated for k = 1, . . . , 10 and the best fittingset of parameters is chosen. Note that this methodology uses all availabledata points to evaluate fitting but could probably be simplified further. Forexample, by using a simpler method for choosing c given k, only use a subsetof available points to asses accuracy or, with sufficient experimentation findsome suitable heuristic for choosing the appropriate value of k. Since thewaveforms are given as data rather than explicit functions the D-optimalpoints have been calculated and then the closest available data points havebeen chosen. In these examples the coefficients in the linear sums can benegative.

3.3.5 Modelling of ESD currents

The IEC 61000-4-2 standard current waveshape

All ESD generators used in testing of equipment and devices must be ableto reproduce the same ESD current waveshape. The requirements for thiswaveshape are given in the IEC 61000-4-2 Standard, [132].

The IEC 61000-4-2 Standard gives a graphical representation of the typ-ical ESD current, Figure 3.10, and also defines, for a given test level voltage,required values of ESD current’s key parameters.

The values of the ESD currents key parameters are listed in Table 3.2for the case of the contact discharge, where Ipeak is the ESD current initialpeak, tr is the rising time defined as the difference between time momentscorresponding to 10% and 90% of the current peak Ipeak, I30 and I60 are theESD current values calculated for time periods of 30 and 60 ns, respectively,starting from the time point corresponding to 10% of Ipeak.

In this section we present the results of fitting a 2-peak AEF to theStandard ESD current given in IEC 61000-4-2. Data points which are usedin the optimization procedure are manually sampled from the graphicallygiven Standard [132] current function, Figure 3.10. The peak currents andcorresponding times are also extracted, and the results of D-optimal inter-polation with two peaks are illustrated, see Figure 3.11. The parameters arelisted in Table 3.3. In the illustrated examples a fairly good fit is found buttypically areas with steeply rising and decaying parts are somewhat moredifficult to fit with good accuracy than the other parts of the waveform.

150

152


D-OPTIMAL DESIGN

Figure 3.10: IEC 61000-4-2 Standard ESDcurrent waveform with parameters, [132](image slightly modified for clarity).

U [kV] Ipeak [A] tr [ns]

2 7.5± 15% 0.8± 25%

4 15.0± 15% 0.8± 25%

6 22.5± 15% 0.8± 25%

8 30.0± 15% 0.8± 25%

U [kV] I30 [A] I60 [A]

2 4.0± 30% 2.0± 30%

4 8.0± 30% 4.0± 30%

6 12.0± 30% 6.0± 30%

8 16.0± 30% 8.0± 30%

Table 3.2: IEC 61000-4-2standard ESD current parame-ters [132].

t [s] #10-8

0 2 4 6 8

i(t)

0

5

10

15IEC 61000-4-22-peaked AEFPeaksInterpolated points

Figure 3.11: 2-peaked AEF interpo-lated on a D-optimal design represent-ing the IEC 61000-4-2 Standard ESDcurrent waveshape for 4 kV.

Local maxima and minima and

corresponding times extracted

from IEC 61000-4-2, [132]

Peak current [A] Peak time [ns]

Imax1 = 15 tmax1 = 6.89

Imin1 = 7.1484 tmin1 = 12.85

Imax2 = 9.0921 tmax2 = 25.54

Parameters of interpolated AEF

Interval n k c

0 ≤ t ≤ tmax1 3 1 0.01385

tmax1 ≤ t ≤ tmax2 3 4 2.025

tmax2 < t 5 10 2.395

Table 3.3: Parameters’ values of2-peaked AEF representing the IEC61000-4-2 Standard ESD currentwaveshape for 4 kV.

151


D-OPTIMAL DESIGN

Figure 3.10: IEC 61000-4-2 Standard ESDcurrent waveform with parameters, [132](image slightly modified for clarity).

U [kV] Ipeak [A] tr [ns]

2 7.5± 15% 0.8± 25%

4 15.0± 15% 0.8± 25%

6 22.5± 15% 0.8± 25%

8 30.0± 15% 0.8± 25%

U [kV] I30 [A] I60 [A]

2 4.0± 30% 2.0± 30%

4 8.0± 30% 4.0± 30%

6 12.0± 30% 6.0± 30%

8 16.0± 30% 8.0± 30%

Table 3.2: IEC 61000-4-2standard ESD current parame-ters [132].

t [s] #10-8

0 2 4 6 8

i(t)

0

5

10

15IEC 61000-4-22-peaked AEFPeaksInterpolated points

Figure 3.11: 2-peaked AEF interpo-lated on a D-optimal design represent-ing the IEC 61000-4-2 Standard ESDcurrent waveshape for 4 kV.

Local maxima and minima and

corresponding times extracted

from IEC 61000-4-2, [132]

Peak current [A] Peak time [ns]

Imax1 = 15 tmax1 = 6.89

Imin1 = 7.1484 tmin1 = 12.85

Imax2 = 9.0921 tmax2 = 25.54

Parameters of interpolated AEF

Interval n k c

0 ≤ t ≤ tmax1 3 1 0.01385

tmax1 ≤ t ≤ tmax2 3 4 2.025

tmax2 < t 5 10 2.395

Table 3.3: Parameters’ values of2-peaked AEF representing the IEC61000-4-2 Standard ESD currentwaveshape for 4 kV.

151

153


3-peaked AEF representing measured current from ESD

In this section we present the results of fitting a 3-peaked AEF to a waveformfrom experimental measurements from [151]. The result is also compared toa common type of function used for modelling ESD current, also from [151].

In Figures 3.12 and 3.13 the results of the interpolation of D-optimalpoints are shown together with the measured data, as well as a sum of twoHeidler functions that was fitted to the experimental data in [151]. Thisfunction is given by

i(t) = I1

(tτ1

)nH1 +

(tτ1

)nH e− tτ2 + I2

(tτ3

)nH1 +

(tτ3

)nH e− tτ4 ,

I1 = 31.365 A, I2 = 6.854 A, nH = 4.036,

τ1 = 1.226 ns, τ2 = 1.359 ns,

τ3 = 3.982 ns, τ4 = 28.817 ns.

Note that this function does not reproduce the second local minimumbut that all three AEF functions can reproduce all local minima and maxima(to a modest degree of accuracy) when suitable values for the n, k and cparameters are chosen. In Figure 3.13 we can see that even small bumps inthe rising part are successfully reproduced.

3.3.6 Modelling of lightning discharge currents

IEC 61312-1 standard current waveshape

In this section we use the scheme to represent the IEC 61312-1 Standardcurrent wave shape as it is described in [117]. Rather than being givengraphically, as the IEC 61000-4-2 Standard current waveform, the shape isdescribed using a Heidler function,

i(t) =Ipeakη

(tT

)n1 +

(tT

)n e− tτ (137)

whose parameters are chosen according to Table 3.4.In Figures 3.14 and 3.15 the results of fitting an AEF by interpolating

on a D-optimal design to the first stroke of a protection level I IEC 61312-1Standard waveshape are shown. The parameters of the fitted AEF are givenin Table 3.6. In this case the waveshape can be reproduced fairly well butgives a relatively complicated expression compared to (137).

152


3-peaked AEF representing measured current from ESD

In this section we present the results of fitting a 3-peaked AEF to a waveformfrom experimental measurements from [151]. The result is also compared toa common type of function used for modelling ESD current, also from [151].

In Figures 3.12 and 3.13 the results of the interpolation of D-optimalpoints are shown together with the measured data, as well as a sum of twoHeidler functions that was fitted to the experimental data in [151]. Thisfunction is given by

i(t) = I1

(tτ1

)nH1 +

(tτ1

)nH e− tτ2 + I2

(tτ3

)nH1 +

(tτ3

)nH e− tτ4 ,

I1 = 31.365 A, I2 = 6.854 A, nH = 4.036,

τ1 = 1.226 ns, τ2 = 1.359 ns,

τ3 = 3.982 ns, τ4 = 28.817 ns.

Note that this function does not reproduce the second local minimumbut that all three AEF functions can reproduce all local minima and maxima(to a modest degree of accuracy) when suitable values for the n, k and cparameters are chosen. In Figure 3.13 we can see that even small bumps inthe rising part are successfully reproduced.

3.3.6 Modelling of lightning discharge currents

IEC 61312-1 standard current waveshape

In this section we use the scheme to represent the IEC 61312-1 Standardcurrent wave shape as it is described in [117]. Rather than being givengraphically, as the IEC 61000-4-2 Standard current waveform, the shape isdescribed using a Heidler function,

i(t) =Ipeakη

(tT

)n1 +

(tT

)n e− tτ (137)

whose parameters are chosen according to Table 3.4.In Figures 3.14 and 3.15 the results of fitting an AEF by interpolating

on a D-optimal design to the first stroke of a protection level I IEC 61312-1Standard waveshape are shown. The parameters of the fitted AEF are givenin Table 3.6. In this case the waveshape can be reproduced fairly well butgives a relatively complicated expression compared to (137).

152

154


D-OPTIMAL DESIGN

Modelling a measured lightning discharge current

In this section we fit a 13-peaked AEF function both with free parameters (asin [145]) and using interpolation on a D-optimal design, to data extractedfrom [130] that comes from measurements of a lightning strike on MountSantis in Switzerland [245].

The results are shown in Figures 3.16 (a)–3.16 (d). It can be seen thatin most cases the AEF with free parameters gives a closer fit but the versioninterpolated on a D-optimal design is often comparable. Parameters for theD-optimal fitting can be found in Table 3.7.

Modelling the lightning discharge current derivative

Here we present some results when attempting to reproduce the derivative ofthe waveshape of the lightning discharge current using the AEF interpolatedon a D-optimal design. We also compare the result of this fitting scheme tothe results in [190] where the parameters of the AEF are chosen freely andfitted using the Marquardt Least Squares Method.

The method for fitting an AEF described here is applied to the modellingof lightning current derivative signals measured at the CN Tower [130]. Theresults of the fitting can be seen in Figure 3.17. From these figures it isclear that in this case of several peaks and few terms in each interval thetwo schemes for fitting the AEF are often similar in quality but sometimesthe extra flexibility offered when letting all the exponents in the AEF bechosen individually can give a significantly better fit, an example of this canbe seen in Figure 3.17. A possible explanation for this in this case is that inthe scheme for D-optimal fitting you need many terms in order to have bothsmall and large exponents. In Figure 3.18 we examine how well the differentfitting schemes model the current when they are integrated. Here we can seethat the free parameter version gives a considerably better matching to thenumerically integrated measured values than the D-optimal fitting version.

Protection level Parameter First stroke Subsequent stroke

n 10 10

T 19.0 µs 0.454 µs

τ 485 µs 143 µs

η 0.930 0.993

I Ipeak 200 kA 50 kA

II Ipeak 150 kA 37.5 kA

III-IV Ipeak 100 kA 25 kA

Table 3.4: IEC 61312-1 standard current key parameters, [134].

153


D-OPTIMAL DESIGN

Modelling a measured lightning discharge current

In this section we fit a 13-peaked AEF function both with free parameters (asin [145]) and using interpolation on a D-optimal design, to data extractedfrom [130] that comes from measurements of a lightning strike on MountSantis in Switzerland [245].

The results are shown in Figures 3.16 (a)–3.16 (d). It can be seen thatin most cases the AEF with free parameters gives a closer fit but the versioninterpolated on a D-optimal design is often comparable. Parameters for theD-optimal fitting can be found in Table 3.7.

Modelling the lightning discharge current derivative

Here we present some results when attempting to reproduce the derivative ofthe waveshape of the lightning discharge current using the AEF interpolatedon a D-optimal design. We also compare the result of this fitting scheme tothe results in [190] where the parameters of the AEF are chosen freely andfitted using the Marquardt Least Squares Method.

The method for fitting an AEF described here is applied to the modellingof lightning current derivative signals measured at the CN Tower [130]. Theresults of the fitting can be seen in Figure 3.17. From these figures it isclear that in this case of several peaks and few terms in each interval thetwo schemes for fitting the AEF are often similar in quality but sometimesthe extra flexibility offered when letting all the exponents in the AEF bechosen individually can give a significantly better fit, an example of this canbe seen in Figure 3.17. A possible explanation for this in this case is that inthe scheme for D-optimal fitting you need many terms in order to have bothsmall and large exponents. In Figure 3.18 we examine how well the differentfitting schemes model the current when they are integrated. Here we can seethat the free parameter version gives a considerably better matching to thenumerically integrated measured values than the D-optimal fitting version.

Protection level Parameter First stroke Subsequent stroke

n 10 10

T 19.0 µs 0.454 µs

τ 485 µs 143 µs

η 0.930 0.993

I Ipeak 200 kA 50 kA

II Ipeak 150 kA 37.5 kA

III-IV Ipeak 100 kA 25 kA

Table 3.4: IEC 61312-1 standard current key parameters, [134].

153

155


t [s] #10-8

0 1 2 3 4 5 6 7 8 9

i(t)

0

2

4

6

Measured data3-peaked AEFTwo Heidler functionPeaksInterpolated points

Figure 3.12: 3-peaked AEF interpolated to a D-optimal design from measuredESD current from [151, Figure 3] compared with an approxima-tion suggested in [151]. Parameters are given in Table 3.5.

t [s] #10-9

0 2 4

i(t)

0

2

4

6


Figure 3.13: Close-up of the risingpart of a 3-peaked AEF interpolatedto a D-optimal design from measuredESD current from [151, Figure 3].Parameters are given in Table 3.5.

Local maxima and corresponding times extracted from [151, Figure 3]

Imax1 = 7.37 [A] Imax2 = 5.02 [A] Imax3 = 3.82 [A]

tmax1 = 1.23 [ns] tmax2 = 6.39 [ns] tmax3 = 15.5 [ns]

Parameters of interpolated AEF shown in Figure 3.12

Interval n k c

0 ≤ t ≤ tmax1 5 5 0.05750

tmax1 ≤ t ≤ tmax2 3 1 0.4920

tmax2 ≤ t ≤ tmax3 4 2 0.5967

tmax3 < t 6 1 1.019

Table 3.5: Parameters’ values of AEF with 3 peaks representing measured ESDcurrent from [151, Figure 3].

154


t [s] #10-8

0 1 2 3 4 5 6 7 8 9

i(t)

0

2

4

6


Figure 3.12: 3-peaked AEF interpolated to a D-optimal design from measuredESD current from [151, Figure 3] compared with an approxima-tion suggested in [151]. Parameters are given in Table 3.5.

t [s] #10-9

0 2 4

i(t)

0

2

4

6


Figure 3.13: Close-up of the risingpart of a 3-peaked AEF interpolatedto a D-optimal design from measuredESD current from [151, Figure 3].Parameters are given in Table 3.5.

Local maxima and corresponding times extracted from [151, Figure 3]

Imax1 = 7.37 [A] Imax2 = 5.02 [A] Imax3 = 3.82 [A]

tmax1 = 1.23 [ns] tmax2 = 6.39 [ns] tmax3 = 15.5 [ns]


Interval n k c

0 ≤ t ≤ tmax1 5 5 0.05750

tmax1 ≤ t ≤ tmax2 3 1 0.4920

tmax2 ≤ t ≤ tmax3 4 2 0.5967

tmax3 < t 6 1 1.019

Table 3.5: Parameters’ values of AEF with 3 peaks representing measured ESDcurrent from [151, Figure 3].

154

156


D-OPTIMAL DESIGN

t [s] #10-3

0 0.5 1 1.5 2 2.5 3

i(t)

0

20

40

60

80

IEC 61312-1D-optimal AEFPeakInterpolated sample points

Figure 3.14: AEF with 1 peak fitted by interpolating D-optimal points sampledfrom the Heidler function describing the IEC 61312-1 waveshapegiven by (137). Parameters are given in Table 3.6.

t [s] #10-5

0 1 2 3

i(t)

0

20

40

60

80

IEC 61312-1D-optimal AEFPeakInterpolated points Figure 3.15: Close-up of the rising part of

the AEF with 1 peak fitted by interpolatingD-optimal points samples from the Heidlerfunction describing the IEC61312-1 wave-shape given by (137).Parameters are given in Table 3.6.

Chosen peak time and current

tmax = 28.14 [µs] I = 92.54 [kA]


Interval n k c

0 ≤ t ≤ tmax 4 10 0.7565tmax < t 5 1 41.82

Table 3.6: Parameters’ values of AEF representing the IEC 61312-1 standardwaveshape.

155


D-OPTIMAL DESIGN

t [s] #10-3

0 0.5 1 1.5 2 2.5 3

i(t)

0

20

40

60

80

IEC 61312-1D-optimal AEFPeakInterpolated sample points

Figure 3.14: AEF with 1 peak fitted by interpolating D-optimal points sampledfrom the Heidler function describing the IEC 61312-1 waveshapegiven by (137). Parameters are given in Table 3.6.

t [s] #10-5

0 1 2 3

i(t)

0

20

40

60

80

IEC 61312-1D-optimal AEFPeakInterpolated points Figure 3.15: Close-up of the rising part of

the AEF with 1 peak fitted by interpolatingD-optimal points samples from the Heidlerfunction describing the IEC61312-1 wave-shape given by (137).Parameters are given in Table 3.6.

Chosen peak time and current

tmax = 28.14 [µs] I = 92.54 [kA]


Interval n k c

0 ≤ t ≤ tmax 4 10 0.7565tmax < t 5 1 41.82

Table 3.6: Parameters’ values of AEF representing the IEC 61312-1 standardwaveshape.

155

157


t [s] #10-4

0 2 4 6 8

i(t)

0

2

4

6

8

10Measured dataD-optimal AEFFree parameter AEFPeaksInterpolated sample points

(a) Comparison of data and AEFs fromt = −0.3437 µs to t = 888.1 µs.

t [s] #10-4

0 2 4 6 8

i(t)

0

0.2

0.4

0.6

0.8

1

1.2Residual D-optimal AEFResidual MLSM AEF .

(b) Residuals for the AEFs and data fromt = −0.3437 µs to t = 888.1 µs.

t [s] #10-6

0 2 4 6 8

i(t)

0

2

4

6

8

10

Measured dataD-optimal AEFFree parameter AEFPeaksInterpolated sample points

(c) Comparison of data and AEFs fromt = −0.3437 µs to t = 9.280 µs.

t [s] #10-6

0 2 4 6 8

i(t)

0

0.2

0.4

0.6

0.8

1


(d) Residuals for the AEFs and data fromt = −0.3437 µs to t = 9.280 µs.

Figure 3.16: Comparison of two AEFs with 13 peaks and 2 terms in eachinterval fitted to measured lightning discharge current derivativefrom [69]. One is fitted by interpolation on D-optimal points andthe other is fitted with free parameters using the MLSM method.Parameters of the D-optimal version are given in Table 3.7.

156


t [s] #10-4

0 2 4 6 8

i(t)

0

2

4

6

8

10Measured dataD-optimal AEFFree parameter AEFPeaksInterpolated sample points

(a) Comparison of data and AEFs fromt = −0.3437 µs to t = 888.1 µs.

t [s] #10-4

0 2 4 6 8

i(t)

0

0.2

0.4

0.6

0.8

1


(b) Residuals for the AEFs and data fromt = −0.3437 µs to t = 888.1 µs.

t [s] #10-6

0 2 4 6 8

i(t)

0

2

4

6

8

10

Measured dataD-optimal AEFFree parameter AEFPeaksInterpolated sample points

(c) Comparison of data and AEFs fromt = −0.3437 µs to t = 9.280 µs.

t [s] #10-6

0 2 4 6 8

i(t)

0

0.2

0.4

0.6

0.8

1


(d) Residuals for the AEFs and data fromt = −0.3437 µs to t = 9.280 µs.

Figure 3.16: Comparison of two AEFs with 13 peaks and 2 terms in eachinterval fitted to measured lightning discharge current derivativefrom [69]. One is fitted by interpolation on D-optimal points andthe other is fitted with free parameters using the MLSM method.Parameters of the D-optimal version are given in Table 3.7.

156

158


D-OPTIMAL DESIGN

Peak times and currents Parameters of fitted AEF

t [µs] I [µs] Interval n k c

t1 = 0.3998 I1 = 8.159 0 ≤ t ≤ t1 2 2 0.4773

t2 = 0.9468 I2 = 10.96 t1 ≤ t ≤ t2 2 10 2.148

t3 = 1.458 I3 = 11.14 t2 ≤ t ≤ t3 2 1 0.3964

t4 = 1.873 I4 = 10.26 t3 ≤ t ≤ t4 2 1 0.2210

t5 = 2.475 I5 = 10.07 t4 ≤ t ≤ t5 2 10 1.695

t6 = 2.904 I6 = 9.819 t5 ≤ t ≤ t6 2 1 0.4591

t7 = 3.533 I7 = 8.519 t6 ≤ t ≤ t7 2 1 0.3503

t8 = 3.985 I8 = 9.097 t7 ≤ t ≤ t8 2 10 3.716

t9 = 5.036 I9 = 8.485 t8 ≤ t ≤ t9 2 1 0.6963

t10 = 6.168 I10 = 8.310 t9 ≤ t ≤ t10 2 1 0.2954

t11 = 8.472 I11 = 8.413 t10 ≤ t ≤ t11 2 6 3.074

t12 = 20.48 I12 = 8.576 t11 ≤ t ≤ t12 2 1 0.2784

t13 = 137.5 I13 = 4.178 t12 ≤ t ≤ t13 2 1 0.6456

t13 < t 4 1 0.3559

Table 3.7: Parameters’ values of AEF with 13 peaks representing measureddata for a lightning discharge current from [245]. Local maximaand corresponding times extracted from [69, Figures 6, 7 and 8] aredenoted t and I and other parameters correspond to the fitted AEFshown in Figures 3.16 (a), 3.16 (b) and 3.16 (c).

157


D-OPTIMAL DESIGN



t1 = 0.3998 I1 = 8.159 0 ≤ t ≤ t1 2 2 0.4773

t2 = 0.9468 I2 = 10.96 t1 ≤ t ≤ t2 2 10 2.148

t3 = 1.458 I3 = 11.14 t2 ≤ t ≤ t3 2 1 0.3964

t4 = 1.873 I4 = 10.26 t3 ≤ t ≤ t4 2 1 0.2210

t5 = 2.475 I5 = 10.07 t4 ≤ t ≤ t5 2 10 1.695

t6 = 2.904 I6 = 9.819 t5 ≤ t ≤ t6 2 1 0.4591

t7 = 3.533 I7 = 8.519 t6 ≤ t ≤ t7 2 1 0.3503

t8 = 3.985 I8 = 9.097 t7 ≤ t ≤ t8 2 10 3.716

t9 = 5.036 I9 = 8.485 t8 ≤ t ≤ t9 2 1 0.6963

t10 = 6.168 I10 = 8.310 t9 ≤ t ≤ t10 2 1 0.2954

t11 = 8.472 I11 = 8.413 t10 ≤ t ≤ t11 2 6 3.074

t12 = 20.48 I12 = 8.576 t11 ≤ t ≤ t12 2 1 0.2784

t13 = 137.5 I13 = 4.178 t12 ≤ t ≤ t13 2 1 0.6456

t13 < t 4 1 0.3559

Table 3.7: Parameters’ values of AEF with 13 peaks representing measureddata for a lightning discharge current from [245]. Local maximaand corresponding times extracted from [69, Figures 6, 7 and 8] aredenoted t and I and other parameters correspond to the fitted AEFshown in Figures 3.16 (a), 3.16 (b) and 3.16 (c).

157

159


t [s] #10-6

0 2 4 6

di/d

t[kA

/s]

0

10

20

30

Measured dataD-optimal 12-peaked AEFFree parameter AEFPeaksInterpolated sample points

Figure 3.17: Comparison of two AEFswith 12 peaks and 2 terms in each in-terval fitted to measured lightning dis-charge current derivative from [130].Parameters are given in Table 3.8.

t [s] #10-6

0 2 4 6

i [k

A]

0

5

10

Measured dataD-optimal AEFFree parameter AEF

Figure 3.18: Comparison of results ofintegrating the approximating functionshown in Figure 3.17.



t0 = −0.3437 I0 = 0 t0 ≤ t ≤ t1 2 10 0.06099

t1 = 0.9468 I1 = 36.65 t1 ≤ t ≤ t2 2 1 0.4506

t2 = 0.5001 I2 = −2.208 t2 ≤ t ≤ t3 3 1 0.04772

t3 = 0.9215 I3 = 6.89 t3 ≤ t ≤ t4 2 1 0.4502

t4 = 1.212 I4 = −7.322 t4 ≤ t ≤ t5 3 1 0.2590

t5 = 1.714 I5 = 3.402 t5 ≤ t ≤ t6 3 2 0.9067

t6 = 2.103 I6 = 1.319 t6 ≤ t ≤ t7 3 1 0.3333

t7 = 2.730 I7 = −1.844 t7 ≤ t ≤ t8 3 1 0.03732

t8 = 3.416 I8 = 16.08 t8 ≤ t ≤ t9 2 4 3.3793

t9 = 4.005 I9 = −5.787 t9 ≤ t ≤ t10 2 1 1.4912

t10 = 4.216 I10 = −0.1268 t10 ≤ t ≤ t11 2 2 0.09448

t11 = 4.875 I11 = 1.972 t11 ≤ t ≤ t12 2 6 2.288

t12 = 5.538 I12 = 1.683 t13 < t 3 1 0.001705

Table 3.8: Parameters’ value of AEF with 12 peaks representing measured datafor a lightning discharge current derivative from [130]. Chosen peaktimes are denoted t and I and other parameters correspond to thefitted AEF shown in Figure 3.17.

158


t [s] #10-6

0 2 4 6

di/d

t[kA

/s]

0

10

20

30

Measured dataD-optimal 12-peaked AEFFree parameter AEFPeaksInterpolated sample points

Figure 3.17: Comparison of two AEFswith 12 peaks and 2 terms in each in-terval fitted to measured lightning dis-charge current derivative from [130].Parameters are given in Table 3.8.

t [s] #10-6

0 2 4 6

i [k

A]

0

5

10

Measured dataD-optimal AEFFree parameter AEF

Figure 3.18: Comparison of results ofintegrating the approximating functionshown in Figure 3.17.



t0 = −0.3437 I0 = 0 t0 ≤ t ≤ t1 2 10 0.06099

t1 = 0.9468 I1 = 36.65 t1 ≤ t ≤ t2 2 1 0.4506

t2 = 0.5001 I2 = −2.208 t2 ≤ t ≤ t3 3 1 0.04772

t3 = 0.9215 I3 = 6.89 t3 ≤ t ≤ t4 2 1 0.4502

t4 = 1.212 I4 = −7.322 t4 ≤ t ≤ t5 3 1 0.2590

t5 = 1.714 I5 = 3.402 t5 ≤ t ≤ t6 3 2 0.9067

t6 = 2.103 I6 = 1.319 t6 ≤ t ≤ t7 3 1 0.3333

t7 = 2.730 I7 = −1.844 t7 ≤ t ≤ t8 3 1 0.03732

t8 = 3.416 I8 = 16.08 t8 ≤ t ≤ t9 2 4 3.3793

t9 = 4.005 I9 = −5.787 t9 ≤ t ≤ t10 2 1 1.4912

t10 = 4.216 I10 = −0.1268 t10 ≤ t ≤ t11 2 2 0.09448

t11 = 4.875 I11 = 1.972 t11 ≤ t ≤ t12 2 6 2.288

t12 = 5.538 I12 = 1.683 t13 < t 3 1 0.001705

Table 3.8: Parameters’ value of AEF with 12 peaks representing measured datafor a lightning discharge current derivative from [130]. Chosen peaktimes are denoted t and I and other parameters correspond to thefitted AEF shown in Figure 3.17.

158

160


D-OPTIMAL DESIGN

3.3.7 Summary of ESD modelling

Here we examined a mathematical model for representation of ESD currents,either from the IEC 61000-4-2 Standard [132], or experimentally measuredones. The model has been proposed and successfully applied to lightningcurrent modelling in Section 3.2 and [183] and named the multi-peakedanalytically extended function (AEF).

It conforms to the requirements for the ESD current and its first deriva-tive, which are imposed by the Standard [132] stating that they must beequal to zero at moment t = 0. Furthermore, the AEF function is time-integrable, see Section 3.1.1, which is necessary for numerical calculation ofradiated fields originating from the ESD current.

We also consider how the model can be fitted to a waveform by restrictingthe exponents in the AEF to an arithmetic sequence and then interpolatepoints of the function we wish to approximate chosen according to a D-optimal design. This makes the modelling less flexible than the case whereall exponents can be chosen freely but gives a scheme for fitting the functionthat scales better to many data points than the MLSM fitting scheme usedin [145,183,184,188].

We apply the resulting methodology to some realistic cases, either takenfrom standards, see Section 3.3.5 and 3.3.6, or measured data, see Sections3.3.5, 3.3.6 and 3.3.6. The methodology can give fairly accurate results evenwith a modest number of interpolated points but strategies for choosing someof the involved parameters should be further investigated. The decayingpart of the waveforms are consistently difficult to fit and if the models areused in a context where significant error propagation appears a more flexibleapproach can be desirable.

159


D-OPTIMAL DESIGN

3.3.7 Summary of ESD modelling

Here we examined a mathematical model for representation of ESD currents,either from the IEC 61000-4-2 Standard [132], or experimentally measuredones. The model has been proposed and successfully applied to lightningcurrent modelling in Section 3.2 and [183] and named the multi-peakedanalytically extended function (AEF).

It conforms to the requirements for the ESD current and its first deriva-tive, which are imposed by the Standard [132] stating that they must beequal to zero at moment t = 0. Furthermore, the AEF function is time-integrable, see Section 3.1.1, which is necessary for numerical calculation ofradiated fields originating from the ESD current.

We also consider how the model can be fitted to a waveform by restrictingthe exponents in the AEF to an arithmetic sequence and then interpolatepoints of the function we wish to approximate chosen according to a D-optimal design. This makes the modelling less flexible than the case whereall exponents can be chosen freely but gives a scheme for fitting the functionthat scales better to many data points than the MLSM fitting scheme usedin [145,183,184,188].

We apply the resulting methodology to some realistic cases, either takenfrom standards, see Section 3.3.5 and 3.3.6, or measured data, see Sections3.3.5, 3.3.6 and 3.3.6. The methodology can give fairly accurate results evenwith a modest number of interpolated points but strategies for choosing someof the involved parameters should be further investigated. The decayingpart of the waveforms are consistently difficult to fit and if the models areused in a context where significant error propagation appears a more flexibleapproach can be desirable.

159

161

162

Chapter 4

Comparison of models ofmortality rate

This chapter is based on Papers H, and I


Paper I Andromachi Boulougari, Karl Lundengard, Milica Rancic,Sergei Silvestrov, Belinda Strass and Samya Suleiman.Application of a power-exponential function based model tomortality rates forecasting.Communications in Statistics: Case Studies, Data Analysis andApplications, Volume 5, Issue 1, pages 3 – 10, 2019.

Chapter 4

Comparison of models ofmortality rate

This chapter is based on Papers H, and I


Paper I Andromachi Boulougari, Karl Lundengard, Milica Rancic,Sergei Silvestrov, Belinda Strass and Samya Suleiman.Application of a power-exponential function based model tomortality rates forecasting.Communications in Statistics: Case Studies, Data Analysis andApplications, Volume 5, Issue 1, pages 3 – 10, 2019.

163


4.1 Modelling and forecasting mortality rates

In this chapter an overview of models of mortality rate found in literature willbe given and three new models introduced. The different models will thenbe compared to each other by fitting the models to the central mortality ratefor men in various countries and computing the corresponding AIC values.

After that we will examine what happens when the central mortalityrate is replaced with mortality rates given by a fitted model in the Lee–Carter method of forecasting described in Section 1.6.1. We will attempt tocharacterize the behaviours of a few models using data from a few differentcountries by looking at how well the values produced by the different mod-els match the assumptions of the Lee–Carter method and how reliable theparameters in the forecasting model are when based on different data sets.

4.2 Overview of models

Modelling all the factors that can affect the lifespan of an individual is notfeasible, instead highly simplified models are used. We will consider somepreviously introduced models, see Table 4.1. Many of these models appearunder different names, can be written in different ways and have severalvariants. For the comparison only the form written in the table will beconsidered. For all models all parameters are real-valued and non-negativeexcept for Hannerz model where the parameters can be negative.

In Section 1.6 the basic properties of survival functions and mortalityrate were discussed. Here we will quickly summarize them for convenience.

The survival function Sx(∆x) gives the probability that an individualsurvives another ∆x units of time. It is related to µ as follows

µx = −dS0dx

S0(x), Sx(∆x) = exp

(−∫ x+∆x

xµ(t) dt

).

There are three conditions that a survival function must satisfy to have areasonable interpretation in terms of lifespan, Sx(0) = 1, lim

∆x→∞Sx(∆x) = 0

and Sx(∆x) must be non-increasing. See page 65 for motivation.There are four patterns that are commonly observed when examining

central mortality rate for developed countries:

1. Rapid decrease in mortality rate for young ages, µ(x) ∼ 1x for small x.

2. A ’hump’ for young adults with a rapid increase that levels off andremains constant or slowly decreases for some years.

3. Exponential growth for higher ages, µ(x) ∼ ecx for large x.4. Deceleration of mortality rate growth for the highest ages.

In Table 4.1 the Heligman–Pollard 1 model and all subsequent modelscan describe all these patterns to some extent. The models that cannot haseither been important historically or can describe some age interval well.

162


4.1 Modelling and forecasting mortality rates

In this chapter an overview of models of mortality rate found in literature willbe given and three new models introduced. The different models will thenbe compared to each other by fitting the models to the central mortality ratefor men in various countries and computing the corresponding AIC values.

After that we will examine what happens when the central mortalityrate is replaced with mortality rates given by a fitted model in the Lee–Carter method of forecasting described in Section 1.6.1. We will attempt tocharacterize the behaviours of a few models using data from a few differentcountries by looking at how well the values produced by the different mod-els match the assumptions of the Lee–Carter method and how reliable theparameters in the forecasting model are when based on different data sets.

4.2 Overview of models

Modelling all the factors that can affect the lifespan of an individual is notfeasible, instead highly simplified models are used. We will consider somepreviously introduced models, see Table 4.1. Many of these models appearunder different names, can be written in different ways and have severalvariants. For the comparison only the form written in the table will beconsidered. For all models all parameters are real-valued and non-negativeexcept for Hannerz model where the parameters can be negative.

In Section 1.6 the basic properties of survival functions and mortalityrate were discussed. Here we will quickly summarize them for convenience.

The survival function Sx(∆x) gives the probability that an individualsurvives another ∆x units of time. It is related to µ as follows

µx = −dS0dx

S0(x), Sx(∆x) = exp

(−∫ x+∆x

xµ(t) dt

).

There are three conditions that a survival function must satisfy to have areasonable interpretation in terms of lifespan, Sx(0) = 1, lim

∆x→∞Sx(∆x) = 0

and Sx(∆x) must be non-increasing. See page 65 for motivation.There are four patterns that are commonly observed when examining

central mortality rate for developed countries:

1. Rapid decrease in mortality rate for young ages, µ(x) ∼ 1x for small x.

2. A ’hump’ for young adults with a rapid increase that levels off andremains constant or slowly decreases for some years.

3. Exponential growth for higher ages, µ(x) ∼ ecx for large x.4. Deceleration of mortality rate growth for the highest ages.

In Table 4.1 the Heligman–Pollard 1 model and all subsequent modelscan describe all these patterns to some extent. The models that cannot haseither been important historically or can describe some age interval well.

162

164

4.2. OVERVIEW OF MODELS

Gompertz–Makeham [92] µ(x) = a+ becx

Weibull [289] µ(x) =a

b

(xb

)a−1

Logistic [29] µ(x) =aebx

1 +ac

b(ebx − 1)

Modified Perks [41] µ(x) =a

1 + eb−cx+ d

Gompertz inverse Gaussian [41] µ(x) =ea−bx√

1 + e−c+bx

Double Geometric [92] µ(x) = a+ b1bx2 + c1c

x2

Thiele [272] µ(x) = a1e−b1x + a2e

−b2 (x−c)22 + a3e

b3x

Heligman–Pollard 1 [122] µ(x) = a(x+a2)a31 + b1e

−b2 ln(xb3

)2

+ c1cx2


−b2 ln(xb3

)2

+c1c

x2

1 + c1cx2


−b2 ln(xb3

)2

+c1c

x2

1 + c3c1cx2


−b2 ln(xb3

)2

+c1c

xc32

1 + c1cxc3

2

Hannerz [114]

µ(x) =f(x)

1 + F (x)with f(x) = α

g1(x)eG1(x)

(1 + eG1(x))2+ (1− α)

g2(x)eG2(x)

(1 + eG2(x))2,

F (x) = αeG1(x)

1 + eG1(x)+ (1− α)

eG2(x)

1 + eG2(x),

g1(x) =a1

x2+ a2x+ a3e

cx, G1(x) = a0 −a1

x+a2x

2

2+a3

cecx,

g2(x) =a5

x2+ a2x+ a3e

cx and G2(x) = a4 −a5

x+a2x

2

2+a3

cecx

First Time Exit Model: SKI-6 [139]

µ(x) =g(x)∫ ∞

xg(t) dt

with g(x) = k√x3

exp(−H2

x2x

), H(x) = a1 + ax4 − b

√x+ lx2 − cx3

First Time Exit Model: Fractional 1st order approximation [261]

µ(x) =g(x)∫ ∞

xg(t) dt

where g(x) =2|l + (c− 1)(bx)c|

σ√

2πx3exp

(−−(l − (bx)c)2

2σ2x

)First Time Exit Model: Fractional 2nd order approximation [261]

µ(x) =g(x)∫ ∞

xg(t) dt

where

g(x) =2

σ√

2πx

(2|l + (c− 1)(bx)c|

σ√

2πx+ k

c(c− 1)(bx)c

2|l + (c− 1)(bx)c|

)exp

(−−(l − (bx)c)2

2σ2x

)Table 4.1: List of the models of mortality rate previously suggested in literature

that are considered in this paper. The references gives a source witha more detailed description of the model, not necessarily the originalsource of the model.

163

4.2. OVERVIEW OF MODELS

Gompertz–Makeham [92] µ(x) = a+ becx

Weibull [289] µ(x) =a

b

(xb

)a−1

Logistic [29] µ(x) =aebx

1 +ac

b(ebx − 1)

Modified Perks [41] µ(x) =a

1 + eb−cx+ d

Gompertz inverse Gaussian [41] µ(x) =ea−bx√

1 + e−c+bx

Double Geometric [92] µ(x) = a+ b1bx2 + c1c

x2

Thiele [272] µ(x) = a1e−b1x + a2e

−b2 (x−c)22 + a3e

b3x


−b2 ln(xb3

)2

+ c1cx2


−b2 ln(xb3

)2

+c1c

x2

1 + c1cx2


−b2 ln(xb3

)2

+c1c

x2

1 + c3c1cx2


−b2 ln(xb3

)2

+c1c

xc32

1 + c1cxc3

2

Hannerz [114]

µ(x) =f(x)

1 + F (x)with f(x) = α

g1(x)eG1(x)

(1 + eG1(x))2+ (1− α)

g2(x)eG2(x)

(1 + eG2(x))2,

F (x) = αeG1(x)

1 + eG1(x)+ (1− α)

eG2(x)

1 + eG2(x),

g1(x) =a1

x2+ a2x+ a3e

cx, G1(x) = a0 −a1

x+a2x

2

2+a3

cecx,

g2(x) =a5

x2+ a2x+ a3e

cx and G2(x) = a4 −a5

x+a2x

2

2+a3

cecx

First Time Exit Model: SKI-6 [139]

µ(x) =g(x)∫ ∞

xg(t) dt

with g(x) = k√x3

exp(−H2

x2x

), H(x) = a1 + ax4 − b

√x+ lx2 − cx3

First Time Exit Model: Fractional 1st order approximation [261]

µ(x) =g(x)∫ ∞

xg(t) dt

where g(x) =2|l + (c− 1)(bx)c|

σ√

2πx3exp

(−−(l − (bx)c)2

2σ2x

)First Time Exit Model: Fractional 2nd order approximation [261]

µ(x) =g(x)∫ ∞

xg(t) dt

where

g(x) =2

σ√

2πx

(2|l + (c− 1)(bx)c|

σ√

2πx+ k

c(c− 1)(bx)c

2|l + (c− 1)(bx)c|

)exp

(−−(l − (bx)c)2

2σ2x

)Table 4.1: List of the models of mortality rate previously suggested in literature

that are considered in this paper. The references gives a source witha more detailed description of the model, not necessarily the originalsource of the model.

163

165


4.3 Power-exponential mortality rate models


In this section we will construct a phenomenological model of the mor-tality rate and show how to easily interpret its parameters that in terms ofpatterns 1–3 in Section 4.2 and uses the power-exponential function definedin Section 4.3.

Let A ⊂ R3 with n nonnegative elements. If a ∈ A we write

a = (a1, a2, a3).

We will use the following expression to approximate the mortality rate

µ(x) =c1

xe−c2x+∑a∈A

a1

(xe−a2x

)a3 .The survival function is given by Sx(t) = exp

(−∫ t

0µ(x+ s) ds

). Con-

sider the change of variable u = a2a3(x+ s)

− ln(Sx(t)) =

∫ t

0

c1

(x+ s)e−c2(x+s)ds+

∑a∈A

a1

((x+ s)e−a2(x+s)

)a3ds

=

∫ t

0

c1


∑a∈A

a1

(a2a3)a3+1

∫ a2a3(x+t)

a2a3xua3 e−a3u ds

= c1

(Ei(c2(x+ t)

)− Ei(c2 x)

)+∑a∈A

a1

(a2a3)a3+1(γ(a3 + 1, a2a3(x+ t))− γ(a3 + 1, a2a3x))

where γ(a, t) =

∫ t

0xa−1e−x dx is the lower incomplete Gamma function and

Ei(x) = −∫ ∞−x

e−s

sds is the exponential integral [2]. Thus

Sx(t) = exp(c1

(Ei(c2x)− Ei

(c2 (x+ t)

))· exp

(−∑a∈A

a1

(a2a3)a3+1

(γ(a3 + 1, a2a3(x+ t)

)− γ(a3 + 1, a2a3x)

))= exp

(c1

(Ei(c2x)− Ei

(c2 (x+ t)

))·∏a∈A

exp

(a1

(a2a3)a3+1

(γ(a3 + 1, a2a3x)− γ

(a3 + 1, a2a3(x+ t)

))).

164


4.3 Power-exponential mortality rate models


In this section we will construct a phenomenological model of the mor-tality rate and show how to easily interpret its parameters that in terms ofpatterns 1–3 in Section 4.2 and uses the power-exponential function definedin Section 4.3.

Let A ⊂ R3 with n nonnegative elements. If a ∈ A we write

a = (a1, a2, a3).

We will use the following expression to approximate the mortality rate

µ(x) =c1

xe−c2x+∑a∈A

a1

(xe−a2x

)a3 .The survival function is given by Sx(t) = exp

(−∫ t

0µ(x+ s) ds

). Con-

sider the change of variable u = a2a3(x+ s)

− ln(Sx(t)) =

∫ t

0

c1


∑a∈A

a1

((x+ s)e−a2(x+s)

)a3ds

=

∫ t

0

c1


∑a∈A

a1

(a2a3)a3+1

∫ a2a3(x+t)

a2a3xua3 e−a3u ds

= c1

(Ei(c2(x+ t)

)− Ei(c2 x)

)+∑a∈A

a1

(a2a3)a3+1(γ(a3 + 1, a2a3(x+ t))− γ(a3 + 1, a2a3x))

where γ(a, t) =

∫ t

0xa−1e−x dx is the lower incomplete Gamma function and

Ei(x) = −∫ ∞−x

e−s

sds is the exponential integral [2]. Thus

Sx(t) = exp(c1

(Ei(c2x)− Ei

(c2 (x+ t)

))· exp

(−∑a∈A

a1

(a2a3)a3+1

(γ(a3 + 1, a2a3(x+ t)

)− γ(a3 + 1, a2a3x)

))= exp

(c1

(Ei(c2x)− Ei

(c2 (x+ t)

))·∏a∈A

exp

(a1

(a2a3)a3+1

(γ(a3 + 1, a2a3x)− γ

(a3 + 1, a2a3(x+ t)

))).

164

166

4.3. POWER-EXPONENTIAL MORTALITY RATE MODELS

Since exp (γ(a, x)− γ(a, x+ t)) is non-increasing with respect to t thedifference Ei(c2x) − Ei

(c2 (x + t)

)is also non-increasing with respect to t

and thus Sx(t) is non-increasing with respect to t.Thus Sx(t) is non-decreasing and continuous, Sx(0) = 1 and lim

t→∞Sx(t) =

1 so this mortality rate models gives a reasonable survival function. InSections 4.3.3 and 4.3.4 two more models that use similar expressions willbe introduced, that these models also have the desired properties can beshown in an analogous way.

4.3.1 Multiple humps

The model is defined so that there can be an arbitrary number of humpsby adjusting the number of terms. See Figure 4.1 for an illustration of this.There are instances where several humps have been observed in mortalityrate data but the examples that the authors are aware of are all in theearly 19th century and therefore of low interest for the type of modellingconsidered here.

10 20 30 40 50 60 70

age, x, years

-6

-5

-4

-3

-2

ln(

(x))

Sweden 1808

central mortality ratefirst termsecond termthird termmodel mortality rate

10 20 30 40 50 60 70

age, x, years

-6

-5

-4

-3

-2

ln(

(x))

Sweden 1751

central mortality ratemodel mortality rate

Figure 4.1: Examples of mortality rate curves with multiple humps. Thesemodels are hand-fitted and are intended to illustrate that they canreplicate multiple humps, not show the best possible fit for multiplehumps.

4.3.2 Single hump model

We can model a single hump using

µ(x) =c1

xe−c2x+ a1

(xe−a2x

)a3 . (138)

The parameters c1, c2 and (a1, a2, a3) can easily be interpreted in termsof qualitative properties of the curve.

To interpret the effects of c1 note that c1 µ(x) → c1x when x → 0.

Looking at ln(µ(x)) instead we can note that ddx ln(µ(x))→ c2 when t→∞

165

4.3. POWER-EXPONENTIAL MORTALITY RATE MODELS

Since exp (γ(a, x)− γ(a, x+ t)) is non-increasing with respect to t thedifference Ei(c2x) − Ei

(c2 (x + t)

)is also non-increasing with respect to t

and thus Sx(t) is non-increasing with respect to t.Thus Sx(t) is non-decreasing and continuous, Sx(0) = 1 and lim

t→∞Sx(t) =

1 so this mortality rate models gives a reasonable survival function. InSections 4.3.3 and 4.3.4 two more models that use similar expressions willbe introduced, that these models also have the desired properties can beshown in an analogous way.

4.3.1 Multiple humps

The model is defined so that there can be an arbitrary number of humpsby adjusting the number of terms. See Figure 4.1 for an illustration of this.There are instances where several humps have been observed in mortalityrate data but the examples that the authors are aware of are all in theearly 19th century and therefore of low interest for the type of modellingconsidered here.

10 20 30 40 50 60 70

age, x, years

-6

-5

-4

-3

-2

ln(

(x))

Sweden 1808

central mortality ratefirst termsecond termthird termmodel mortality rate

10 20 30 40 50 60 70

age, x, years

-6

-5

-4

-3

-2

ln(

(x))

Sweden 1751

central mortality ratemodel mortality rate

Figure 4.1: Examples of mortality rate curves with multiple humps. Thesemodels are hand-fitted and are intended to illustrate that they canreplicate multiple humps, not show the best possible fit for multiplehumps.

4.3.2 Single hump model

We can model a single hump using

µ(x) =c1

xe−c2x+ a1

(xe−a2x

)a3 . (138)

The parameters c1, c2 and (a1, a2, a3) can easily be interpreted in termsof qualitative properties of the curve.

To interpret the effects of c1 note that c1 µ(x) → c1x when x → 0.

Looking at ln(µ(x)) instead we can note that ddx ln(µ(x))→ c2 when t→∞

165

167


so c2 gives the approximate slope of ln(µ(x)) for large x. The effects of thetwo terms are illustrated in Figure 4.2 where some examples of fitting themodel to data from various countries are shown.

In the simplest case where A = (a1, a2, a3) we get the expected humpwhen ai > 0 for i = 1, 2, 3. In this case a1

aa32

gives the maximum height of

the hump, 1a2

gives the time for the humps maximum and a3 determinesthe steepness of the hump (the larger a3 the steeper rise and the faster thedecay).

4.3.3 Split power-exponential model

In the power-exponential model the parameter c1 affects the shape of thecurve both for high and low ages and a3 affects both the increasing anddecreasing part of the hump.

To construct a model where this coupling is avoided we can split the twoterms in the model at their respective local extreme points and adjust thevalues so that µ(x) is continuous (since the terms are split at local extremepoints the derivative will also be continuous). We will refer to such a modelas the split power-exponential model and it will give the following expressionfor the mortality rate.

µ(x) =c

xe−c2x+ a1

(xe−a2x

)a+θ

(x− 1

c2

)· c2 · e · (c1 − c3) (139)

where

c =

c1, x ≤ 1

c2

c3, x > 1c2

, a =

a3, x ≤ 1

a2

a4, x > 1a2

, θ(x) =

0, x ≤ 0

1, x > 0.

This increases the total number of parameters from five to seven. To in-terpret the parameters in this model we can reason similarly to the power-exponential model and note that c1 largely determines the slope of the mor-tality rate for low ages, c2 and c3 largely determines the slope for high ages,a1, a2 and a3 determine the maximum height of the hump, a2 determine thelocation of the maximum while a3 and a4 determine the slope before andafter the maximum of the hump.

4.3.4 Adjusted power-exponential model

The split power-exponential model above can give some adjustments to theshape but when comparing the model to data it was found that it still hassome issues, primarily with matching the infant mortality rate. For thisreason we suggest another modification that needs eight parameters.

µ(x) = c1

(ec2x

c2x

)c+ a1

(xe−a2x

)a(140)

166


so c2 gives the approximate slope of ln(µ(x)) for large x. The effects of thetwo terms are illustrated in Figure 4.2 where some examples of fitting themodel to data from various countries are shown.

In the simplest case where A = (a1, a2, a3) we get the expected humpwhen ai > 0 for i = 1, 2, 3. In this case a1

aa32

gives the maximum height of

the hump, 1a2

gives the time for the humps maximum and a3 determinesthe steepness of the hump (the larger a3 the steeper rise and the faster thedecay).

4.3.3 Split power-exponential model

In the power-exponential model the parameter c1 affects the shape of thecurve both for high and low ages and a3 affects both the increasing anddecreasing part of the hump.

To construct a model where this coupling is avoided we can split the twoterms in the model at their respective local extreme points and adjust thevalues so that µ(x) is continuous (since the terms are split at local extremepoints the derivative will also be continuous). We will refer to such a modelas the split power-exponential model and it will give the following expressionfor the mortality rate.

µ(x) =c

xe−c2x+ a1

(xe−a2x

)a+θ

(x− 1

c2

)· c2 · e · (c1 − c3) (139)

where

c =

c1, x ≤ 1

c2

c3, x > 1c2

, a =

a3, x ≤ 1

a2

a4, x > 1a2

, θ(x) =

0, x ≤ 0

1, x > 0.

This increases the total number of parameters from five to seven. To in-terpret the parameters in this model we can reason similarly to the power-exponential model and note that c1 largely determines the slope of the mor-tality rate for low ages, c2 and c3 largely determines the slope for high ages,a1, a2 and a3 determine the maximum height of the hump, a2 determine thelocation of the maximum while a3 and a4 determine the slope before andafter the maximum of the hump.

4.3.4 Adjusted power-exponential model

The split power-exponential model above can give some adjustments to theshape but when comparing the model to data it was found that it still hassome issues, primarily with matching the infant mortality rate. For thisreason we suggest another modification that needs eight parameters.

µ(x) = c1

(ec2x

c2x

)c+ a1

(xe−a2x

)a(140)

166

168

4.4. FITTING AND COMPARING MODELS

where

c =

c3, x ≤ 1

c2,

c4, x > 1c2,

and a =

a3, x ≤ 1

c2,

a4, x > 1c2.

In this model the c2 parameter can be interpreted as the position of theminimum mortality rate if there is no hump. The slope before the minimumis controlled by c3 and after the minimum the slope is controlled by c4. Theremaining parameter in the first term, c1, is an overall scale factor for thenon-hump part of the model. The parameters in the second factor, a1, a2, a3

and a4 can be interpreted the same ways as for the split power-exponentialmodel.

4.4 Fitting and comparing models


In order to compare the different models each model will be fitted tocentral mortality rates, defined on page 66, taken from the Human MortalityDatabase (HMD) [127]. The central mortality at age x for year t is denotedby mx,t and here we will only consider time intervals of one year and thusthe estimates for the central mortality rate mx,t is estimated the same wayas µ(x) so for any given year t we can assume that mx,t ≈ µ(x).

What properties of a model is the most important depends on the in-tended application. Here we will try to examine how well the differentmodels can describe the mortality rate in the entire human lifespan. To dothis we will fit the model to the ages 1–100 years. The age range is chosensince it covers all the common patterns discussed on Page 67 and the HMDhas reliable data for this age-range.

One of the features that is most challenging to model is the hump thathappens for young adults. Since this effect is usually much more pronouncedfor men we will only consider mortality rate data for men in this analysis.

Since the mortality rate for high ages is several orders of magnitudehigher than the mortality rate for low ages in order to recreate featureslike the adolescence hump we will find a least squares fitting of the naturallogarithm of the model and the natural logarithm of the data. Some detailson the methods used for fitting the different model will be discussed inSection 4.4.1.

After the fitting is performed we will compare the quality of the fit ofthe different models using Akaike’s Information Criterion (AIC) describedin Section 1.3.3. If we denote the estimated mortality rate at age x in yeart ∈ 1, . . . , n with mx,t as before and let µ(x) be the fitted model, in other

167


where

c =

c3, x ≤ 1

c2,

c4, x > 1c2,

and a =

a3, x ≤ 1

c2,

a4, x > 1c2.

In this model the c2 parameter can be interpreted as the position of theminimum mortality rate if there is no hump. The slope before the minimumis controlled by c3 and after the minimum the slope is controlled by c4. Theremaining parameter in the first term, c1, is an overall scale factor for thenon-hump part of the model. The parameters in the second factor, a1, a2, a3

and a4 can be interpreted the same ways as for the split power-exponentialmodel.

4.4 Fitting and comparing models


In order to compare the different models each model will be fitted tocentral mortality rates, defined on page 66, taken from the Human MortalityDatabase (HMD) [127]. The central mortality at age x for year t is denotedby mx,t and here we will only consider time intervals of one year and thusthe estimates for the central mortality rate mx,t is estimated the same wayas µ(x) so for any given year t we can assume that mx,t ≈ µ(x).

What properties of a model is the most important depends on the in-tended application. Here we will try to examine how well the differentmodels can describe the mortality rate in the entire human lifespan. To dothis we will fit the model to the ages 1–100 years. The age range is chosensince it covers all the common patterns discussed on Page 67 and the HMDhas reliable data for this age-range.

One of the features that is most challenging to model is the hump thathappens for young adults. Since this effect is usually much more pronouncedfor men we will only consider mortality rate data for men in this analysis.

Since the mortality rate for high ages is several orders of magnitudehigher than the mortality rate for low ages in order to recreate featureslike the adolescence hump we will find a least squares fitting of the naturallogarithm of the model and the natural logarithm of the data. Some detailson the methods used for fitting the different model will be discussed inSection 4.4.1.

After the fitting is performed we will compare the quality of the fit ofthe different models using Akaike’s Information Criterion (AIC) describedin Section 1.3.3. If we denote the estimated mortality rate at age x in yeart ∈ 1, . . . , n with mx,t as before and let µ(x) be the fitted model, in other

167

169


words we have chosen the parameters such that

n∑x=1

(ln(mx,t)− µ(x))2

is minimized, then the maximum of the likelihood function is given by

L = −n2

ln

(n∑x=1

(ln(mx,t)− ln

(µ(x)

))2)+ c where c is a constant that

only depends on the dataset used, not on the fitted model [40]. Since wewill only use the AIC to compare different models we can ignore the constantterm and simply use the following expression for the AIC

AIC = 2k + 2 + n ln

(n∑x=1

(ln(mx,t)− ln

(µ(x)

))2). (141)

Remark: For models with many parameters and small sample sizes it isrecommended to use the AICC , discussed in Section 1.9, given by the ex-pression

AICC = AIC +2(k + 1)(k + 2)

n− k − 2.

In the current investigation the sample size will be n = 100 and the numberof parameters varies between 2 and 9. Thus the second order correction termwill vary between 1

4 and 22089 ≈ 2.47. For the analysis done in this paper, see

Section 4.4.2 and supplementary material, this correction is usually muchsmaller than the differences between the AIC from year to year for thedifferent models and using the AICC rarely changes which model has thelowest information criterion. Since we are mainly looking for trends in whenthe different models are suitable this means that for this analysis using theAIC or the AICC makes no difference in practice. In the supplementarymaterials [193] you can find the computed AICC values as well as the AIC.

4.4.1 Some comments on fitting

The models will be fitted to central mortality rate for men from 36 countriestaken from the HMD. For each country the most recently available centralmortality rates data for was taken for the ages 1–100 years. If more than100 years of data was available for a country the most recent 100 years werechosen, otherwise all available data was used. In some cases a few datapoints were missing, usually the mortality rate for a single age in a singleyear, in these cases the missing data was interpolated as the mean valueof neighbouring values. The models were fitted using software designed fornumerical computations ([206]) and the majority of the models were fittedusing an algorithm based on the interior reflective Newton method describedin [55,56].

168


words we have chosen the parameters such that

n∑x=1

(ln(mx,t)− µ(x))2

is minimized, then the maximum of the likelihood function is given by

L = −n2

ln

(n∑x=1

(ln(mx,t)− ln

(µ(x)

))2)+ c where c is a constant that

only depends on the dataset used, not on the fitted model [40]. Since wewill only use the AIC to compare different models we can ignore the constantterm and simply use the following expression for the AIC

AIC = 2k + 2 + n ln

(n∑x=1

(ln(mx,t)− ln

(µ(x)

))2). (141)

Remark: For models with many parameters and small sample sizes it isrecommended to use the AICC , discussed in Section 1.9, given by the ex-pression

AICC = AIC +2(k + 1)(k + 2)

n− k − 2.

In the current investigation the sample size will be n = 100 and the numberof parameters varies between 2 and 9. Thus the second order correction termwill vary between 1

4 and 22089 ≈ 2.47. For the analysis done in this paper, see

Section 4.4.2 and supplementary material, this correction is usually muchsmaller than the differences between the AIC from year to year for thedifferent models and using the AICC rarely changes which model has thelowest information criterion. Since we are mainly looking for trends in whenthe different models are suitable this means that for this analysis using theAIC or the AICC makes no difference in practice. In the supplementarymaterials [193] you can find the computed AICC values as well as the AIC.

4.4.1 Some comments on fitting

The models will be fitted to central mortality rate for men from 36 countriestaken from the HMD. For each country the most recently available centralmortality rates data for was taken for the ages 1–100 years. If more than100 years of data was available for a country the most recent 100 years werechosen, otherwise all available data was used. In some cases a few datapoints were missing, usually the mortality rate for a single age in a singleyear, in these cases the missing data was interpolated as the mean valueof neighbouring values. The models were fitted using software designed fornumerical computations ([206]) and the majority of the models were fittedusing an algorithm based on the interior reflective Newton method describedin [55,56].

168

170


The three first time exit models (SKI-6, Fractional 1st order and Frac-tional 2nd order) were fitted using the methodology described in their orig-inal sources where instead of fitting the logarithm of µ(x) directly g(x) is

approximated from the central mortality rates as g(x) ≈n∑k=1

mx,t and the

result fit the parameters of the g(x) function (using the interior reflectiveNewton method and σ = 1) and then the mortality rate is computed as

µ(x) = g(x)

(∫ ∞x

g(t) dt

)−1

using high accuracy numerical integration.

The power-exponential model given by (138) caused numerical instabil-ities in the fitting procedure so it was rewritten on the form

µ(x) =c1

xe−c2x+ ea3−a1

(a2xe

−a2x)a3instead that resulted in a much more stable behaviour.

The two variations of the power-exponential model given by (139) and(140) had similar issues and could not be reliably fitted using the interiorreflective Newton method. In order to fit the them were rewritten on thefollowing forms:

Split power-exponential

µ(x) =c

xe−c22x

+ ea−a21

(a2

2xe−a22x

)a+ θ

(x− 1

c22

)· c2

2 · e ·(c2

1 − c23

)where c =

c21, x ≤ 1

c22

c23, x > 1

c22

, a =

a23, x ≤ 1

a22

a24, x > 1

a22

, θ(x) =

0, x ≤ 0

1, x > 0,

Adjusted power-exponential model

µ(x) = e−c21−c(ec

22x

c22x

)c+ ea

21−a

(a2

2xe−a22x

)awhere c =

c23, x ≤ 1

c22,

c4, x > 1c22,

and a =

a23, x ≤ 1

c22,

a4, x > 1a22.

The parameters for these forms were then found using the Marquardt–Levenberg method [174,202]. This fitting can produce negative parametersbut since they are squared in the formulation of the models they can bereplaced with their absolute value. For both the split power-exponentialmodel and the adjusted power-exponential model the parameters can bechosen such that the resulting model is equivalent to the power-exponentialmodel. To get a good fit with these models it is a good idea to first fit thepower-exponential model and then use those parameters as initial values forthe fitting of the two modified models.

169


The three first time exit models (SKI-6, Fractional 1st order and Frac-tional 2nd order) were fitted using the methodology described in their orig-inal sources where instead of fitting the logarithm of µ(x) directly g(x) is

approximated from the central mortality rates as g(x) ≈n∑k=1

mx,t and the

result fit the parameters of the g(x) function (using the interior reflectiveNewton method and σ = 1) and then the mortality rate is computed as

µ(x) = g(x)

(∫ ∞x

g(t) dt

)−1

using high accuracy numerical integration.

The power-exponential model given by (138) caused numerical instabil-ities in the fitting procedure so it was rewritten on the form

µ(x) =c1

xe−c2x+ ea3−a1

(a2xe

−a2x)a3instead that resulted in a much more stable behaviour.

The two variations of the power-exponential model given by (139) and(140) had similar issues and could not be reliably fitted using the interiorreflective Newton method. In order to fit the them were rewritten on thefollowing forms:

Split power-exponential

µ(x) =c

xe−c22x

+ ea−a21

(a2

2xe−a22x

)a+ θ

(x− 1

c22

)· c2

2 · e ·(c2

1 − c23

)where c =

c21, x ≤ 1

c22

c23, x > 1

c22

, a =

a23, x ≤ 1

a22

a24, x > 1

a22

, θ(x) =

0, x ≤ 0

1, x > 0,

Adjusted power-exponential model

µ(x) = e−c21−c(ec

22x

c22x

)c+ ea

21−a

(a2

2xe−a22x

)awhere c =

c23, x ≤ 1

c22,

c4, x > 1c22,

and a =

a23, x ≤ 1

c22,

a4, x > 1a22.

The parameters for these forms were then found using the Marquardt–Levenberg method [174,202]. This fitting can produce negative parametersbut since they are squared in the formulation of the models they can bereplaced with their absolute value. For both the split power-exponentialmodel and the adjusted power-exponential model the parameters can bechosen such that the resulting model is equivalent to the power-exponentialmodel. To get a good fit with these models it is a good idea to first fit thepower-exponential model and then use those parameters as initial values forthe fitting of the two modified models.

169

171


When performing the least squares fitting it is assumed that the residu-als are normally distributed with mean zero and that the variance does notdepend on age. This is not always an accurate set of assumptions for severalreasons. One reason is that some models cannot recreate all expected fea-tures in the data, e.g. the Gompertz–Makeham, Weibull or Modified Perksmodel cannot describe infant mortality accurately. Another reason is thatsome age intervals are consistently noisier than others. This is most no-ticeable in the intervals with the lowest and the highest mortality rates incountries with small to medium-sized populations. For these countries theages with the lowest mortality rates are estimated from few deaths and thehighest mortality rates are estimated from a small population.

In Figure 4.3 some examples of quantile-quantile plots (Q-Q plots, seeSection 1.3.2) for the residuals of models fitted to data are shown. We expectthe residuals to mostly follow a straight line in the Q-Q plot apart from somerandom noise. For the models and data sets we have examined in this paperthere is some deviation for all models and data sets. In most cases this is acombination of some difference in trend and outliers. The patterns also varyfrom model to model and country to country so when applying a model itmight be worth considering adapting the fitting method to account for this.

Here we still use the simple least squares fitting since choosing a moreaccurate distribution for the residuals would require more information aboutthe considered countries (e.g. population numbers) and evaluating how wellthe models perform based on a very simple case can be informative aboutthe practicality of the models. Anyone who intends to use models like theseshould evaluate them based on the particular data set they intend to usethem for and what properties of the fitted model that are considered themost important for the intended application. The purpose of a systematiccomparison like the one in this paper is to help identify some small numberappropriate models and to evaluate and compare newly constructed modelsto existing models.

In some cases models that allow a high degree of plasticity for the locationand height of the hump (Hannerz, Heligman–Pollard and power-exponentialmodels) are susceptible to overfitting when the central mortality rates arenoisy, for examples see Figure 4.4. Choosing the model with lowest AICoften helps avoiding overfitting [40] but if the location and size of the humpis important extra care must be taken when fitting these models. For thesemodels it is also possible to identify a specific part of the expression thatcontrols the shape of the hump, so adding appropriate constraints to theparameters of the model can help mitigate overfittting.

170


When performing the least squares fitting it is assumed that the residu-als are normally distributed with mean zero and that the variance does notdepend on age. This is not always an accurate set of assumptions for severalreasons. One reason is that some models cannot recreate all expected fea-tures in the data, e.g. the Gompertz–Makeham, Weibull or Modified Perksmodel cannot describe infant mortality accurately. Another reason is thatsome age intervals are consistently noisier than others. This is most no-ticeable in the intervals with the lowest and the highest mortality rates incountries with small to medium-sized populations. For these countries theages with the lowest mortality rates are estimated from few deaths and thehighest mortality rates are estimated from a small population.

In Figure 4.3 some examples of quantile-quantile plots (Q-Q plots, seeSection 1.3.2) for the residuals of models fitted to data are shown. We expectthe residuals to mostly follow a straight line in the Q-Q plot apart from somerandom noise. For the models and data sets we have examined in this paperthere is some deviation for all models and data sets. In most cases this is acombination of some difference in trend and outliers. The patterns also varyfrom model to model and country to country so when applying a model itmight be worth considering adapting the fitting method to account for this.

Here we still use the simple least squares fitting since choosing a moreaccurate distribution for the residuals would require more information aboutthe considered countries (e.g. population numbers) and evaluating how wellthe models perform based on a very simple case can be informative aboutthe practicality of the models. Anyone who intends to use models like theseshould evaluate them based on the particular data set they intend to usethem for and what properties of the fitted model that are considered themost important for the intended application. The purpose of a systematiccomparison like the one in this paper is to help identify some small numberappropriate models and to evaluate and compare newly constructed modelsto existing models.

In some cases models that allow a high degree of plasticity for the locationand height of the hump (Hannerz, Heligman–Pollard and power-exponentialmodels) are susceptible to overfitting when the central mortality rates arenoisy, for examples see Figure 4.4. Choosing the model with lowest AICoften helps avoiding overfitting [40] but if the location and size of the humpis important extra care must be taken when fitting these models. For thesemodels it is also possible to identify a specific part of the expression thatcontrols the shape of the hump, so adding appropriate constraints to theparameters of the model can help mitigate overfittting.

170

172


age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

USA 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Canada 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Sweden 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Switzerland 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Ukraine 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Japan 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Taiwan 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Australia 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Chile 1995

central mortality ratefirst termsecond termmodel mortality rate

Figure 4.2: Examples of the power-exponential model fitted to the central mor-tality rate for various countries with the role of the two terms il-lustrated.

171


age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

USA 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Canada 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Sweden 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Switzerland 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Ukraine 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Japan 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Taiwan 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Australia 1995

age, x, years0 10 20 30 40 50 60

ln(7

(x))

-10

-8

-6

-4

Chile 1995

central mortality ratefirst termsecond termmodel mortality rate

Figure 4.2: Examples of the power-exponential model fitted to the central mor-tality rate for various countries with the role of the two terms il-lustrated.

171

173


-2 -1 0 1 2

Standard normal quantiles

-0.5

0

0.5

1

Qua

ntile

s of

res

idua

ls

Hannerz model fitted to centralmortality rate for USA 2017

-2 -1 0 1 2


-1

-0.5

0

0.5

1

Qua

ntile

s of

res

idua

ls

First Time Exit: Fractional 2nd order approx.fitted to central mortality rate for USA 2017

-2 -1 0 1 2


-0.6

-0.4

-0.2

0

0.2

0.4

Qua

ntile

s of

res

idua

ls

Adjusted power-exponential model fittedto central mortality rate for USA 2017

-2 -1 0 1 2


-0.6

-0.4

-0.2

0

0.2

0.4

Qua

ntile

s of

res

idua

ls

Heligman-Pollard 4 model fitted tocentral mortality rate for USA 2017

Figure 4.3: Examples of quantile-quantile plots for the residuals of some mod-els that fit the central mortality rate for USA 2017 well. The closerthe residuals are to the dashed line the better the residuals matchthe expected result from a normal distribution. All models con-sidered in this chapter show some degree of deviation, but morecomplicated models generally deviate less.

172


-2 -1 0 1 2


-0.5

0

0.5

1

Qua

ntile

s of

res

idua

ls

Hannerz model fitted to centralmortality rate for USA 2017

-2 -1 0 1 2


-1

-0.5

0

0.5

1

Qua

ntile

s of

res

idua

ls

First Time Exit: Fractional 2nd order approx.fitted to central mortality rate for USA 2017

-2 -1 0 1 2


-0.6

-0.4

-0.2

0

0.2

0.4

Qua

ntile

s of

res

idua

ls

Adjusted power-exponential model fittedto central mortality rate for USA 2017

-2 -1 0 1 2


-0.6

-0.4

-0.2

0

0.2

0.4

Qua

ntile

s of

res

idua

ls

Heligman-Pollard 4 model fitted tocentral mortality rate for USA 2017

Figure 4.3: Examples of quantile-quantile plots for the residuals of some mod-els that fit the central mortality rate for USA 2017 well. The closerthe residuals are to the dashed line the better the residuals matchthe expected result from a normal distribution. All models con-sidered in this chapter show some degree of deviation, but morecomplicated models generally deviate less.

172

174


0 20 40 60 80 100

age (years)

-10

-8

-6

-4

-2

0

ln(

)

Estonia 1960

mx,t

Heligman-Pollard 3

0 20 40 60 80 100

age (years)

-12

-10

-8

-6

-4

-2

0

ln(

)

Denmark 2015

mx,t

Heligman-Pollard HP4

0 20 40 60 80 100

age (years)

-10

-8

-6

-4

-2

0

ln(

)

Estonia 2015

mx,t

Power-exponential

0 20 40 60 80 100

age (years)

-10

-8

-6

-4

-2

0

ln(

)

Canada 1965

mx,t

Adjusted power-exponential

Figure 4.4: Examples of instances of overfitting with a few different models.Overfitting around the hump happens occasionally for most of themodels where the hump is controlled by a separate term in theexpression for the mortality rate. Here mx,t refers to the centralmortality rate for men taken from the Human Mortality Database.

173


0 20 40 60 80 100

age (years)

-10

-8

-6

-4

-2

0

ln(

)

Estonia 1960

mx,t

Heligman-Pollard 3

0 20 40 60 80 100

age (years)

-12

-10

-8

-6

-4

-2

0

ln(

)

Denmark 2015

mx,t

Heligman-Pollard HP4

0 20 40 60 80 100

age (years)

-10

-8

-6

-4

-2

0

ln(

)

Estonia 2015

mx,t

Power-exponential

0 20 40 60 80 100

age (years)

-10

-8

-6

-4

-2

0

ln(

)

Canada 1965

mx,t

Adjusted power-exponential

Figure 4.4: Examples of instances of overfitting with a few different models.Overfitting around the hump happens occasionally for most of themodels where the hump is controlled by a separate term in theexpression for the mortality rate. Here mx,t refers to the centralmortality rate for men taken from the Human Mortality Database.

173

175


4.4.2 Results and discussion

In this section we will present and discuss some representative examples ofthe results of fitting the models to central mortality rates for men in 36countries. The parameter values for the fitted models and computed AICscan be found in the supplementary material.

The three models introduced in Section 4.3 fit the data to varying de-grees. Usually the models can receive a good fit past the peak of the humpbut the power-exponential model often has issues fitting the mortality at lowages and some times for the rising part of the hump. The split-exponentialmodel often improves the fitting on the rising part of the hump but still can-not properly fit the mortality at low ages. The adjusted power-exponentialconsistently outperforms the other two models when fitting the mortalityfor young ages. See Figure 4.5 for some representative examples.

0 20 40 60 80 100

Ages (years)

-10

-8

-6

-4

-2

0

ln(

)

Sweden 1970

Central mortality ratePower-exponential

0 20 40 60 80 100

Ages (years)

-12

-10

-8

-6

-4

-2

0

ln(

)

Sweden 2014


0 20 40 60 80 100

Ages (years)

-10

-8

-6

-4

-2

0

ln(

)

Japan 1970

Central mortality rateSplit power-exponential

0 20 40 60 80 100

Ages (years)

-12

-10

-8

-6

-4

-2

0

ln(

)

Japan 2014


0 20 40 60 80 100

Ages (years)

-10

-8

-6

-4

-2

0

ln(

)

USA 1970

Central mortality rateAdjusted power-exponential

0 20 40 60 80 100

Ages (years)

-10

-8

-6

-4

-2

0

ln(

)

USA 2014


Figure 4.5: Some examples of the three models introduced in Section 4.3 fittedto central mortality rate for men taken from the Human MortalityDatabase.

174


4.4.2 Results and discussion

In this section we will present and discuss some representative examples ofthe results of fitting the models to central mortality rates for men in 36countries. The parameter values for the fitted models and computed AICscan be found in the supplementary material.

The three models introduced in Section 4.3 fit the data to varying de-grees. Usually the models can receive a good fit past the peak of the humpbut the power-exponential model often has issues fitting the mortality at lowages and some times for the rising part of the hump. The split-exponentialmodel often improves the fitting on the rising part of the hump but still can-not properly fit the mortality at low ages. The adjusted power-exponentialconsistently outperforms the other two models when fitting the mortalityfor young ages. See Figure 4.5 for some representative examples.

0 20 40 60 80 100

Ages (years)

-10

-8

-6

-4

-2

0

ln(

)

Sweden 1970


0 20 40 60 80 100

Ages (years)

-12

-10

-8

-6

-4

-2

0

ln(

)

Sweden 2014


0 20 40 60 80 100

Ages (years)

-10

-8

-6

-4

-2

0

ln(

)

Japan 1970


0 20 40 60 80 100

Ages (years)

-12

-10

-8

-6

-4

-2

0

ln(

)

Japan 2014


0 20 40 60 80 100

Ages (years)

-10

-8

-6

-4

-2

0

ln(

)

USA 1970


0 20 40 60 80 100

Ages (years)

-10

-8

-6

-4

-2

0

ln(

)

USA 2014


Figure 4.5: Some examples of the three models introduced in Section 4.3 fittedto central mortality rate for men taken from the Human MortalityDatabase.

174

176


Comparing the AIC of the models to each other shows that the modelswith eight or more parameters (Hannerz, Heligman–Pollard 1-4 and adjustedpower-exponential) often perform better than the other models. In Table 4.2we show the AIC for all the different models fitted to the central mortalityrate for Switzerland for ten different years. In Figure 4.6 the computedAIC values are shown for seven different countries and all models except theWeibull model that is excluded since it gives very large AIC values comparedto the other models. Note that in Figure 4.6 the Weibull model has beenomitted since it consistently performs much worse than the other models.The values in Table 4.2 and the graphs in Figure 4.6 show trends that areobserved to some extent in all of the examined countries.

1916 1926 1936 1946 1956 1966 1976 1986 1996 2006 2016

Weibull 520.5 523.3 529.8 542.4 572.6 573.3 570.4 564.1 560.8 570.7 574.6

Gompertz–Makeham 289.7 296.4 285.9 287.0 323.0 319.1 301.3 299.9 300.2 328.6 320.2

Gompertz Inv. G. 401.0 399.8 400.1 411.4 428.6 432.8 421.5 413.6 396.8 412.8 411.3

Logistic 388.5 387.5 387.6 399.3 417.3 421.9 410.6 402.9 385.6 402.8 401.3

Modified Perks 290.0 295.2 283.7 286.2 323.4 315.1 302.7 301.9 302.2 330.6 322.2

Thiele 270.3 301.9 311.8 157.8 297.9 291.8 209.4 343.4 351.3 384.1 319.0

Double Geometric 171.3 352.1 341.1 376.9 470.2 438.1 351.8 485.8 416.6 475.5 504.3

Hannerz 144.3 106.3 144.5 103.0 137.0 150.9 177.9 132.3 159.5 242.6 230.2

Power-exponential 240.7 237.2 237.2 256.5 286.8 298.0 277.5 272.8 260.6 291.6 293.2

Split power-exp. 215.9 220.2 210.6 217.8 268.7 272.3 245.6 240.3 242.5 286.3 281.7

Adjusted power-exp. 89.5 74.9 102.9 99.2 134.4 165.0 151.8 134.3 168.5 246.1 220.1

SKI-6 170.1 197.3 179.4 152.7 135.0 176.6 216.3 193.9 201.7 288.8 285.5

Fractional 1st order 291.7 306.3 305.3 309.1 268.9 304.9 344.4 400.1 397.4 402.7 404.5

Fractional 2nd order 228.7 225.6 224.8 230.2 201.2 245.1 279.1 284.7 293.5 331.8 337.9

Heligman–Pollard 1 123.2 98.3 129.8 123.5 124.9 180.2 168.0 145.7 174.3 280.4 230.3




Table 4.2: Computed AIC values for the different models fitted to the centralmortality rate for men for Switzerland for ten different years. Ineach column the lowest AIC for that year is marked in bold.

The Gompertz–Makeham model and modified Perks model are usuallyvery similar and outperforms most other models with four parameters orless. The exception is the Fractional 2nd order approximation of the FirstTime Exit model that has four (free) parameters and does considerablybetter than the classical models in many cases.

The performance of the Double geometric and Thiele models is incon-

175


Comparing the AIC of the models to each other shows that the modelswith eight or more parameters (Hannerz, Heligman–Pollard 1-4 and adjustedpower-exponential) often perform better than the other models. In Table 4.2we show the AIC for all the different models fitted to the central mortalityrate for Switzerland for ten different years. In Figure 4.6 the computedAIC values are shown for seven different countries and all models except theWeibull model that is excluded since it gives very large AIC values comparedto the other models. Note that in Figure 4.6 the Weibull model has beenomitted since it consistently performs much worse than the other models.The values in Table 4.2 and the graphs in Figure 4.6 show trends that areobserved to some extent in all of the examined countries.

1916 1926 1936 1946 1956 1966 1976 1986 1996 2006 2016

Weibull 520.5 523.3 529.8 542.4 572.6 573.3 570.4 564.1 560.8 570.7 574.6

Gompertz–Makeham 289.7 296.4 285.9 287.0 323.0 319.1 301.3 299.9 300.2 328.6 320.2

Gompertz Inv. G. 401.0 399.8 400.1 411.4 428.6 432.8 421.5 413.6 396.8 412.8 411.3

Logistic 388.5 387.5 387.6 399.3 417.3 421.9 410.6 402.9 385.6 402.8 401.3

Modified Perks 290.0 295.2 283.7 286.2 323.4 315.1 302.7 301.9 302.2 330.6 322.2

Thiele 270.3 301.9 311.8 157.8 297.9 291.8 209.4 343.4 351.3 384.1 319.0

Double Geometric 171.3 352.1 341.1 376.9 470.2 438.1 351.8 485.8 416.6 475.5 504.3

Hannerz 144.3 106.3 144.5 103.0 137.0 150.9 177.9 132.3 159.5 242.6 230.2

Power-exponential 240.7 237.2 237.2 256.5 286.8 298.0 277.5 272.8 260.6 291.6 293.2

Split power-exp. 215.9 220.2 210.6 217.8 268.7 272.3 245.6 240.3 242.5 286.3 281.7

Adjusted power-exp. 89.5 74.9 102.9 99.2 134.4 165.0 151.8 134.3 168.5 246.1 220.1

SKI-6 170.1 197.3 179.4 152.7 135.0 176.6 216.3 193.9 201.7 288.8 285.5

Fractional 1st order 291.7 306.3 305.3 309.1 268.9 304.9 344.4 400.1 397.4 402.7 404.5

Fractional 2nd order 228.7 225.6 224.8 230.2 201.2 245.1 279.1 284.7 293.5 331.8 337.9





Table 4.2: Computed AIC values for the different models fitted to the centralmortality rate for men for Switzerland for ten different years. Ineach column the lowest AIC for that year is marked in bold.

The Gompertz–Makeham model and modified Perks model are usuallyvery similar and outperforms most other models with four parameters orless. The exception is the Fractional 2nd order approximation of the FirstTime Exit model that has four (free) parameters and does considerablybetter than the classical models in many cases.

The performance of the Double geometric and Thiele models is incon-

175

177


sistent but usually relatively poor. For the other models with five to sevenparameters the power-exponential and split power-exponential usually dobetter than models with fewer parameters but generally not as well as theSKI-6 model. The performance of the SKI-6 model also varies significantly,this might be a consequence of the different fitting scheme applied to theFirst Time Exit models. In fact, the SKI-6 model occasionally performs justas well as the best models in the set of tested models.

Among the models with eight parameters or more there is no model thatis clearly superior to the others but sometimes there is a model that clearlyworks better for a specific country, e.g. the adjusted power-exponentialperforms very well for the USA after 1950 and Hannerz model performs verywell for Japan after 1960. Based on the countries and models considered herea good default method for describing the entire lifespan would be Heligman–Pollard 3 or Heligman–Pollard 4 since they consistently perform well andwere comparatively easy to fit.

For many countries the AIC for the best performing models tends to riseover time thus decreasing the difference between the best performing modelsand the other models. This effect is more pronounced in countries withsmaller populations. A possible explanation is that overall the logarithm ofcentral mortality rate is decreasing and getting more noisy. This change canbe seen in Figure 4.5, the effect is strongest in Sweden where the populationwas approximately eight to ten million in 1970–2014, the upwards trends andchanges in noise are less noticeable in Japan (population 100–126 million)and in the USA the (population 209–320 million), where no upwards trendfor the AIC could be seen in Figure 4.6, there is no increase in noise level. Asthe noise level increases the best possible fit is reduced and the advantagesof the more complicated models are reduced. For some countries the humpalso becomes less distinct over time, e.g. due to a decreasing number oflethal traffic accidents (one of the major causes of death for young men),and when the hump becomes less distinct the accuracy of the model nearthe hump matters less.

There are several ways that this work can be continued. Since differentmodels give mortality rates with different qualitative properties it couldalso be valuable to do a similar analysis where the models are only fittedto a shorter age interval, for instance only higher ages or only lower ages.The fitting procedure can probably also be improved, for instance takinginto account how the noisiness of the central mortality rates varies with ageand population size or finding a fitting procedure that is based on a moreaccurate assumption on the properties of the residuals.

176


sistent but usually relatively poor. For the other models with five to sevenparameters the power-exponential and split power-exponential usually dobetter than models with fewer parameters but generally not as well as theSKI-6 model. The performance of the SKI-6 model also varies significantly,this might be a consequence of the different fitting scheme applied to theFirst Time Exit models. In fact, the SKI-6 model occasionally performs justas well as the best models in the set of tested models.

Among the models with eight parameters or more there is no model thatis clearly superior to the others but sometimes there is a model that clearlyworks better for a specific country, e.g. the adjusted power-exponentialperforms very well for the USA after 1950 and Hannerz model performs verywell for Japan after 1960. Based on the countries and models considered herea good default method for describing the entire lifespan would be Heligman–Pollard 3 or Heligman–Pollard 4 since they consistently perform well andwere comparatively easy to fit.

For many countries the AIC for the best performing models tends to riseover time thus decreasing the difference between the best performing modelsand the other models. This effect is more pronounced in countries withsmaller populations. A possible explanation is that overall the logarithm ofcentral mortality rate is decreasing and getting more noisy. This change canbe seen in Figure 4.5, the effect is strongest in Sweden where the populationwas approximately eight to ten million in 1970–2014, the upwards trends andchanges in noise are less noticeable in Japan (population 100–126 million)and in the USA the (population 209–320 million), where no upwards trendfor the AIC could be seen in Figure 4.6, there is no increase in noise level. Asthe noise level increases the best possible fit is reduced and the advantagesof the more complicated models are reduced. For some countries the humpalso becomes less distinct over time, e.g. due to a decreasing number oflethal traffic accidents (one of the major causes of death for young men),and when the hump becomes less distinct the accuracy of the model nearthe hump matters less.

There are several ways that this work can be continued. Since differentmodels give mortality rates with different qualitative properties it couldalso be valuable to do a similar analysis where the models are only fittedto a shorter age interval, for instance only higher ages or only lower ages.The fitting procedure can probably also be improved, for instance takinginto account how the noisiness of the central mortality rates varies with ageand population size or finding a fitting procedure that is based on a moreaccurate assumption on the properties of the residuals.

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HannerzDouble GeometricThieleSKI-6Fractional 1st order Fractional 2nd orderHeligman-Pollard 1Heligman-Pollard 2Heligman-Pollard 3Heligman-Pollard 4

Gompertz-MakehamGompertz Inverse GaussianLogisticModified Perks ModelPower-exponentialSplit power-exponentialDouble power-exponential

Figure 4.6: AIC for seven countries and seventeen models.

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HannerzDouble GeometricThieleSKI-6Fractional 1st order Fractional 2nd orderHeligman-Pollard 1Heligman-Pollard 2Heligman-Pollard 3Heligman-Pollard 4

Gompertz-MakehamGompertz Inverse GaussianLogisticModified Perks ModelPower-exponentialSplit power-exponentialDouble power-exponential

Figure 4.6: AIC for seven countries and seventeen models.

177

179


The models considered here are simple parametric models. There areother ways to mathematically model mortality rates, e.g. the dynamicalmodel methods described in [260], and comparing them to the parametricmodels considered here could be interesting.

How well the models fit data is only one of the properties that are im-portant for the model. There are many other properties that could deter-mine which models is most suitable depending on the intended application.Analysing how the same list of models interact with different methods forforecasting mortality rates (see [33] for example of this), computation ofestimates like life expectancy or other demography related key values, e.g.the method of estimating the expected healthy lifetime described in [259],how the parameters of the models can be interpreted and measured indepen-dently of the mortality rate, simplicity of expressions for survival functionsand similar theoretical concerns, and so on, could all be useful complementswhen considering what model to choose in given situation. When identifyingthat a certain model works better for certain countries than others it couldalso be very interesting to try and find some way to explain why a certainmodel works well for a certain data set and try to see if those propertiescould be used to create helpful indicators for what model is suitable fora data set that has not been examined before, or if they can be useful toidentify what feature of the group that the data set describes that makes itbehave in a particular way.

4.5 Comparison of parametric models appliedto mortality rate forecasting


In Section 1.6.1 we described the Lee–Carter method for forecasting thelogarithm of central mortality rates. In this Section we will examine howthe results of using this method is affected if the central mortality rate isreplaced by mortality rates given by a mathematical model.

The Lee–Carter method is based on the assumption that central mortal-ity rates can be fairly accurately approximated by

ln(mx,t) = ax + bxkt + εx,t,

where ax, bx and kt are computed from historical central mortality rate asdescribed in Section 1.6.1. In this section we will discuss two ways of char-acterising the reliability of the forecast by examining the mortality indiceskt. We will then compare the results of applying the Lee–Carter method todata either taken from the Human Mortality database (HMD) or given bya simple mathematical model fitted to the central mortality rate from theHMD.

178


The models considered here are simple parametric models. There areother ways to mathematically model mortality rates, e.g. the dynamicalmodel methods described in [260], and comparing them to the parametricmodels considered here could be interesting.

How well the models fit data is only one of the properties that are im-portant for the model. There are many other properties that could deter-mine which models is most suitable depending on the intended application.Analysing how the same list of models interact with different methods forforecasting mortality rates (see [33] for example of this), computation ofestimates like life expectancy or other demography related key values, e.g.the method of estimating the expected healthy lifetime described in [259],how the parameters of the models can be interpreted and measured indepen-dently of the mortality rate, simplicity of expressions for survival functionsand similar theoretical concerns, and so on, could all be useful complementswhen considering what model to choose in given situation. When identifyingthat a certain model works better for certain countries than others it couldalso be very interesting to try and find some way to explain why a certainmodel works well for a certain data set and try to see if those propertiescould be used to create helpful indicators for what model is suitable fora data set that has not been examined before, or if they can be useful toidentify what feature of the group that the data set describes that makes itbehave in a particular way.

4.5 Comparison of parametric models appliedto mortality rate forecasting


In Section 1.6.1 we described the Lee–Carter method for forecasting thelogarithm of central mortality rates. In this Section we will examine howthe results of using this method is affected if the central mortality rate isreplaced by mortality rates given by a mathematical model.

The Lee–Carter method is based on the assumption that central mortal-ity rates can be fairly accurately approximated by

ln(mx,t) = ax + bxkt + εx,t,

where ax, bx and kt are computed from historical central mortality rate asdescribed in Section 1.6.1. In this section we will discuss two ways of char-acterising the reliability of the forecast by examining the mortality indiceskt. We will then compare the results of applying the Lee–Carter method todata either taken from the Human Mortality database (HMD) or given bya simple mathematical model fitted to the central mortality rate from theHMD.

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180

4.5. COMPARISON OF PARAMETRIC MODELS APPLIEDTO MORTALITY RATE FORECASTING

First we will consider errors of forecasted mortality index, εt, which rep-resent noise that causes deviation from the expected linear change. Thisterm is normally distributed with mean 0 and variance σ2

error. Note thatthe ks are not independent, they have successive innovations that are inde-pendent and the variance of error between them is estimated as

(see)2 =1

t− 2

T∑t−2

(kt − kt−1 − θ)2 (142)

which is used to calculate the uncertainty in forecasting kt over any givenhorizon.

Then the associated error variance of forecasted values is given as

var(kt) = (see)2 ·∆t (143)

and the square root of this gives the standard error estimate for the firstforecast horizon

SD(kt) = see ·√

∆t,

where ∆t is the forecast horizon which shows that standard deviation in-creases with the square root of the increasing distance to the forecast, andfrom standard deviation of kt we can calculate the confidence band using95% confidence interval with t-factor of 1.96 as;

(kt)± 1.96(SD(kt)).

To forecast two periods ahead, we just substitute for the definition of kt−1

moved back in time one period and plug in the estimate of the drift param-eter θ

kt−1 = kt−2 + θ + εt−1

then kt becomes

kt = kt−1 + θ + εt

= kt−2 + θ + εt−1 + (θ + εt)

= kt−2 + 2θ + (εt−1 + εt).

To forecast kt at time T + ∆t with the data available up to T , we follow thesame procedure iteratively ∆t times and obtain

kT+∆t = kT + ∆tθ +∆t∑n=1

εT + n− 1

= kT + ∆tθ +√

∆tεt (144)

179


First we will consider errors of forecasted mortality index, εt, which rep-resent noise that causes deviation from the expected linear change. Thisterm is normally distributed with mean 0 and variance σ2

error. Note thatthe ks are not independent, they have successive innovations that are inde-pendent and the variance of error between them is estimated as

(see)2 =1

t− 2

T∑t−2

(kt − kt−1 − θ)2 (142)

which is used to calculate the uncertainty in forecasting kt over any givenhorizon.

Then the associated error variance of forecasted values is given as

var(kt) = (see)2 ·∆t (143)

and the square root of this gives the standard error estimate for the firstforecast horizon

SD(kt) = see ·√

∆t,

where ∆t is the forecast horizon which shows that standard deviation in-creases with the square root of the increasing distance to the forecast, andfrom standard deviation of kt we can calculate the confidence band using95% confidence interval with t-factor of 1.96 as;

(kt)± 1.96(SD(kt)).

To forecast two periods ahead, we just substitute for the definition of kt−1

moved back in time one period and plug in the estimate of the drift param-eter θ

kt−1 = kt−2 + θ + εt−1

then kt becomes

kt = kt−1 + θ + εt

= kt−2 + θ + εt−1 + (θ + εt)

= kt−2 + 2θ + (εt−1 + εt).

To forecast kt at time T + ∆t with the data available up to T , we follow thesame procedure iteratively ∆t times and obtain

kT+∆t = kT + ∆tθ +∆t∑n=1

εT + n− 1

= kT + ∆tθ +√

∆tεt (144)

179

181


ignoring the error term since its mean is 0 and assumed to be indepen-dent with the same variance, we get forecast point estimates which follow astraight line as a function of ∆t with slope θ

kT+∆t = kT + ∆tθ.

The forecast of the logarithm of the mortality rate can then be computedusing the forecasted mortality index ln(µx,T+∆t) = ax + bxkT+∆t.

In Figure 4.7 an example is shown where the L–C method was appliedusing the original data and data given by two different models. A thirty yearperiod (1970–2000) was used to compute mortality indices and a predictionwas made 10 years in the future and compared to the measured mortalityrate at that time.

4.5.1 Comparison of models

To compare the different models to each other the mortality indices willbe computed over a period of time and then we will use the same RWDto forecast kt in the interval t ∈ [1, T ] and use the result to estimate thevariance of εt in (144). The variance of εt will be estimated by computingthe quantity l∆t that is found by removing the drift term, constant termand the scaling of the stochastic term in (144),

l∆t =k1+∆t −∆tθ√

∆t.

We can then consider l∆t to be measurements of l∆t = εt ∈ N (0, var(εt))and then var(εt) can be estimated using a standard maximum likelihoodestimation of lt.

Since different models will give different kt they will also give differentvar(εt) and a lower variance indicates a more suitable model.

The second way to characterise the suitability of the model for forecastingis to compare the associated error variance of the forecasted values, givenby (143). This is characterized by the standard error estimate given by(142). Thus computing and comparing the standard error estimates alsogives some idea of the comparative suitability for forecasting of differentmodels. An example of how the mortality indices can change for differentmodels is illustrated in Figure 4.8.

4.5.2 Results, discussion and further work

Here we will present results of applying the methodology described in previ-ous sections using five models and data from six countries. The models areModified Perks, Logistic, Heligman–Pollard HP4 given in Table 4.1 as well asthe power-exponential model and split power-exponential model described

180


ignoring the error term since its mean is 0 and assumed to be indepen-dent with the same variance, we get forecast point estimates which follow astraight line as a function of ∆t with slope θ

kT+∆t = kT + ∆tθ.

The forecast of the logarithm of the mortality rate can then be computedusing the forecasted mortality index ln(µx,T+∆t) = ax + bxkT+∆t.

In Figure 4.7 an example is shown where the L–C method was appliedusing the original data and data given by two different models. A thirty yearperiod (1970–2000) was used to compute mortality indices and a predictionwas made 10 years in the future and compared to the measured mortalityrate at that time.

4.5.1 Comparison of models

To compare the different models to each other the mortality indices willbe computed over a period of time and then we will use the same RWDto forecast kt in the interval t ∈ [1, T ] and use the result to estimate thevariance of εt in (144). The variance of εt will be estimated by computingthe quantity l∆t that is found by removing the drift term, constant termand the scaling of the stochastic term in (144),

l∆t =k1+∆t −∆tθ√

∆t.

We can then consider l∆t to be measurements of l∆t = εt ∈ N (0, var(εt))and then var(εt) can be estimated using a standard maximum likelihoodestimation of lt.

Since different models will give different kt they will also give differentvar(εt) and a lower variance indicates a more suitable model.

The second way to characterise the suitability of the model for forecastingis to compare the associated error variance of the forecasted values, givenby (143). This is characterized by the standard error estimate given by(142). Thus computing and comparing the standard error estimates alsogives some idea of the comparative suitability for forecasting of differentmodels. An example of how the mortality indices can change for differentmodels is illustrated in Figure 4.8.

4.5.2 Results, discussion and further work

Here we will present results of applying the methodology described in previ-ous sections using five models and data from six countries. The models areModified Perks, Logistic, Heligman–Pollard HP4 given in Table 4.1 as well asthe power-exponential model and split power-exponential model described

180

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0 20 40 60 80 100

ln(μ

)

-10

-5

0Original data

Central mortality rate 2000.....Central mortality rate 2010.....Forecasted mortality rate 2010

0 20 40 60 80 100

ln(μ

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0

age (years)

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ln(μ

)

-10

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Lower 95% confidence intervalUpper 95% confidence interval

Logistic

Power-exponential

Figure 4.7: Example of central and forecasted mortality rates for Australia withoriginal data and two different models. The mortality indices werecomputed using data generated in the period 1970–2000 and thelogarithm of the mortality was forecasted 10 years into the future.The forecasted mortality rate 2010 is compared to the initial mor-tality rate (measured mortality rate 2000) and the measured value(measured mortality rate 2010). The three models demonstratehow the quality of the prediction can depend on the model. Whenusing the original data the forecast differs relatively much in theage range 20–60 years. When using the logistic model the predic-tion and the central mortality rate are very similar but the modeldoes not describe the actual shape of the mortality rate curve well.When using the power-exponential model the prediction and cen-tral mortality rate are very similar except around the peak of thehump.

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0 20 40 60 80 100

ln(μ

)

-10

-5

0Original data

Central mortality rate 2000.....Central mortality rate 2010.....Forecasted mortality rate 2010

0 20 40 60 80 100

ln(μ

)

-10

-5

0

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ln(μ

)

-10

-5

0

age (years)

Lower 95% confidence intervalUpper 95% confidence interval

Logistic

Power-exponential

Figure 4.7: Example of central and forecasted mortality rates for Australia withoriginal data and two different models. The mortality indices werecomputed using data generated in the period 1970–2000 and thelogarithm of the mortality was forecasted 10 years into the future.The forecasted mortality rate 2010 is compared to the initial mor-tality rate (measured mortality rate 2000) and the measured value(measured mortality rate 2010). The three models demonstratehow the quality of the prediction can depend on the model. Whenusing the original data the forecast differs relatively much in theage range 20–60 years. When using the logistic model the predic-tion and the central mortality rate are very similar but the modeldoes not describe the actual shape of the mortality rate curve well.When using the power-exponential model the prediction and cen-tral mortality rate are very similar except around the peak of thehump.

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183


year1980 1990 2000 2010 2020 2030 2040 2050

k t

-15

-10

-5

0

5Mortality index Australia

Original dataLogisticPower-exponential

Figure 4.8: Example of estimated and forecasted mortality indices for Australiawith three different models along with their 95% confidence inter-vals. Note that the three different models forecast slightly differenttrendlines and that the confidence intervals have slightly differentwidths. In Section 4.5.1, two ways of characterising the reliabilityin the measured interval (1970-2010) and the forecasted interval(2011-2050), respectively, are described.

in Section 4.3. The data is taken from the Human Mortality Database(HMD) [128], and gives the mortality rate for ages 1–100 in 1970–2010 forUSA, Canada, Switzerland, Japan, Taiwan and Australia. The choice ofyears and countries is primarily driven by practical consideration, we wanteda set of countries with varying properties with respect to geographical po-sition, populations size and population density while also being developedenough to be qualitatively similar and have reliable data. The year range waschosen so that there would be no obvious major trend changes in mortalityin either of the countries, this is a necessary requirement for the require-ments of the L–C model to be considered reasonable. We also wanted ascenario where the fitting method worked efficiently and reliably and theassumptions behind L–C forecasting were relatively reasonable.

Note: For some of the countries data was missing for certain years and ages.In these cases the missing data was replaced with an average of neighbouringvalues.

The results are shown in Tables 4.3 and 4.4. In both tables a lowervalue indicates a more reliable forecasting (assuming there are no majordevelopments that significantly affect the mortality of certain populationparts).

Examining Table 4.3 we see that most of the time using the measureddata for forecasting is the most desirable, the exception are Switzerland andAustralia where the Split power-exponential model gives the best results

182


year1980 1990 2000 2010 2020 2030 2040 2050

k t

-15

-10

-5

0

5Mortality index Australia

Original dataLogisticPower-exponential

Figure 4.8: Example of estimated and forecasted mortality indices for Australiawith three different models along with their 95% confidence inter-vals. Note that the three different models forecast slightly differenttrendlines and that the confidence intervals have slightly differentwidths. In Section 4.5.1, two ways of characterising the reliabilityin the measured interval (1970-2010) and the forecasted interval(2011-2050), respectively, are described.

in Section 4.3. The data is taken from the Human Mortality Database(HMD) [128], and gives the mortality rate for ages 1–100 in 1970–2010 forUSA, Canada, Switzerland, Japan, Taiwan and Australia. The choice ofyears and countries is primarily driven by practical consideration, we wanteda set of countries with varying properties with respect to geographical po-sition, populations size and population density while also being developedenough to be qualitatively similar and have reliable data. The year range waschosen so that there would be no obvious major trend changes in mortalityin either of the countries, this is a necessary requirement for the require-ments of the L–C model to be considered reasonable. We also wanted ascenario where the fitting method worked efficiently and reliably and theassumptions behind L–C forecasting were relatively reasonable.

Note: For some of the countries data was missing for certain years and ages.In these cases the missing data was replaced with an average of neighbouringvalues.

The results are shown in Tables 4.3 and 4.4. In both tables a lowervalue indicates a more reliable forecasting (assuming there are no majordevelopments that significantly affect the mortality of certain populationparts).

Examining Table 4.3 we see that most of the time using the measureddata for forecasting is the most desirable, the exception are Switzerland andAustralia where the Split power-exponential model gives the best results

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184


Estimated Countryvariance of εt USA Canada Switzerland Japan Taiwan Australia

Measured data 0.111 0.123 0.123 0.143 0.113 0.0607Logistic 0.124 0.131 0.140 0.154 0.125 0.0704Modified Perks 0.122 0.128 0.132 0.149 0.118 0.0695Power-exponential 0.123 0.130 0.129 0.149 0.141 0.0615Split power-exp. 0.115 0.125 0.120 0.143 0.135 0.0602HP4 0.116 0.134 0.128 0.142 0.120 0.0647

Table 4.3: Estimated variance of εt found in the way described on page 180.The bold values are the lowest values in each column.

Standard Countryerror estimate USA Canada Switzerland Japan Taiwan Australia


Table 4.4: Standard error estimates of forecasted mortality indices.

(for Japan slightly better results are obtained by HP4). Note that using asimple parametrized model can bring advantages compared to using mea-sured data so it is also interesting to only compare the different models witheach other. Here the results vary significantly from country to country butusually estimated variance is lower for the more accurate models using moreparameters.

Examining Table 4.4 we can see that for Switzerland and Taiwan themodels tend to give a smaller standard error estimate than the measureddata, but otherwise the measured data gives standard error estimates thatare as good or better than the other models. There is also greater variationin which model seems most reliable with respect to forecasting compared toTable 4.3.

To understand if the variations here are indicative of trends for differentclasses of models a larger number of models should be applied to a largernumber of datasets, note that more sophisticated fitting and forecastingmethod should probably also be employed to ensure that the comparisonactually compares scenarios where each method is used in an appropriateway.

There are also other aspects and applications that require a different

183


Estimated Countryvariance of εt USA Canada Switzerland Japan Taiwan Australia


Table 4.3: Estimated variance of εt found in the way described on page 180.The bold values are the lowest values in each column.

Standard Countryerror estimate USA Canada Switzerland Japan Taiwan Australia


Table 4.4: Standard error estimates of forecasted mortality indices.

(for Japan slightly better results are obtained by HP4). Note that using asimple parametrized model can bring advantages compared to using mea-sured data so it is also interesting to only compare the different models witheach other. Here the results vary significantly from country to country butusually estimated variance is lower for the more accurate models using moreparameters.

Examining Table 4.4 we can see that for Switzerland and Taiwan themodels tend to give a smaller standard error estimate than the measureddata, but otherwise the measured data gives standard error estimates thatare as good or better than the other models. There is also greater variationin which model seems most reliable with respect to forecasting compared toTable 4.3.

To understand if the variations here are indicative of trends for differentclasses of models a larger number of models should be applied to a largernumber of datasets, note that more sophisticated fitting and forecastingmethod should probably also be employed to ensure that the comparisonactually compares scenarios where each method is used in an appropriateway.

There are also other aspects and applications that require a different

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185


kind of analysis, for instance if we wanted to compare the models suitabilityfor pricing life insurance or modelling pensions we would use different ageranges since life insurance is mostly intended for those who die at a rela-tively young age (but not children) while suitable pension planning requiresunderstanding of how many individuals will live to a high age.

We have also only considered a few simple parametrized models and nottaken the relative explanatory power of the models into account. More ofthe models from Table 4.1 could be considered as well as other types ofmortality modelling, for instance dynamical model methods [260].

In other words, every aspect of the comparison could be improved but webelieve that some sort of systematic comparison of these type of models oneasily available but relevant corpus of data could be a useful and informativetool for researchers and professionals.

184


kind of analysis, for instance if we wanted to compare the models suitabilityfor pricing life insurance or modelling pensions we would use different ageranges since life insurance is mostly intended for those who die at a rela-tively young age (but not children) while suitable pension planning requiresunderstanding of how many individuals will live to a high age.

We have also only considered a few simple parametrized models and nottaken the relative explanatory power of the models into account. More ofthe models from Table 4.1 could be considered as well as other types ofmortality modelling, for instance dynamical model methods [260].

In other words, every aspect of the comparison could be improved but webelieve that some sort of systematic comparison of these type of models oneasily available but relevant corpus of data could be a useful and informativetool for researchers and professionals.

184

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[273] Henry C. Thode. Testing Normality. Marcel Dekker, Inc. New York,2002.

[274] Joseph F. Traub. Associated polynomials and uniform methods forthe solution of linear problems. SIAM Review, 8(3):277–301, 1966.

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[284] Maryna S. Viazovska. The sphere packing problem in dimension 8.Annals of Mathematics, 185(2):991–1015, 2017.

[285] Abraham Wald. On the efficient design of statistical investigations.The Annals of Mathematical Statistics, 14(2):134–140, June 1943.

[286] Kai Wang, D. Pommerenke, R. Chundru, T. Van Doren, J. L. Drew-niak, and A. Shashindranath. Numerical modeling of electrostaticdischarge generators. IEEE Transactions on Electromagnetic Com-patibility, 45(2):258–271, 2003.

[287] Ke Wang, Jinshan Wang, and Xiaodong Wang. Four order electro-static discharge circuit model and its simulation. TELKOMNIKA,10(8):2006–2012, 2012.

[288] Edward Waring. Problems concerning interpolations. PhilosophicalTransactions of the Royal Society of London, 69:59–67, 1779.

[289] Waloddi Weibull. A statistical distribution function of wide applica-bility. Journal of Applied Mechanics, 18:293–297, 1951.

[290] Tim Williams. EMC for Product Designers. Newnes, 3rd edition,2001.

[291] JR Wilmoth. Mortality projections for Japan: A comparison of fourmethods. health and mortality among elderly population. In GraziellaCaselli and Alan D. Lopez, editors, Health and mortality among elderlypopulations. Clarendon Press, 1996.

[292] John Wishart. The generalised product moment distribution in sam-ples from a normal multivariate population. Biometrika, 20A(1/2):32–52, July 1928.

[293] Kenneth Wolsson. A condition equivalent to linear dependence forfunctions with vanishing Wronskian. Linear Algebra and its Applica-tions, 116:1–8, 1989.

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[297] Yuhong Yang. Can the strengths of AIC and BIC be shared? A conflictbetween model indentification and regression estimation. Biometrika,92(4):937–950, 2005.

[298] Natthasurang Yasungnoen and P. Sattayatham. Forecasting thai mor-tality by using the Lee–Carter model. Asia-Pacific Journal of Risk andInsurance, 10(1):91–105, 2015.

[299] Chen Yazhou, Liu Shanghe, Wu Xiaorong, and Zhang Feizhou. A newkind of channel-base current function. In 3rd International symposiumon Electromagnetic Compatibility, pages 304–646, 2002.

[300] Bernard Ycart. A case of mathematical eponymy: the Vandermondedeterminant. Revue d’Histoire des Mathematiques, 9(1):43–77, 2013.

[301] Zhiyong Yuan, Tun Li, Jinliang He, Shuiming Chen, and Rong Zeng.New mathematical descriptions of ESD current waveform based on thepolynomial of pulse function. IEEE Transactions on ElectromagneticCompatibility, 48(3):589–591, 2006.

[302] Changqing Zhu, Sanghe Liu, and Ming Wei. Analytic expressionand numerical solution of ESD current. High Voltage Engineering,31(7):22–24, 2005. in Chinese.

208

210

Index

accident hump, 69AEF, see analytically extended func-

tionAIC, 56, 171, 176

second order correction, 59, 169AICC , see second order correction of

the AICAkaike Information Criterion, see AICalternant matrix, 28analytically extended function, 128,

135, 141

central mortality rate, 68Coulomb gas, 35Coulombian interaction, 32curve fitting, 39

D-optimal design, 61death rate, see mortality ratedeterminant, 23

Vandermonde, 24digamma function, 139divided differences, 44

electromagnetic compatibility, 62electromagnetic disturbance, 62electromagnetic interference, see elec-

tromagnetic disturbanceelectrostatic discharge, 63EMC, see electromagnetic compati-

bilityESD, see electrostatic dischargeexponential integral, 165

Fisher information matrix, 60force of mortality, see mortality rate

G-optimal design, 60

Gamma function, 35, 134, 139incomplete, 132, 165

generalized divided differences, 46Grobner basis, 84Grobner basis, 84, 86

hazard rate, see mortality rateHeidler function, 65Hermite polynomial, 96

interpolation, 39Hermite, 43Newton, 44polynomial, 40

Jacobi polynomial, 149Jacobian matrix, 29, 136

Kullback–Leibler divergence, 56

Lagrange interpolation, 41Lagrange multipliers, 84, 85, 87, 89,

94, 95, 105, 113, 118, 120,121, 123, 149

least squares method, 47Lee–Carter method, 70, 179lightning discharge, 64, 142likelihood function, 52linear model, 40

Marquardt least squares method, 49,135, 141

maximum likelihood estimation, 52Meijer G-function, 139MLE, see maximum likelihood esti-

mationMLSM, see Marquardt least squares

method

209

Index

accident hump, 69AEF, see analytically extended func-

tionAIC, 56, 171, 176

second order correction, 59, 169AICC , see second order correction of

the AICAkaike Information Criterion, see AICalternant matrix, 28analytically extended function, 128,

135, 141

central mortality rate, 68Coulomb gas, 35Coulombian interaction, 32curve fitting, 39

D-optimal design, 61death rate, see mortality ratedeterminant, 23

Vandermonde, 24digamma function, 139divided differences, 44

electromagnetic compatibility, 62electromagnetic disturbance, 62electromagnetic interference, see elec-

tromagnetic disturbanceelectrostatic discharge, 63EMC, see electromagnetic compati-

bilityESD, see electrostatic dischargeexponential integral, 165

Fisher information matrix, 60force of mortality, see mortality rate

G-optimal design, 60

Gamma function, 35, 134, 139incomplete, 132, 165

generalized divided differences, 46Grobner basis, 84Grobner basis, 84, 86

hazard rate, see mortality rateHeidler function, 65Hermite polynomial, 96

interpolation, 39Hermite, 43Newton, 44polynomial, 40

Jacobi polynomial, 149Jacobian matrix, 29, 136

Kullback–Leibler divergence, 56

Lagrange interpolation, 41Lagrange multipliers, 84, 85, 87, 89,

94, 95, 105, 113, 118, 120,121, 123, 149

least squares method, 47Lee–Carter method, 70, 179lightning discharge, 64, 142likelihood function, 52linear model, 40

Marquardt least squares method, 49,135, 141

maximum likelihood estimation, 52Meijer G-function, 139MLE, see maximum likelihood esti-

mationMLSM, see Marquardt least squares

method

209

211


mortality rate, 67central, 68models, 163

orthogonal polynomialHermite, 96Jacobi, 149

overfitting, 55, 171

power-exponential function, 127, 165

Q-Q plot, 54, 171quantile-quantile plot, see Q-Q plot

regression, 52Runge’s phenomenon, 41, 62

Schur polynomials, 32survival function, 67, 165

VandermondeAlexandre Theophile, 21determinant, 23, 24matrix, 21

generalized, 31inverse, 27, 41

Wronskian matrix, 29

210


mortality rate, 67central, 68models, 163

orthogonal polynomialHermite, 96Jacobi, 149

overfitting, 55, 171

power-exponential function, 127, 165

Q-Q plot, 54, 171quantile-quantile plot, see Q-Q plot

regression, 52Runge’s phenomenon, 41, 62

Schur polynomials, 32survival function, 67, 165

VandermondeAlexandre Theophile, 21determinant, 23, 24matrix, 21

generalized, 31inverse, 27, 41

Wronskian matrix, 29

210

212

List of Figures

1.1 Illustration of the most significant connections in the thesis. . 18

1.2 Some examples of different interpolating curves. The set ofred points are interpolated by a polynomial (left), a self-affinefractal (middle) and a Lissajous curve (right). . . . . . . . . . 38

1.3 Illustration of Lagrange interpolation of 4 data points. The

red dots are the data set and p(x) =4∑

k=1

ykp(xk) is the inter-

polating polynomial. . . . . . . . . . . . . . . . . . . . . . . . 40

1.4 Illustration of Runge’s phenomenon. Here we attempt to ap-proximate a function (dashed line) by polynomial interpola-tion (solid line). With 7 equidistant sample points (left figure)the approximation is poor near the edges of the interval andincreasing the number of sample points to 14 (center) and 19(right) clearly reduces accuracy at the edges further. . . . . . 41

1.5 The basic iteration step of the Marquardt least squares method,definitions of computed quantities are given in (21), (22) and(23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.6 Comparison of different functions representing the StandardESD current waveshape for 4kV. . . . . . . . . . . . . . . . . 64

1.7 Examples of central mortality rate curves for men demon-strating the typical patterns of rapidly decreasing mortalityrate for very young ages followed by a ’hump’ for young adultand a rapid increase for high ages. . . . . . . . . . . . . . . . 67

2.5 Illustration of the ellipsoid defined byx2

9+y2

4+ z2 = 0 with

the extreme points of the Vandermonde determinant marked.Displayed in Cartesian coordinates on the right and in ellip-soidal coordinates on the left. . . . . . . . . . . . . . . . . . . 85

2.6 Illustration of the cylinder defined by y2 +16

25z2 = 1 with

the extreme points of the Vandermonde determinant marked.Displayed in Cartesian coordinates on the right and in cylin-drical coordinates on the left. . . . . . . . . . . . . . . . . . . 86

211

List of Figures

1.1 Illustration of the most significant connections in the thesis. . 18

1.2 Some examples of different interpolating curves. The set ofred points are interpolated by a polynomial (left), a self-affinefractal (middle) and a Lissajous curve (right). . . . . . . . . . 38

1.3 Illustration of Lagrange interpolation of 4 data points. The

red dots are the data set and p(x) =4∑

k=1

ykp(xk) is the inter-

polating polynomial. . . . . . . . . . . . . . . . . . . . . . . . 40

1.4 Illustration of Runge’s phenomenon. Here we attempt to ap-proximate a function (dashed line) by polynomial interpola-tion (solid line). With 7 equidistant sample points (left figure)the approximation is poor near the edges of the interval andincreasing the number of sample points to 14 (center) and 19(right) clearly reduces accuracy at the edges further. . . . . . 41

1.5 The basic iteration step of the Marquardt least squares method,definitions of computed quantities are given in (21), (22) and(23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.6 Comparison of different functions representing the StandardESD current waveshape for 4kV. . . . . . . . . . . . . . . . . 64

1.7 Examples of central mortality rate curves for men demon-strating the typical patterns of rapidly decreasing mortalityrate for very young ages followed by a ’hump’ for young adultand a rapid increase for high ages. . . . . . . . . . . . . . . . 67

2.5 Illustration of the ellipsoid defined byx2

9+y2

4+ z2 = 0 with

the extreme points of the Vandermonde determinant marked.Displayed in Cartesian coordinates on the right and in ellip-soidal coordinates on the left. . . . . . . . . . . . . . . . . . . 85

2.6 Illustration of the cylinder defined by y2 +16

25z2 = 1 with

the extreme points of the Vandermonde determinant marked.Displayed in Cartesian coordinates on the right and in cylin-drical coordinates on the left. . . . . . . . . . . . . . . . . . . 86

211

213


2.7 Illustration of the ellipsoid defined by (45) with the extremepoints of the Vandermonde determinant marked. Displayedin Cartesian coordinates on the right and in ellipsoidal coor-dinates on the left. . . . . . . . . . . . . . . . . . . . . . . . . 89

2.12 Illustration of S2p for p = 2, p = 4, p = 6, p = 8, and p = ∞

with a section cut out. The outer cube corresponds to p = 0and p = 2 corresponds to the sphere in the middle. . . . . . . 108

3.1 An illustration of how the steepness of the power exponentialfunction varies with β. . . . . . . . . . . . . . . . . . . . . . . 125

3.2 Illustration of the AEF (solid line) and its derivative (dashedline) with different βq,k-parameters but the same Imq and tmq .(a) 0 < βq,k < 1, (b) 4 < βq,k < 5,12 < βq,k < 13, (d) a mixture of large and small βq,k-parameters.128

3.3 An example of a two-peaked AEF where some of the ηq,k-parameters are negative, so that it has points where the firstderivative changes sign between two peaks. The solid line isthe AEF and the dashed lines is the derivative of the AEF. . 129

3.4 Schematic description of the parameter estimation algorithm. 135

3.5 First-positive stroke represented by the AEF function. Hereit is fitted with respect to both the data points as well as Q0

and W0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.6 First-negative stroke represented by the AEF function. Hereit is fitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.7 Fast-decaying waveshape represented by the AEF function.Here it is fitted with the extra constraint 0 ≤ η ≤ 1 for allη-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.8 AEF fitted to measurements from [257]. Here the peaks havebeen chosen to correspond to local maxima in the measureddata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.9 AEF fitted to measurements from [257]. Here the peaks havebeen chosen to correspond to local maxima and minima inthe measured data. . . . . . . . . . . . . . . . . . . . . . . . . 144

3.10 IEC 61000-4-2 Standard ESD current waveform with param-eters, [132] (image slightly modified for clarity). . . . . . . . . 151

3.11 2-peaked AEF interpolated on a D-optimal design represent-ing the IEC 61000-4-2 Standard ESD current waveshape for4 kV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3.12 3-peaked AEF interpolated to a D-optimal design from mea-sured ESD current from [151, Figure 3] compared with anapproximation suggested in [151]. Parameters are given inTable 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

212


2.7 Illustration of the ellipsoid defined by (45) with the extremepoints of the Vandermonde determinant marked. Displayedin Cartesian coordinates on the right and in ellipsoidal coor-dinates on the left. . . . . . . . . . . . . . . . . . . . . . . . . 89

2.12 Illustration of S2p for p = 2, p = 4, p = 6, p = 8, and p = ∞

with a section cut out. The outer cube corresponds to p = 0and p = 2 corresponds to the sphere in the middle. . . . . . . 108

3.1 An illustration of how the steepness of the power exponentialfunction varies with β. . . . . . . . . . . . . . . . . . . . . . . 125

3.2 Illustration of the AEF (solid line) and its derivative (dashedline) with different βq,k-parameters but the same Imq and tmq .(a) 0 < βq,k < 1, (b) 4 < βq,k < 5,12 < βq,k < 13, (d) a mixture of large and small βq,k-parameters.128

3.3 An example of a two-peaked AEF where some of the ηq,k-parameters are negative, so that it has points where the firstderivative changes sign between two peaks. The solid line isthe AEF and the dashed lines is the derivative of the AEF. . 129

3.4 Schematic description of the parameter estimation algorithm. 135

3.5 First-positive stroke represented by the AEF function. Hereit is fitted with respect to both the data points as well as Q0

and W0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.6 First-negative stroke represented by the AEF function. Hereit is fitted with the extra constraint 0 ≤ η ≤ 1 for all η-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.7 Fast-decaying waveshape represented by the AEF function.Here it is fitted with the extra constraint 0 ≤ η ≤ 1 for allη-parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.8 AEF fitted to measurements from [257]. Here the peaks havebeen chosen to correspond to local maxima in the measureddata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.9 AEF fitted to measurements from [257]. Here the peaks havebeen chosen to correspond to local maxima and minima inthe measured data. . . . . . . . . . . . . . . . . . . . . . . . . 144

3.10 IEC 61000-4-2 Standard ESD current waveform with param-eters, [132] (image slightly modified for clarity). . . . . . . . . 151

3.11 2-peaked AEF interpolated on a D-optimal design represent-ing the IEC 61000-4-2 Standard ESD current waveshape for4 kV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3.12 3-peaked AEF interpolated to a D-optimal design from mea-sured ESD current from [151, Figure 3] compared with anapproximation suggested in [151]. Parameters are given inTable 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

212

214

LIST OF FIGURES

3.13 Close-up of the rising part of a 3-peaked AEF interpolated toa D-optimal design from measured ESD current from [151,Figure 3]. Parameters are given in Table 3.5. . . . . . . . . . 154

3.14 AEF with 1 peak fitted by interpolating D-optimal pointssampled from the Heidler function describing the IEC 61312-1 waveshape given by (137). Parameters are given in Table 3.6.155

3.15 Close-up of the rising part of the AEF with 1 peak fittedby interpolating D-optimal points samples from the Heidlerfunction describing the IEC61312-1 waveshape given by (137).Parameters are given in Table 3.6. . . . . . . . . . . . . . . . 155

3.16 Comparison of two AEFs with 13 peaks and 2 terms in eachinterval fitted to measured lightning discharge current deriva-tive from [69]. One is fitted by interpolation on D-optimalpoints and the other is fitted with free parameters using theMLSM method. Parameters of the D-optimal version aregiven in Table 3.7. . . . . . . . . . . . . . . . . . . . . . . . . 156

3.17 Comparison of two AEFs with 12 peaks and 2 terms in eachinterval fitted to measured lightning discharge current deriva-tive from [130]. Parameters are given in Table 3.8. . . . . . . 158

3.18 Comparison of results of integrating the approximating func-tion shown in Figure 3.17. . . . . . . . . . . . . . . . . . . . . 158

4.1 Examples of mortality rate curves with multiple humps. Thesemodels are hand-fitted and are intended to illustrate that theycan replicate multiple humps, not show the best possible fitfor multiple humps. . . . . . . . . . . . . . . . . . . . . . . . . 165

4.2 Examples of the power-exponential model fitted to the centralmortality rate for various countries with the role of the twoterms illustrated. . . . . . . . . . . . . . . . . . . . . . . . . . 171

4.3 Examples of quantile-quantile plots for the residuals of somemodels that fit the central mortality rate for USA 2017 well.The closer the residuals are to the dashed line the better theresiduals match the expected result from a normal distribu-tion. All models considered in this chapter show some degreeof deviation, but more complicated models generally deviateless. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.4 Examples of instances of overfitting with a few different mod-els. Overfitting around the hump happens occasionally formost of the models where the hump is controlled by a sepa-rate term in the expression for the mortality rate. Here mx,t

refers to the central mortality rate for men taken from theHuman Mortality Database. . . . . . . . . . . . . . . . . . . . 173

213

LIST OF FIGURES

3.13 Close-up of the rising part of a 3-peaked AEF interpolated toa D-optimal design from measured ESD current from [151,Figure 3]. Parameters are given in Table 3.5. . . . . . . . . . 154

3.14 AEF with 1 peak fitted by interpolating D-optimal pointssampled from the Heidler function describing the IEC 61312-1 waveshape given by (137). Parameters are given in Table 3.6.155

3.15 Close-up of the rising part of the AEF with 1 peak fittedby interpolating D-optimal points samples from the Heidlerfunction describing the IEC61312-1 waveshape given by (137).Parameters are given in Table 3.6. . . . . . . . . . . . . . . . 155

3.16 Comparison of two AEFs with 13 peaks and 2 terms in eachinterval fitted to measured lightning discharge current deriva-tive from [69]. One is fitted by interpolation on D-optimalpoints and the other is fitted with free parameters using theMLSM method. Parameters of the D-optimal version aregiven in Table 3.7. . . . . . . . . . . . . . . . . . . . . . . . . 156

3.17 Comparison of two AEFs with 12 peaks and 2 terms in eachinterval fitted to measured lightning discharge current deriva-tive from [130]. Parameters are given in Table 3.8. . . . . . . 158

3.18 Comparison of results of integrating the approximating func-tion shown in Figure 3.17. . . . . . . . . . . . . . . . . . . . . 158

4.1 Examples of mortality rate curves with multiple humps. Thesemodels are hand-fitted and are intended to illustrate that theycan replicate multiple humps, not show the best possible fitfor multiple humps. . . . . . . . . . . . . . . . . . . . . . . . . 165

4.2 Examples of the power-exponential model fitted to the centralmortality rate for various countries with the role of the twoterms illustrated. . . . . . . . . . . . . . . . . . . . . . . . . . 171

4.3 Examples of quantile-quantile plots for the residuals of somemodels that fit the central mortality rate for USA 2017 well.The closer the residuals are to the dashed line the better theresiduals match the expected result from a normal distribu-tion. All models considered in this chapter show some degreeof deviation, but more complicated models generally deviateless. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.4 Examples of instances of overfitting with a few different mod-els. Overfitting around the hump happens occasionally formost of the models where the hump is controlled by a sepa-rate term in the expression for the mortality rate. Here mx,t

refers to the central mortality rate for men taken from theHuman Mortality Database. . . . . . . . . . . . . . . . . . . . 173

213

215


4.5 Some examples of the three models introduced in Section 4.3fitted to central mortality rate for men taken from the HumanMortality Database. . . . . . . . . . . . . . . . . . . . . . . . 174

4.6 AIC for seven countries and seventeen models. . . . . . . . . 1774.7 Example of central and forecasted mortality rates for Aus-

tralia with original data and two different models. The mor-tality indices were computed using data generated in the pe-riod 1970–2000 and the logarithm of the mortality was fore-casted 10 years into the future. The forecasted mortality rate2010 is compared to the initial mortality rate (measured mor-tality rate 2000) and the measured value (measured mortalityrate 2010). The three models demonstrate how the qualityof the prediction can depend on the model. When using theoriginal data the forecast differs relatively much in the agerange 20–60 years. When using the logistic model the pre-diction and the central mortality rate are very similar butthe model does not describe the actual shape of the mortalityrate curve well. When using the power-exponential model theprediction and central mortality rate are very similar exceptaround the peak of the hump. . . . . . . . . . . . . . . . . . . 181

4.8 Example of estimated and forecasted mortality indices forAustralia with three different models along with their 95%confidence intervals. Note that the three different modelsforecast slightly different trendlines and that the confidenceintervals have slightly different widths. In Section 4.5.1, twoways of characterising the reliability in the measured interval(1970-2010) and the forecasted interval (2011-2050), respec-tively, are described. . . . . . . . . . . . . . . . . . . . . . . . 182

214


4.5 Some examples of the three models introduced in Section 4.3fitted to central mortality rate for men taken from the HumanMortality Database. . . . . . . . . . . . . . . . . . . . . . . . 174

4.6 AIC for seven countries and seventeen models. . . . . . . . . 1774.7 Example of central and forecasted mortality rates for Aus-

tralia with original data and two different models. The mor-tality indices were computed using data generated in the pe-riod 1970–2000 and the logarithm of the mortality was fore-casted 10 years into the future. The forecasted mortality rate2010 is compared to the initial mortality rate (measured mor-tality rate 2000) and the measured value (measured mortalityrate 2010). The three models demonstrate how the qualityof the prediction can depend on the model. When using theoriginal data the forecast differs relatively much in the agerange 20–60 years. When using the logistic model the pre-diction and the central mortality rate are very similar butthe model does not describe the actual shape of the mortalityrate curve well. When using the power-exponential model theprediction and central mortality rate are very similar exceptaround the peak of the hump. . . . . . . . . . . . . . . . . . . 181

4.8 Example of estimated and forecasted mortality indices forAustralia with three different models along with their 95%confidence intervals. Note that the three different modelsforecast slightly different trendlines and that the confidenceintervals have slightly different widths. In Section 4.5.1, twoways of characterising the reliability in the measured interval(1970-2010) and the forecasted interval (2011-2050), respec-tively, are described. . . . . . . . . . . . . . . . . . . . . . . . 182

214

216

List of Tables

2.1 Table of some determinants of generalized Vandermonde ma-trices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.2 Polynomials, Pnp , whose roots give the coordinates of the ex-treme points of the Vandermonde determinant on the spheredefined by the p-norm in n dimensions. . . . . . . . . . . . . . 117

3.1 AEF function’s parameters for some current waveshapes . . . 143

3.2 IEC 61000-4-2 standard ESD current parameters [132]. . . . . 151

3.3 Parameters’ values of 2-peaked AEF representing the IEC61000-4-2 Standard ESD current waveshape for 4 kV. . . . . 151

3.4 IEC 61312-1 standard current key parameters, [134]. . . . . . 153

3.5 Parameters’ values of AEF with 3 peaks representing mea-sured ESD current from [151, Figure 3]. . . . . . . . . . . . . 154

3.6 Parameters’ values of AEF representing the IEC 61312-1 stan-dard waveshape. . . . . . . . . . . . . . . . . . . . . . . . . . 155

3.7 Parameters’ values of AEF with 13 peaks representing mea-sured data for a lightning discharge current from [245]. Localmaxima and corresponding times extracted from [69, Figures6, 7 and 8] are denoted t and I and other parameters corre-spond to the fitted AEF shown in Figures 3.16 (a), 3.16 (b)and 3.16 (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3.8 Parameters’ value of AEF with 12 peaks representing mea-sured data for a lightning discharge current derivative from[130]. Chosen peak times are denoted t and I and other pa-rameters correspond to the fitted AEF shown in Figure 3.17. 158

4.1 List of the models of mortality rate previously suggested inliterature that are considered in this paper. The referencesgives a source with a more detailed description of the model,not necessarily the original source of the model. . . . . . . . . 163

215

List of Tables

2.1 Table of some determinants of generalized Vandermonde ma-trices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.2 Polynomials, Pnp , whose roots give the coordinates of the ex-treme points of the Vandermonde determinant on the spheredefined by the p-norm in n dimensions. . . . . . . . . . . . . . 117

3.1 AEF function’s parameters for some current waveshapes . . . 143

3.2 IEC 61000-4-2 standard ESD current parameters [132]. . . . . 151

3.3 Parameters’ values of 2-peaked AEF representing the IEC61000-4-2 Standard ESD current waveshape for 4 kV. . . . . 151

3.4 IEC 61312-1 standard current key parameters, [134]. . . . . . 153

3.5 Parameters’ values of AEF with 3 peaks representing mea-sured ESD current from [151, Figure 3]. . . . . . . . . . . . . 154

3.6 Parameters’ values of AEF representing the IEC 61312-1 stan-dard waveshape. . . . . . . . . . . . . . . . . . . . . . . . . . 155

3.7 Parameters’ values of AEF with 13 peaks representing mea-sured data for a lightning discharge current from [245]. Localmaxima and corresponding times extracted from [69, Figures6, 7 and 8] are denoted t and I and other parameters corre-spond to the fitted AEF shown in Figures 3.16 (a), 3.16 (b)and 3.16 (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3.8 Parameters’ value of AEF with 12 peaks representing mea-sured data for a lightning discharge current derivative from[130]. Chosen peak times are denoted t and I and other pa-rameters correspond to the fitted AEF shown in Figure 3.17. 158

4.1 List of the models of mortality rate previously suggested inliterature that are considered in this paper. The referencesgives a source with a more detailed description of the model,not necessarily the original source of the model. . . . . . . . . 163

215

217

4.2 Computed AIC values for the different models fitted to thecentral mortality rate for men for Switzerland for ten differentyears. In each column the lowest AIC for that year is markedin bold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.3 Estimated variance of εt found in the way described on page180. The bold values are the lowest values in each column. . . 183

4.4 Standard error estimates of forecasted mortality indices. . . . 183

List of Definitions

Definition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Definition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Definition 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Definition 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Definition 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Definition 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Definition 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Definition 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Definition 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Definition 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Definition 1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Definition 1.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Definition 1.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Definition 1.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Definition 1.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Definition 1.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Definition 1.17 (The G-optimality criterion) . . . . . . . . . . . 58Definition 1.18 (The D-optimality criterion) . . . . . . . . . . . 59Definition 1.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Definition 1.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Definition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Definition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Definition 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Definition 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Definition 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Definition 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Definition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Definition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

216

4.2 Computed AIC values for the different models fitted to thecentral mortality rate for men for Switzerland for ten differentyears. In each column the lowest AIC for that year is markedin bold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.3 Estimated variance of εt found in the way described on page180. The bold values are the lowest values in each column. . . 183

4.4 Standard error estimates of forecasted mortality indices. . . . 183

List of Definitions

Definition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Definition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Definition 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Definition 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Definition 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Definition 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Definition 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Definition 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Definition 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Definition 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Definition 1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Definition 1.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Definition 1.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Definition 1.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Definition 1.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Definition 1.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Definition 1.17 (The G-optimality criterion) . . . . . . . . . . . 58Definition 1.18 (The D-optimality criterion) . . . . . . . . . . . 59Definition 1.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Definition 1.20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Definition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Definition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Definition 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Definition 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Definition 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Definition 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Definition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Definition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

216

218

List of Theorems

Theorem 1.1 (Leibniz formula for determinants) . . . . . . . . . 22Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Theorem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Theorem 1.7 (Kiefer–Wolfowitz equivalence theorem) . . . . . . 59

Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Theorem 2.6 (Vieta’s formula) . . . . . . . . . . . . . . . . . . . 112Theorem 2.7 (Newton–Girard formulae) . . . . . . . . . . . . . 112Theorem 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Theorem 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Theorem 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

217

List of Theorems

Theorem 1.1 (Leibniz formula for determinants) . . . . . . . . . 22Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Theorem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Theorem 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Theorem 1.7 (Kiefer–Wolfowitz equivalence theorem) . . . . . . 59

Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Theorem 2.6 (Vieta’s formula) . . . . . . . . . . . . . . . . . . . 112Theorem 2.7 (Newton–Girard formulae) . . . . . . . . . . . . . 112Theorem 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Theorem 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Theorem 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

217

219

List of Lemmas

Lemma 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Lemma 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Lemma 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Lemma 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Lemma 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Lemma 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Lemma 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Lemma 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Lemma 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Lemma 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Lemma 1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Lemma 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Lemma 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Lemma 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Lemma 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Lemma 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Lemma 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Lemma 2.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Lemma 2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Lemma 2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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List of Lemmas

Lemma 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Lemma 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Lemma 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Lemma 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Lemma 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Lemma 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Lemma 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Lemma 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Lemma 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Lemma 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Lemma 1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Lemma 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Lemma 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Lemma 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Lemma 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Lemma 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Lemma 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Lemma 2.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Lemma 2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Lemma 2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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ISBN 978-91-7485-431-2ISSN 1651-4238

Address: P.O. Box 883, SE-721 23 Västerås. SwedenAddress: P.O. Box 325, SE-631 05 Eskilstuna. SwedenE-mail: [email protected] Web: www.mdh.se

Extreme points of the Vandermonde determinant and phenomenological modelling with power exponential functionsKarl Lundengård

Mälardalen University Doctoral Dissertation 293

Documents

Extreme points of the Vandermonde determinant and ...mdh.diva-portal.org/smash/get/diva2:1329454/FULLTEXT01.pdfM i;j - Element on the ith row and jth column of M . M;j, M i; - Column