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Research Matters
February 25 2009
Nick HighamDirector of Research
School of Mathematics
1 6
Rehabilitating Correlations AvoidingInversion and Extracting Roots
Nick HighamSchool of Mathematics
The University of Manchester
NicholasJHighammanchesteracuknhigham nickhighamwordpresscom
httpwwwmathsmanchesteracuk~highamtalks
The Actuarial ProfessionMarch 20 2013 London
Precision
SingleDoubleQuadVariable
AccuracyEstimatesGuarantees
Efficiency
SpeedParallelismEnergy efficiency
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 3 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Question from London Fund Management Company (2000)ldquoGiven a real symmetric matrix A which is almost acorrelation matrix
What is the best approximating (in Frobenius norm)correlation matrixIs it uniqueCan we compute it
Typically we are working with 1400times 1400 at the momentbut this will probably grow to 6500times 6500rdquo
Correlations Matrix Inversion Matrix Roots
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Actuarial Matrix Computations 6 37
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Precision
SingleDoubleQuadVariable
AccuracyEstimatesGuarantees
Efficiency
SpeedParallelismEnergy efficiency
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 3 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Question from London Fund Management Company (2000)ldquoGiven a real symmetric matrix A which is almost acorrelation matrix
What is the best approximating (in Frobenius norm)correlation matrixIs it uniqueCan we compute it
Typically we are working with 1400times 1400 at the momentbut this will probably grow to 6500times 6500rdquo
Correlations Matrix Inversion Matrix Roots
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Actuarial Matrix Computations 6 37
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 3 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Question from London Fund Management Company (2000)ldquoGiven a real symmetric matrix A which is almost acorrelation matrix
What is the best approximating (in Frobenius norm)correlation matrixIs it uniqueCan we compute it
Typically we are working with 1400times 1400 at the momentbut this will probably grow to 6500times 6500rdquo
Correlations Matrix Inversion Matrix Roots
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Actuarial Matrix Computations 6 37
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Question from London Fund Management Company (2000)ldquoGiven a real symmetric matrix A which is almost acorrelation matrix
What is the best approximating (in Frobenius norm)correlation matrixIs it uniqueCan we compute it
Typically we are working with 1400times 1400 at the momentbut this will probably grow to 6500times 6500rdquo
Correlations Matrix Inversion Matrix Roots
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Actuarial Matrix Computations 6 37
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Question from London Fund Management Company (2000)ldquoGiven a real symmetric matrix A which is almost acorrelation matrix
What is the best approximating (in Frobenius norm)correlation matrixIs it uniqueCan we compute it
Typically we are working with 1400times 1400 at the momentbut this will probably grow to 6500times 6500rdquo
Correlations Matrix Inversion Matrix Roots
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Actuarial Matrix Computations 6 37
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Correlation Matrix
An n times n symmetric matrix A is a correlation matrix ifIt has ones on the diagonalAll its eigenvalues are nonnegative
Is this a correlation matrix1 1 01 1 10 1 1
Spectrum minus04142 10000 24142
University of Manchester Nick Higham Actuarial Matrix Computations 4 37
Question from London Fund Management Company (2000)ldquoGiven a real symmetric matrix A which is almost acorrelation matrix
What is the best approximating (in Frobenius norm)correlation matrixIs it uniqueCan we compute it
Typically we are working with 1400times 1400 at the momentbut this will probably grow to 6500times 6500rdquo
Correlations Matrix Inversion Matrix Roots
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Actuarial Matrix Computations 6 37
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Question from London Fund Management Company (2000)ldquoGiven a real symmetric matrix A which is almost acorrelation matrix
What is the best approximating (in Frobenius norm)correlation matrixIs it uniqueCan we compute it
Typically we are working with 1400times 1400 at the momentbut this will probably grow to 6500times 6500rdquo
Correlations Matrix Inversion Matrix Roots
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Actuarial Matrix Computations 6 37
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
How to Proceed
times Make ad hoc modifications to matrix eg shiftnegative ersquovals up to zero then diagonally scale
radicPlug the gaps in the missing data then compute anexact correlation matrix
radicCompute the nearest correlation matrix in theweighted Frobenius norm (A2 =
sumij wiwja2
ij )Given approx correlation matrix A find correlationmatrix C to minimize Aminus C
Constraint set is a closed convex set so uniqueminimizer
University of Manchester Nick Higham Actuarial Matrix Computations 6 37
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Development
Derived theory and algorithmN J Higham Computing the Nearest CorrelationMatrixmdashA Problem from Finance IMA J NumerAnal 22 329ndash343 2002
Extensions inCraig Lucas Computing Nearest Covariance andCorrelation Matrices MSc Thesis University ofManchester 2001
University of Manchester Nick Higham Actuarial Matrix Computations 7 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Alternating Projections Algorithmvon Neumann (1933) for subspacesDykstra (1983) corrections for closed convex sets
S1
S2
Easy to implementGuaranteed convergence at a linear rateCan add further constraintsprojections
University of Manchester Nick Higham Actuarial Matrix Computations 8 37
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Unexpected Applications
Some recent papers
Applying stochastic small-scale damage functionsto German winter storms (2012)
Estimating variance components and predictingbreeding values for eventing disciplines andgrades in sport horses (2012)
Characterisation of tool marks on cartridge casesby combining multiple images (2012)
Experiments in reconstructing twentieth-centurysea levels (2011)
University of Manchester Nick Higham Actuarial Matrix Computations 9 37
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Newton Method
Qi amp Sun (2006) Newton method based on theory ofstrongly semismooth matrix functions
Applies Newton to dual (unconstrained) ofmin 1
2Aminus X2F problem
Globally and quadratically convergent
H amp Borsdorf (2010) improve efficiency and reliability
use minres for Newton equationJacobi preconditionerreliability improved by line search tweaksextra scaling step to ensure unit diagonal
University of Manchester Nick Higham Actuarial Matrix Computations 11 37
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Alternating Projections vs Newton
Matrix n tol Code Time (s) Iters1 Random 100 1e-10 g02aa 0023 4
alt proj 0052 152 Random 500 1e-10 g02aa 048 4
alt proj 301 263 Real-life 1399 1e-4 g02aa 68 5
alt proj 1006 68
University of Manchester Nick Higham Actuarial Matrix Computations 14 37
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Performance of NAG Codes
University of Manchester Nick Higham Actuarial Matrix Computations 15 37
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Factor Model (1)
ξ = X︸︷︷︸ntimesk
η︸︷︷︸ktimes1
+ F︸︷︷︸ntimesn
ε︸︷︷︸ntimes1
ηi εi isin N(01)
where var(ξi) equiv 1 F = diag(fii) Implies
ksumj=1
x2ij le 1 i = 1 n
ldquoMultifactor normal copula modelrdquoCollateralized debt obligations (CDOs)Multivariate time series
University of Manchester Nick Higham Actuarial Matrix Computations 16 37
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Factor Model (2)
Yields correlation matrix of form
C(X ) = D + XX T = D +ksum
j=1
xjxTj
D = diag(I minus XX T ) X = [x1 xk ]
C(X ) has k factor correlation matrix structure
C(X ) =
1 yT
1 y2 yT1 yn
yT1 y2 1
yT
nminus1yn
yT1 yn yT
nminus1yn 1
yi isin Rk
University of Manchester Nick Higham Actuarial Matrix Computations 17 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswer
Algorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Factor Structure
Nearest correlation matrix with factor structure
Principal factors method (Andersen et al 2003) has noconvergence theory and can converge to an incorrectanswerAlgorithm based on spectral projected gradientmethod (Borsdorf H amp Raydan 2010)
Respects the constraints exploits their convexityand converges to a feasible stationary pointNAG routine g02aef (Mark 23 2012)
University of Manchester Nick Higham Actuarial Matrix Computations 18 37
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Variations
Specific rank or bound on correlations
Block structure
Bounds on individual correlations
Other requirements
University of Manchester Nick Higham Actuarial Matrix Computations 19 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 20 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Avoiding Inversion
Fundamental Tenet of Numerical AnalysisDonrsquot invert matrices
Ax = b use Gaussian elimination (with pivoting) notx = Aminus1bxT Aminus1y = xT (Aminus1y)
(Aminus1)ii = eTi Aminus1ei
trace(Aminus1) asymp mminus1summk=1 vT
k Aminus1vk vk sim uniformminus11 (Bekas et al 2007)
University of Manchester Nick Higham Actuarial Matrix Computations 21 37
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
How to Invert (1)
Use a condition estimator to warn when A is nearlysingular
gtgt A = hilb(16)gtgt X = inv(A)Warning Matrix is close to singular orbadly scaled Results may be inaccurateRCOND = 9721674e-19
University of Manchester Nick Higham Actuarial Matrix Computations 22 37
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
How to Invert (2)
To compute variancendashcovariance matrix (X T X )minus1 ofleast squares estimator use QR factorization X = QR((X T X )minus1 = Rminus1RminusT ) and do not explicitly form X T X
Quality of computed inverse Y asymp Aminus1 measured by
YAminus IYA
orAY minus IYA
but not both
University of Manchester Nick Higham Actuarial Matrix Computations 23 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Computing the Sample Variance
Sample variance of x1 xn
s2n =
1n minus 1
nsumi=1
(xi minus x)2 where x =1n
nsumi=1
xi (1)
Can compute using one-pass formula
s2n =
1n minus 1
( nsumi=1
x2i minus
1n
( nsumi=1
xi
)2) (2)
For x = (100001000110002) using 8-digit arithmetic(1) gives 10 (2) gives 00
(2) can even give negative results in floating pointarithmetic
University of Manchester Nick Higham Actuarial Matrix Computations 24 37
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Casio fx-992VB
University of Manchester Nick Higham Actuarial Matrix Computations 25 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Spreadsheets in the CloudStandard deviation of x = [n n + 1 n + 2]T
n Exact Google Sheet107 1 1108 1 0
McCullough amp Yalta (2013)
University of Manchester Nick Higham Actuarial Matrix Computations 26 37
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Outline
1 Rehabilitating Correlations
2 Matrix Inversion
3 Matrix Roots
University of Manchester Nick Higham Actuarial Matrix Computations 27 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester
Term ldquomatrixrdquo coined in 1850by James Joseph SylvesterFRS (1814ndash1897)
Matrix algebra developed byArthur Cayley FRS (1821ndash1895)Memoir on the Theory of Ma-trices (1858)
University of Manchester Nick Higham Actuarial Matrix Computations 28 37
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir
Tony Crilly Arthur Cayley Mathemati-cian Laureate of the Victorian Age2006
Sylvester (1883) gave first defini-tion of f (A) for general f
Karen Hunger Parshall James JosephSylvester Jewish Mathematician in aVictorian World 2006
University of Manchester Nick Higham Actuarial Matrix Computations 29 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Matrix Roots in Markov Models
Let vectors v2011 v2010 represent risks credit ratings orstock prices in 2011 and 2010Assume a Markov model v2011 = Pv2010 where P is atransition probability matrixP12 enables predictions to be made at 6-monthlyintervals
P12 is matrix X such that X 2 = P What are P23 P09
Ps = exp(s log P)
Problem log P P1k may have wrong sign patternsrArrldquoregularizerdquo
University of Manchester Nick Higham Actuarial Matrix Computations 30 37
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Chronic Disease Example
Estimated 6-month transition matrixFour AIDS-free states and 1 AIDS state2077 observations (Charitos et al 2008)
P =
08149 00738 00586 00407 0012005622 01752 01314 01169 0014303606 01860 01521 02198 0081501676 00636 01444 04652 01592
0 0 0 0 1
Want to estimate the 1-month transition matrix
Λ(P) = 1096440498001493minus00043
H amp Lin (2011)Lin (2011 for survey of regularization methods
University of Manchester Nick Higham Actuarial Matrix Computations 31 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Powers
gtgt A = [1 1e-8 0 1]A =10000e+000 10000e-008
0 10000e+000
gtgt A^01ans =
1 00 1
gtgt expm(01logm(A))ans =
10000e+000 10000e-0090 10000e+000
University of Manchester Nick Higham Actuarial Matrix Computations 32 37
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
MATLAB Arbitrary Power
New Schur algorithm (H amp Lin 2011 2013) reliablycomputes Ap for any real pNew backward-error based inverse scaling andsquaring alg for matrix logarithm (Al-Mohy H ampRelton 2012)mdashfaster and more accurateAlternative Newton-based algorithms available for A1q
with q an integer eg for
Xk+1 =1q[(q + 1)Xk minus X q+1
k A] X0 = A
Xk rarr Aminus1q
University of Manchester Nick Higham Actuarial Matrix Computations 33 37
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Knowledge Transfer Partnership
University of Manchester and NAG (2010ndash2013)funded by EPSRC NAG and TSB
Developing suite of NAG Library codes for matrixfunctions
Extensive set of new codes included in Mark 23 (2012)Mark 24 (2013)
Improvements to existing state of the art faster andmore accurate
My work also supported by curren2M ERC Advanced Grant
University of Manchester Nick Higham Actuarial Matrix Computations 34 37
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
Correlations Matrix Inversion Matrix Roots
Final Remarks
Fast moving developments in numerical linear algebraalgorithms
Numerical reliability is essential
Partnership with NAG enables rapid inclusion of ouralgorithms in the NAG Library
Keen to hear about your matrix problems
University of Manchester Nick Higham Actuarial Matrix Computations 37 37
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
References I
A H Al-Mohy and N J HighamImproved inverse scaling and squaring algorithms forthe matrix logarithmSIAM J Sci Comput 34(4)C153ndashC169 2012
A H Al-Mohy N J Higham and S D ReltonComputing the Freacutechet derivative of the matrix logarithmand estimating the condition numberMIMS EPrint 201272 Manchester Institute forMathematical Sciences The University of ManchesterUK July 201217 ppRevised December 2012
University of Manchester Nick Higham Actuarial Matrix Computations 1 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
References II
L Anderson J Sidenius and S BasuAll your hedges in one basketRisk pages 67ndash72 Nov 2003|wwwrisknet|
C Bekas E Kokiopoulou and Y SaadAn estimator for the diagonal of a matrixAppl Numer Math 571214ndash1229 2007
R Borsdorf N J Higham and M RaydanComputing a nearest correlation matrix with factorstructureSIAM J Matrix Anal Appl 31(5)2603ndash2622 2010
University of Manchester Nick Higham Actuarial Matrix Computations 2 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
References III
T Charitos P R de Waal and L C van der GaagComputing short-interval transition matrices of adiscrete-time Markov chain from partially observeddataStatistics in Medicine 27905ndash921 2008
T CrillyArthur Cayley Mathematician Laureate of the VictorianAgeJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8011-4xxi+610 pp
University of Manchester Nick Higham Actuarial Matrix Computations 3 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
References IV
P Glasserman and S SuchintabandidCorrelation expansions for CDO pricingJournal of Banking amp Finance 311375ndash1398 2007
C-H Guo and N J HighamA SchurndashNewton method for the matrix pth root and itsinverseSIAM J Matrix Anal Appl 28(3)788ndash804 2006
University of Manchester Nick Higham Actuarial Matrix Computations 4 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
References V
N J HighamAccuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied MathematicsPhiladelphia PA USA second edition 2002ISBN 0-89871-521-0xxx+680 pp
N J HighamComputing the nearest correlation matrixmdashA problemfrom financeIMA J Numer Anal 22(3)329ndash343 2002
University of Manchester Nick Higham Actuarial Matrix Computations 5 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
References VI
N J Higham and L LinOn pth roots of stochastic matricesLinear Algebra Appl 435(3)448ndash463 2011
N J Higham and L LinAn improved SchurndashPadeacute algorithm for fractionalpowers of a matrix and their Freacutechet derivativesMIMS EPrint 20131 Manchester Institute forMathematical Sciences The University of ManchesterUK Jan 201320 pp
University of Manchester Nick Higham Actuarial Matrix Computations 6 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
References VII
L LinRoots of Stochastic Matrices and Fractional MatrixPowersPhD thesis The University of Manchester ManchesterUK 2010117 ppMIMS EPrint 20119 Manchester Institute forMathematical Sciences
B D McCullough and A T YaltaSpreadsheets in the cloudmdashnot ready yetJournal of Statistical Software 521ndash14 2013
University of Manchester Nick Higham Actuarial Matrix Computations 7 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
References VIII
K H ParshallJames Joseph Sylvester Jewish Mathematician in aVictorian WorldJohns Hopkins University Press Baltimore MD USA2006ISBN 0-8018-8291-5xiii+461 pp
H-D Qi and D SunA quadratically convergent Newton method forcomputing the nearest correlation matrixSIAM J Matrix Anal Appl 28(2)360ndash385 2006
University of Manchester Nick Higham Actuarial Matrix Computations 8 8
![Research Matters Nick Higham School of Mathematics The …higham/talks/writing... · 2019. 4. 25. · Better: as shown by Vaughan and Williams [2]. Nick Higham How to Write Scientific](https://img.pdfslide.us/doc/110x75/608a984d889371198b4dbed9/research-matters-nick-higham-school-of-mathematics-the-highamtalkswriting.jpg)
