Multivariate Resolution in Chemistry Lecture 1 Roma Tauler IIQAB-CSIC, Spain e-mail: [email protected]

Multivariate Resolution in Chemistry

Lecture 1

Roma TaulerRoma TaulerIIQAB-CSIC, Spain

e-mail: [email protected]

Lecture 1

• Introduction to data structures and soft-modelling methods.

• Factor Analysis of two-way data: Bilinear models.

• Rotation and intensity ambiguities.

• Pseudo-rank, local rank and rank deficiency.

• Evolving Factor Analysis.

Chemical sensors and analytical data structures

one variable x1 e.g. pH

two variables x1,x2 e.g pH i T

three variables x1, x2 and x3

e.g. pH, T i P

n variables ?????

* ***** * ** *** *

pH

T

pH

pH

T

P

*

***

**

*

*

x; one sample gives one scalar (tensor 0th order)

Examples: - selective electrodes, pH- absorption at one wavelength - height/area chromatographic peak

Assumptions: - total selectivity- known lineal response

Tools: - univariate algebra and statistics

Advantages: - simple and easy to understand

Disadvantages: - only one compound information- total selectivity

- one sensor for every analyte- low information content

c

h

Time

xx

x

x

x x

xx

x

ci

hi

Data Structures: Zero order Zero-way data

x1, x2, ....., xn; one sample gives one vector (tensors of order 1)Examples:

- matrix of sensors- absorption at many (spectra)- chromatograms at a single - current intensities at many E- readings with time (kinetics)-..................

Assumptions: - known lineal responses - different and independent responses

Tools: - linear algebra - multivariate statistics - spectral analysis - chemometrics (PCA,MLR, PCR, PLS...)

Advantages: - Calibration in presence of interferences is possible - Multicomponent analysis is possible

Disadvantages: Interferences should be present in calibration samples

Spectrum

Wavelength (nm)

Abso

rban

ce

min10 20 30 40

0

Time

Chromatogram

Data Structures: First order One-way data

xij; each sample gives a datatable/matrix; tensor of order 2X = xkyk

T

Examples: - LC-DAD; LC-FTIR; GC-MS; LC-MS; FIA-DAD; CE-MS,.. (hyphenated techniques)- esp. excitation/emission (fluorescence)- MS/MS, NMR 2D, GCxGC-MS ...- spectroscopic/voltammetric monitoring of chemical reactions/processes with pH, time, T, etc.

Assumptions: - linear responses- sufficient rank (of the data matrices)

Tools: - linear algebra- chemometrics

Advantages: - calibration for the analyte in the presence of interferences not modelled in calibration samples is possible- full characterization of the analyte and interferents may be possible- few calibration samples are needed (only one sample calibration)

Wavelengths

Elu

tio

nti

me

s Spectrum

Ch

rom

ato

gra

m

Wavelengths

Elu

tio

nti

me

s Spectrum

Ch

rom

ato

gra

m

Data Structures: Second order / Two-way data

xijk; each sample gives a datacube; tensor of order 3X = xkykzk

Examples- Several spectroscopic matrices- Several hyphenated chromatographic- Hyphenated multidimensional chromatography (GC x GC / MS)

- excitation/emission/time..............Assumptions: - bilinear/trilinear responses

- sufficient rank (of the data matrices)Tools: - multilinear and tensor algebra

- chemometricsAdvantages: - unique solutions (no ambiguities)

- calibration for the analyte in the presence of interferences not modelled in calibration samples is possible

- full characterization of the analyte and interferents is possible- few calibration samples are needed (only one sample calibration)

Dtime

Run nr.

D

Di

time

Multi-way data analysis(PARAFAC, GRAM)

Extended multivariate resolution

Data Structures: Third order Three-way data

0th order data: ISE, pH,..

1th order data: spectra

2nd order data: LC/DAD GC/MS fluorescence

3rf order data: time/ /excitation/ /emission

ExamplesChemical reaction systems monitored using spectroscopic measurements (even at femtosecond scale) to follow the evolution of a reaction with time, pH, temperature, etc., and the detection of the formation and disappearance of intermediate and transient species

Monitoring chemical reactions.

T=37oC

Spectrophotometer

Peristalticpump

0.050 ml

-125.3

pHmeter

Autoburette

Computer

PrinterStirrer

Thermostatic bath

D (NR,NC)

NCNRNRNR

NC

NC

pHpHpH

pHpHpH

pHpHpH

ddd

ddd

ddd

,,,

,,,

,,,

..

.....

..

..

21

22212

12111

pH

wavelength

Examples

Quality control and optimisation of industrial batch reactions and processes, where on-line measurements are applied to monitor the process.

Process analysis

Computer Spectrometer

*

*

*

* ***

*

*

* *

probe

D (NR,NC)

NCNRNRNR

NC

NC

ttt

ttt

ttt

ddd

ddd

ddd

,,,

,,,

,,,

..

.....

..

..

21

22212

12111

time

wavelength

Examples Analytical characterisation of complex environmental, industrial and food mixtures using hyphenated (chromatography, continuous flow methods with spectroscopic detection)

Chromatographic Hyphenated techniques

LC-DAD, GC-MS, LC-MS, LC-MS/MS....

D (NR,NC)

NCNRNRNR

NC

NC

ttt

ttt

ttt

ddd

ddd

ddd

,,,

,,,

,,,

..

.....

..

..

21

22212

12111

time

wavelength

Examples FIA-DAD-UV with pH gradient for the analysis of a mixture of drugs.

D (NR,NC)

NCNRNRNR

NC

NC

pHpHpH

pHpHpH

pHpHpH

ddd

ddd

ddd

,,,

,,,

,,,

..

.....

..

..

21

22212

12111

pH

wavelength

Examples Analytical characterisation of complex sea-water samples by means of Excitation-Emission spectra for an unknown with tripheniltin (in the reaction with flavonol)

Excitation emission (fluorescence) EEM techniques

NCNRNRNR

NC

NC

xxx

xxx

xexex

ddd

ddd

ddd

,,,

,,,

,,,

..

.....

..

..

21

22212

12111

D (NR,NC)

emission

exci

tatio

n

Examples Protein folding and dynamic protein-nucleic acid interaction processes. In the post-genomic era, understanding these biochemical complex evolving processes is one of the main challenges of the current proteomics research.Conformation changes

D (NR,NC)

NCNRNRNR

NC

NC

TTT

TTT

TTT

ddd

ddd

ddd

,,,

,,,

,,,

..

.....

..

..

21

22212

12111

Temperature

wavelength

Primarystructure

Secondarystructure

Tertiarystructure

Amino acidsGlobule

formationAssembled subunitsHelix, sheet formation

Val

Leu

Ser

Ala

Asp

Ala

Trp

Gly

Val

His

-helix

-sheet

turn

Random coil

Quaternarystructure

Examples Image analysis of spatially distributed chemicals on 2D surfaces measured using coupled microscopy-spectroscopy techniques in geological samples, biological tissues or food samples.

Spectroscopic Image analysis

0.5

1

1.5

2

2.5

3

3.5

4

10 20 30 40 50 60

10

20

30

40

50

60

x

y

T

ota

l nu

mb

er o

f p

ixe

ls (

x

y)

Data Structures in ChemistryExperimental Data

two orders/ways/modes of measurement

d d d

d d d

d d d

NC

NC

NR NR NR NC

1 1 1 2 1

2 1 2 2 2

1 2

, , ,

, , ,

, , ,

. .

. .

. . . . .

. .

row-order(way,mode)i.e. usuallychange in chemical composition (concentrationorder)

column order (way,mode)i.e usually change in system propertieslike in spectroscopy, voltammetry,...(spectral order)

D(NR,NC)

Chemical data tables (two-way data)

I sp

ectr

a (t

imes

)

J variables (wavelengths)

Data table or matrix

D

Instrumental measurements(spectra,voltammograms,...)concentration

changesmeasurements(time, tempera-ture, pH, ....

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

Plot of spectra(rows)

Plot of elutionprofiles (columns)

Chemical data modelling methods may be divided in:Hard- modelling methods (deterministic)Soft-modelling methods (data driven)Hybrid hard-soft modelling methods

Chemical data modelling

Soft modellingHard modelling

DataPhysical

HardModel

AnalyticalInformation

Data AnalyticalInformation

PhysicalModel

Datadriven

softmodel

?

Hard-modelling approaches for chemical (stationary, dynamic, evolving…) systems are based on an accurate physical description of the system and on the solution of complex systems of (differential) equations fitting the experimental measurements describing the evolution and dynamics of these systems. They are deterministic models.

Hard-modelling methods usually use non-linear least squares regression (Marquardt algorithm) and optimisation methods to find out the best values for the parameters of the model.

Hard-modelling usually deal with univariate data. It has been often used in the past until the advent of modern instrumentation and computers giving large amounts of data outputs.

Hard-modelling is often successful for laboratory experiments, where all the variables are under control and the physicochemical nature of the dynamic model is known and can be fully described using a known mathematical model

Hard-modelling

However, and even at a laboratory level, there are examples where hard-modelling requirements and constraints are not totally fulfilled or no physicochemical model is known to describe the process (e.g. in chromatographic separations or in protein folding experiments).

Data sets obtained from the study of natural and industrial evolving processes are too complex and difficult to analyse using hard-modelling methods. In these cases, there is no known physical model available or it is too complex to be set in a general way.

Advanced hard-modelling in industrial applications has been attempted to model experimental difficulties, such as changes in temperature, pH, ionic strength and activity coefficients. This is a very difficult task!

Hard-modelling

Data Fitting in the Chemical SciencesP. Gans, John Wiley and Sons, New York 1992

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3x 104

Wavelengths

Ab

so

rtiv

itie

s

LS (D, C)(ST)

ST

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

Wavelength

Ab

so

rba

nc

e

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Co

nc

en

tra

tio

nNon-linear model

fittingmin(D(I-CC+)C = f(k1, k2)

D C

Output: C, S and model parameters.

The model should describe all the variation in the experimental measurements.

Hard modelling

Soft-modelling instead, attempts the description of these systems without the need of an a priori physical or (bio)chemical model postulation. The goal of the latter methods is the explanation of the variations observed in the systems using the minimal and softer assumptions about data. They are data driven models.

Soft models usually give an improved analytical description of the analysed process.

Soft modelling needs more data than hard-modelling. Soft modelling methods deal with multivariate data. Its use has augmented in the recent years because of the advent of modern analytical instrumentation and computers providing large amounts of data outputs.

The disadvantage of soft models is their poorer extrapolating capabilities (compared with hard-modelling).

Soft-modelling

A soft model is hardly able to predict the behaviour of the system under very different conditions from which it was derived.

Complex multivariate soft-modelling data analysis methods have been introduced for the study of chemical processes/systems like Factor Analysis derived methods.

Factor Analysis is a multivariate technique for reducing matrices of data to their lowest dimensionality by the use of orthogonal factor space and transformations that yield predictions and/or recognizable factors.

Soft-modelling

Factor Analysis in Chemistry 3rd Edition, E.R.Malinowski, Wiley, New York 2002

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3x 104

Wavelengths

Ab

so

rtiv

itie

s

ST

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

Wavelength

Ab

so

rba

nc

e

D C

Constrained ALS optimisationLS (D,C) S*LS (D,S*) C*min (D –C*S*)

,

Output: C and S.

All absorbing contributions in and out of the process are modelled.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Co

nc

en

tra

tio

n

Soft modelling

Lecture 1






Soft-modelling

1 1 2 21

T

.....

D = U V

n

ij i j i j in nj ik kjk

d u v u v u v u v

In matrix form

data scores loadings

experimental datais modelled as alinear sum of weighted(scores) factors (loadings)

Factor Analysis (Bilinear Model)

D = + + + ... +A B C E

D =

A

+

B

+

C

+ ... + E+ +

BILINEARITY

Assumption: Bilinearity (the contributions of the components in the two orders of measurement

are additive)

Soft-modelling

0 50 100

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 20 40 60

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Recovery of the responses of every component (chemical species) in the different modes of

measurement

GOALS OF BILINEAR MODEL

=

Soft-modelling

Soft-modelling: Factor Analysis

ExperimentalData Matrix

PrincipalComponents

FactorIdentification

Targettesting

ClusterAnalysis

Real FactorModels

Predictions

Covariancematrix

AbstractFactors

New AbstractFactors

Datamatrix

RealFactors

com

bina

tion

matrixmultiplication

abstractreproduction

targettransformation

decomposition

abstract rotation

Soft-modelling: Factor Analysis(traditional approach)

Soft-modelling methods (I)

Factor Analysis methods based on the use of latent variables or eigenvalue/singular value data matrix decompositions. Examples

•PCA, SVD, rotation FA methods•Evolving Factor Analysis methods•Rank Annihilation methods•Window Factor Analysis methods•Heuristic Evolving Latent Projections methods•Subwindow Factor Analysis methods•…..

Multivariate Resolution methods do a data matrix decomposition into their ‘pure’ components without using explicitly latent variables analysis techniques. Examples:

•SIMPLISMA•Orthogonal Projection Approach (OPA),•Positive Matrix Factorization methods (and Multilinear Engine extensions)•Multivariate Curve Resolution-Alternating Least Squares (MCR-ALS)•Gentle•.....

Soft-modelling methods (II)

Three-way and Multiway methods which decompose three-way or multiway data structures. Examples:

•Multiway and multiset extensions of PCA•Genralized rank Annihilation, GRAM; Direct Trilear Decomposition (DTD, TLD)•Multiway and multiset extensions of MCR-ALS methods •PARAFAC-ALS•Tucker3-ALS•.......

Soft-modelling methods (III)

Factor Analysis in Chemistry, 3rd Ed., E.R.Malinowski, John Wiley & Sons, New York, 2002

Principal Component Analysis, I.T. Jollife, 2nd Ed., Springer, Berlin, 2002

Multiway Analysis, Applications in the Chemical Sciences, A.Smilde, R.Bro and P.Geladi, John Wiley & Sons, New York, 2004

Multivariate Image Analysis, P.Geladi, John Wiley and Sons, 1996

Soft modeling of Analytical Data. A.de Juan, E.Casassas and R.Tauler, Encyclopedia of Analytical Chemistry: Instrumentation and Applications, Edited by R.A.Meyers, John Wiley & Sons, 2000, Vol 11, 9800-9837

Soft-modelling

Data structures Type of ModelsOne way data (vectors) Linear and non-linear models

di = b0 + b ci;di = fnon-linear(ci)

Two way data (matrices) Bilinear and non-bilinear models

Non-bilinear data can still belinear in one of the two modes

Three-way data (cubes) Trilinear and non-trilinear models

Non-trilinear data can still be bilinear in two modes

N

ij in nj ijn 1

d c s e

D

I samplesdij

J variables

di

pqr

c

g c

z

zp q r

N

ijk in jn kn ijkn=1

N N N

ijk ip jq krn ijkp=1 q=1 r=1

d = s +e

d = s +ek=1,...,Kconditions

i=1,

...,I

j=1,...,J

I samples

Soft-modelling

N

ij in nj ijn 1

T

d c s e

D CS E

dij is the data measurement (response) of variable j in sample in=1,...,N are the number of components (species, sources...)cin is the concentration of component n in sample i;snj is the response of component n at variable j

Bilinear models for two way data:

D

J

Idij

Soft-modelling

N

DUor

C

VT or ST

E+

J J J

I I

N

N << I or J

Bilinear models for two way data

PCAD = UVT + E

U orthogonal, VT orthonormalVT in the direction of maximum

variance

Unique solutions but without physical meaning

Useful for interpretationbut not for resolution!

MCRD = CST + E

Other constraints (non-negativity,unimodality, local rank,… )

U=C and VT =ST non-negative,...C or ST normalization

Non-unique solutions but with physical meaning

Useful for resolution(and obviously for interpretation)!

I

Soft-modelling

PCA Model (Principal Component Analysis)

X = U VT + E

U ‘scores’ matrix (orthogonal)VT loadings matrix (orthonormal)

SVD Model (Singular Value Decomposition)

D = U* S VT + E

U* ‘scores’ matrix (orthonormal)S diagonal matrix of the singular values s

s = 1/2

eigenvalues of the covariances matrix DDT

VT ‘loadings’ matrix (orthonormal)

,i i is

PCA Model: D = U VT

= +DU

VT

E

D = u1v1T + u2v2

T + ……+ unvnT + E

D u1v1T u2 v2

T unvnT E= + +….+ +

n number of components (<< number of variables in D)

rank 1 rank 1 rank 1

scores

loadings(projections)

unexplained variance

PCA Model

X = U VT + EX = structure + noise

It is an approximation to the experimental data matrix X

• Loadings, Projections: VT relationships between original variables and the principal components (eigenvectors of the covariances matrix). Vectors in VT (loadings) are orthonormals (orthogonal and normalized).

• Scores, Targets: U relationships between the samples (coordinates ofsamples or objects in the space defined by the principal componentsVectors in U (scores) are orthogonal

Noise E Experimental error, non-explained variances

Summary of Principal Component Analysis PCA

1. Formulation of the problem to solve2. Plot of the original data 3. Data pretreatment.

(data centering, autoscaling, logarithmic transformation…)4. Built PCA model. Determination of the number of

components. Graphical inspection of explained/residual plots)

5. Study of the PCA model PCA. Multivariate data exploration- ‘loadings’ plot ==> map of the variables- ‘scores’ plot ==> map of the samples

6. Interpretation of the PCA mode. Identification of themain sources of data variance

7. Analysis of the residuals matrix E = D -U VT

-2-3

-2-10123

PC

2 (

27

%)

Scores plot

-1

-0.8-0.6-0.4-0.2

00.2

PC

2 (

27

%)

Loadings plot

-1 0 1 2 3 4PC1 (41%)

1 2

3

4 5 6

7

8

910

11

12

13141516

17

18

19

2021

22

1

2

3

-1 -0.5 0 0.5 1PC1 (41%)

1

2

3 4 5

6

7 8 910

11

1213141516

1718

1920

21222324

25

2627

282930

313233

34

35

363738

39

40414243444546

47

48495051

52

5354

5556

5758596061

6263

6465

666768697071

727374

75

76

77

78

79

8081

8283

84

858687

8889

9091 929394

9596

B

A

-2 -1 0 1 2 3 4-3

-2

-1

0

1

2

3

PC1 (41%)

PC

2 (

27

%)

Biplot

1 2 3 4 5 6 7 8 91011121314151617 18192021222324

25 26272829303132333435363738

3940414243444546

47484950 515253545556 5758596061 6263646566676869707172737475767778

7980818283 8485868788899091 9293949596

1 2

3

4 5 6

7

8

910

11

12

13141516

17

18

19

20

21

22

Site 1Site 2Site 3

Site 22

Sa

mp

lin

g

sit

es

[org]1 [org]2 [org]3 [org]96

Pollutant concentrationData set

D

Uscores

VT

loadings

PCA

Multivariate Curve Resolution (MCR)

Pure component information

C

ST

sn

s1

c nc 1

WavelengthsRetention times

Pure concentration profiles Chemical model

Process evolutionCompound contribution

relative quantitation

Pure signals

Compound identity

source identification and Interpretation

D

Mixed information

tR

Lecture 1






Factor Analysis Ambiguities in the analysis of a data matrix (two-way data)

Rotation Ambiguities

Factor Analysis (PCA) Data Matrix DecompositionD = U VT + E

‘True’ Data Matrix DecompositionD = C ST + E

Rotation and scale/intensity ambiguities

D = U T T-1 VT + E = C ST + EC = U T; ST = T-1 VT

How to find the rotation matrix T?

Factor Analysis Ambiguities in the analysis of a data matrix (two-way data)

Rotation Ambiguities


D = C ST + E = D* + E

Cnew = C T ( NR,N) (NR,N) (N,N)

STnew = T-1 ST

(N,NC) (N,N) (N,NC)

D* = C ST = CnewSTnew

Matrix decomposition is not unique!T(N,N) is any non-singular matrix

Rotational freedom for any T


=

Cnew,2 Cold,1, Cold,2 t1,2

t2,2

STnew,1

STold,1

STold,2

STnew,2

t-12,2t-1

2,1

STold,1

STold,2

=

=

T

t-11,1 t-1

1,2

T-1

=

Cnew,1 Cold,1, Cold,2 t1,1

t2,1

T


Rotation ambiguities and rotation matrix T(N,N)

Intensity (scale) ambiguities:

d c s c sij in nj

n

in nj

n

k1

k

For any scalar k

Intensity/scale ambiguities make difficuly to obtain quantitative informationWhen they are solved then it is also possible to have quantitative information


Intensity (scale) ambiguities: cold x k = cnew

x x

1/k x sold = snew

cold sold == ( cold x k)(1/k x sold) =

cnew snew


Questions to answer:

1. Is it possible to have unique solutions?

2. What are the conditions to have unique solutions?

3. If total unique solutions are not possible:

a) Is it still possible at least to find out some of the possible solutions?

b) Is it possible to have an estimation of the band or range of possible/feasible solutions?

c) How this range of feasible solutions can be reduced?


Lecture 1




• Pseudo rank, local rank and rank deficiency.


Definitions

Mathematical rank of a data matrix is the minimum number of linearly independent rows or columns describing the variance of the whole data set. Minimum number of basis vectors spanning the row and column vector spaces. It may be obtained by SVD or PCA.

Pseudo-rank or Chemical rank is the mathematical rank in absence of experimental error/noise. Usually it is equal to the number of chemical/physical components contributing to the observed data variance apart from experimental noise/error. Obtained from the number of larger components from PCA, SVD or other FA methods

Local Rank is the chemical rank of data submatrices. Obtained from EFA, EFF, SIMPLISMA, OPA, or other FA submatrix analysis methods

Rank deficiency when chemical rank is lower than the known number of contributions. Rank deficiency may be broken/solved by data matrix augmentation and perturbation strategies.

Rank overlap rank deficiency caused by equal vector profiles of different chemical/physical components in one or more modes.

Pseudo Rank: Number of contributions (factors, components)

Principal Component Analysis

D = U

unu1

VT

vn

v1 Gives an abstract

(orthogonal) bilinear

model to describe

optimally the variation

in our data set.Useful chemical information

Size of the model (chemical rank)

Number of chemical contributions

Principal Component Analysis (SVD algorithm)

D = TPT = USVT

Diagonal matrix (singular values)

Magnitude of singular value

Importance of contribution


Principal Component Analysis (SVD algorithm)

1 2 3 4 5 6 7 8 9 1018

19

20

21

22

23

24

25

Number of components

log

(eig

en

valu

es)

4 contributions Plot log(eigenvalues)

Plot singular values

Eigenvalue = (sing. value)2

D = TPT = USVT

Diagonal matrix (singular values)


2s

small size

large size

Overestimations of rank (overfitting).

– Large overestimation: the measurements may not follow a bilinear model.

– Small overfestimation: presence of structured noise or high

noise levels.

Underestimations of rank (rank deficiency).

Linear dependencies

Contributions with very similar signals or concentration profiles.

Compounds with non-measurable signals.

Minor compounds.


• Are all the signals distinguishable and independent?

• Are all the concentration profiles distinguishable and independent?

No Rank- deficient systems

Rank deficiency

Detectable rank < nr. of process contributions

Examples:1) 2nd order reaction A = B + C, [B] = [C], 3 chemical species/contributions, but Rank =22) Enantiomer conversion monitored by UV and the spectrum D = spectrum L, two chemical species/components but Rank =1 (Rank overlap)

Rank deficiency

• Closed reaction systems. Some concentration profiles are described as linear combinations of others.

System HA / A-, HB / B-

CA = [HA] + [A-]

CB = [HB] + [B-]

CB = kCA

[HA], [HB]

[A-] = CA - [HA]

[B-] = CB - [HB] = kCA - [HB]

[HA], [HB], [A-], [B-] f ([HA], [HB], CA) Rank 3

0

0,02

0,04

0,06

0,08

0,1

0,12

1 6 11

pH

Co

nce

ntr

atio

n

CA

CB

Breaking rank-deficiency by matrix augmentation

• Matrix Augmentation in the rank-deficient direction

Data set

HA

HA / HB

pH

pH

CB kCA

[B-] = CB - [HB] kCA - [HB]

[HA], [HB], [A-], [B-] f ([HA], [HB], CA)

Rank 4

Rank deficiency

Lecture 1




• Pseudo-rank and rank deficiency.

• Local Rank and Evolving Factor Analysis.

Local exploratory analysis

Study of the variation of the number of contributions in the process or system. Study of the rank variation during the process.

Evolving Factor Analysis (EFA)

Fixed Size Moving Window - Evolving Factor Analysis (FSMW-EFA)

Evolving Factor Analysis

• Stepwise chemometric monitoring of a process.

– Forward Evolving FA (from beginning to end)– Backward Evolving FA (from end to beginning)

Working procedure

Display of subsequent PCA analyses along gradually increasing data set windows.


HPLC-DAD example

D

Wavelengths

Ret

enti

on

tim

es

Spectrum

Ch

rom

ato

gra

m


Forward Evolving Factor Analysis

PCA PCA PCA PCA

0 5 10 15 20 25 30 35 40 45 507.5

8

8.5

9

9.5

10

10.5

11

Retention times

log

(eig

enva

lues

)


Forward Evolving Factor Analysis

0 5 10 15 20 25 30 35 40 45 507.5

8

8.5

9

9.5

10

10.5

11

Retention times

log

(eig

enva

lues

)

Total number of compounds (PCA in last window)

Noise level

Location of the emergence of compounds

Selective zone


Backward Evolving Factor Analysis

PCAPCAPCAPCA

0 5 10 15 20 25 30 35 40 45 507.5

8

8.5

9

9.5

10

10.5

11

Retention times

log

(eig

en

valu

es)


Backward Evolving Factor Analysis

0 5 10 15 20 25 30 35 40 45 507.5

8

8.5

9

9.5

10

10.5

11

Retention times

log

(eig

en

valu

es)

Noise level

Location of the disappearance of compounds

Total number of compounds (PCA last total window)

Selective zone


Combined EFA plot (forward and backward EFA)

5 10 15 20 25 30 35 40 45 507.5

8

8.5

9

9.5

10

10.5

11

Retention times

log

(eig

enva

lues

)

Location of emergence and decay of compounds

Detection of selective zones (extremes)

Total number of components (PCA of extreme windows)


Sequential processes Consecutive emergence-decay profiles. No embedded compounds.

Approximate concentration profiles

Noise level

5 10 15 20 25 30 35 40 45 507.5

8

8.5

9

9.5

10

10.5

11

Retention times

log

(eig

enva

lues

)

Concentration windowZero-component windows


Approximate concentration profiles

5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6x 10

8

EFA derived concentration profiles

Real concentration profiles

Fixed Size Moving Window-Evolving FA (FSMW-EFA)

• Local rank map along the process direction or the signal direction.

Working procedure

Subsequent PCA in fixed size windows moving stepwisely along the data set.

Window size min(number of components + 1)

FSMW-EFA

0 5 10 15 20 25 30 35 40 45 503.5

4

4.5

5

5.5

Retention times

log

(eig

enva

lues

)

PCA PCA PCA PCA

FSMW-EFA

0 5 10 15 20 25 30 35 40 45 503.5

4

4.5

5

5.5

Retention times

log

(eig

enva

lues

)

Noise level

Detection of selective zones along the whole process

10 01 12 22

Variation of local rank along the process direction (complexity, degree of overlap among compounds)

0 20 40 6000.20.40.60.8

11.21.41.61.8

2 x 10 -5

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5 x 104

0 20 40 600

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60-3

-2

-1

0

1

2

3

0 10 20 30 40 50-2

-1.5

-1

-0.5

0

0.5

1

0 10 20 30 40 50-2

-1.5

-1

-0.5

0

0.5

1

window size 5 window size 8

Local rank

detection

EFAEFA

FSMW-EFA FSMW-EFA

FSMW-EFA vs. EFA

EFA

• Displays the evolution of the process.

• The compounds are well identified (concentration windows)

• Local rank information is not easily interpreted.

FSMW-EFA

Clear definition of local rank.

Sensitive to detection of minor compounds.

The idea of process evolution is not preserved.

• Detection of the selective windows or regions where only one species exists (total selectivity)

• Detection of zero concentration windows or regions (no species is present)

• Detection of windows or regions where a particular species is not present

• Detection of the concentration windows or regions where one species is present (other species can coexist)

Getting Local rank information from Evolving Factor Analysis methods

References

• EFA– H. Gampp, M. Maeder, C.J. Meyer and A.D.

Zuberbühler. Talanta, 32, 1133-1139 (1985).– M. Maeder. Anal. Chem. 59, 527-530 (1987).

• FSMW-EFA– H.R. Keller and D.L. Massart. Anal. Chim. Acta,

246, 379-390 (1991).

• SIMPLISMA

– W. Windig and J. Guilment. Anal. Chem., 63, 1425-1432 (1991).

Documents

Multivariate Resolution in Chemistry Lecture 1 Roma Tauler IIQAB-CSIC, Spain e-mail: [email protected]