including trilinear and restricted tucker3 models as a constraint in multivariate curve resolution...

Including trilinear and restricted Tucker3 models as a constraint in Multivariate Curve Resolution Alternating Least

Squares

Romà TaulerDepartment of Environmental Chemistry, IIQAB-CSIC, Jordi Girona 18-26, Spain

e-mail: [email protected]

Outline

• Introduction

• MCR-ALS of multiway data

• Example of application: MCR-ALS with trilinearity constraint

• Example of application: MCR-ALS with component interaction constraint

• Conclusions

Motivations of this workMultivariate Curve Resolution (MCR) methods have been shown to be powerful and useful tools to describe multicomponent mixture systems through constrained bilinear models describing the 'pure' contributions of each component in each measurement mode

Pure component information

CST

sn

s1

cnc1

WavelengthsRetention times

Pure signalsCompound identity

source identification and Interpretation

D

Mixed information

tR

PCA X orthogonal, YT orthonormal

YT in the direction of maximum variance

Unique solutions

but without physical meaningIdentification and Interpretation!

MCRC and ST non-negativeC or ST normalization

other constraints (unimodality, closure, local rank,… )Non-unique solutions

but with physical meaningResolution!

N

DXor

C

YT or ST

E+

J J J

I I

N

N << I or JI

Bilinear models for two way data:

TPCA

CSCDmin ˆˆˆ

ˆ T

PCAS

SCDminT

ˆˆˆ

• Optional constraints ( non-negativity, unimodality, closure, local rank …) are applied at each iteration• Initial estimates for C or ST are needed

C and ST are obtained by solving iteratively the two alternating LS equations:

An algorithm to solve Bilinear models using Multivariate Curve Resolution (MCR):

Alternating Least Squares (MCR-ALS)

Constraints applied to resolved profiles have included non-negativity, unimodality, closure, selectivity, local rank and physical and chemical (deterministic) laws and models.

Hard + soft modelling constraints

MCR-ALS hybrid (grey) models

DataMatrix

InitialEstimation

SVDor

PCA

ALSoptimization

ResolvedSpectraprofiles

Resolv

ed

Con

cen

trati

on

pro

file

s

Estimation of the

number of components

Initial estimation ALS optimization

CONSTRAINTSResults of the ALS optimization procedure:

Fit and Diagnostics

E+

Data matrix decomposition according to a bilinear

model

Flowchart of MCR-ALS

DC

ST

TPCA

CSCDmin ˆˆˆ

ˆ T

PCAS

SCDminT

ˆˆˆ

D = C ST + E(bilinear model)

Journal of Chemometrics, 1995, 9, 31-58; Chemomet.Intel. Lab. Systems, 1995, 30, 133-146Journal of Chemometrics, 2001, 15, 749-7; Analytica Chimica Acta, 2003, 500,195-210

Until recentlyMCR-ALS input had to be typed in the MATLAB command line

Troublesome and difficult in complex cases where several data matrices are simultaneously analyzed and/or different constraints are applied to each of them for an optimal resolution

Now!

A new graphical user-friendly interface for MCR-ALS

J. Jaumot, R. Gargallo, A. de Juan and R. Tauler, Chemometrics and Intelligent Laboratory Systems, 2005, 76(1) 101-110

Multivariate Curve Resolution Home Page

http://www.ub.es/gesq/mcr/mcr.htm

D = C ST + E = D* + E

Cnew = C T

STnew = T-1 ST

D* = C ST = C TT-1 ST = CnewSTnew

Reliability of MCR-ALS solutionsMCR solutions are not unique

Identification of sougth solutions => evaluation of rotation ambiguities => calculation of feasible band boundaries

R.Tauler (J.of Chemometrics 2001, 15, 627-46)

Rotation matrix T is not unique. It depends on the constraints.Tmax and Tmin may be found by a non-linear constrained optimization algorithm!!!

•0 •5 •10 •15 •20 •25 •30 •35 •40 •45 •50•0

•0.1

•0.2

•0.3

•0.4

•0.5

•0 •5 •10 •15 •20 •25 •30 •35 •40•0

•0.5

•1

•1.5

Tmax

Tmin

Tmax

Tmin

0 sconstraint subject to

CS

(T)(T)sc of max/min T

kk

T

(T)g

(T)f

k

k

3 4 5 6 7 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mean, bands and confidence range of concentration profiles

pH

Rel

. co

nc

240 250 260 270 280 290 300 310 3200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Mean, bands and confidence range of spectra

Wavelength /nm

Abs

orba

nce

/a.u

.

pK1 pK2

Noise added

ValueStd. dev

ValueStd. dev

0 % 3.6539 2e-14 4.9238 2e-14

0.1 % 3.6540 6e-4 4.9226 0.0022

1 % 3.6592 0.0061 4.9134 0.0264

2 % 3.6656 0.0101 4.9100 0.0409

5 % 4.0754 0.4873 5.3308 1.1217

Montecarlo Simulation

JackknifeNoise Addition

Resampling Methods

TheoreticalData

ExperimentalData

Noise 1%

Reliability of MCR-ALS resultsError estimation of MCR-ALS resolved profiles

Error propagation and Confidence intervalsJ.Jaumot, R.Gargallo and R.Tauler, J. of Chemometrics, 2004, 18, 327–340

Outline

• Introduction




• Conclusions

Extension of Bilinear Models Matrix Augmentation (PCA, MCR, ...)

The same experiment monitored with different techniques

=

D1

D2

D3

D

X1

=

D1

D2

D3

D

Several experiments monitored with the same technique

=

D1 D2 D3

D4 D5 D6

=

D1 D2 DX1

D4 D5 D6

XD Several experiments monitored with several

techniques

Row-wise

Column-wise Row and column-wiseD

=D1 D2 D3

D

=D1 D2 D3

D

=D1 D2 D3D1 D2 D3

Y1T

X

XY2

T Y3T

Y1T Y2

T Y3T

YT

X2

YT

YT

X2

X3

X

Daug

Y

metals

compartments

site

s

F

S

W

F

S

W

contaminants

site

ssi

tes

site

s

1

2

3

4

5

6

MA-PCAMA-MCR-ALS

Bilinear modelling of three-way data(Matrix Augmentation or matricizing, stretching, unfolding )

Xaug

Augmenteddata matrix

Augmentedscores matrix

Loadings

Advantages of MA-MCR-ALS

• Resolution local rank/selectivity conditions are achieved in many situations for well designed experiments (unique solutions!)

• Rank deficiency problems can be more easily solved

• Constraints (local rank/selectivity and natural constraints) can be applied independently to each component and to each individual data matrix.

J,of Chemometrics 1995, 9, 31-58 J.of Chemometrics and Intell. Lab. Systems, 1995, 30, 133

1 2 3 x

i

4 5 6 xii

PCA

1st comp

PCA

1st comp

zi

zii

Scores refolding

strategy!!!(applied to augmented

scores)

Xaug

D

YT

contaminants

compartments

site

s

FS

W

F

S

W

contaminants

site

ssi

tes

site

s

1

2

3

4

5

6

PCAMCR-ALS

Bilinear modelling of three-way data(Matrix Augmentation, matricizing, stretching, unfolding )

X Y

Z

site

s

contaminants

compartments (F,S,W)

x

i

xii z

iz

ii

Loadings recalculationin two modes

from augmentedscores

D

contaminants

compartments

site

s

F

S

W

F

S

W

contaminants

site

ssi

tes

site

s

Xaug

YT

1

2

3

MCR-ALS

TRILINEARITY CONSTRAINT(ALS iteration step)

Selection of species profile

1

2

3

Folding

every augmentedscored wanted tofollow the trilinearmodel is refolded

MA-MCR-ALSTrilinearity constraint

PCA

Substitution ofspecies profile

Rebuilding augmented scores

1’

2’

3’



X YT

contaminants

Z

site

s


This constraintis applied at each stepof the ALS optimization

and independently for each component

individually

D

=

Xaug

Y

contaminants

compartments

site

s

F

S

W

F

S

W

metals

site

ssi

tes

site

s

1

2

3

4

5

6

MCR-ALS

Folding

1 2 3 4 5 6

component interaction constraint

(ALS iteration step)

interacting augmented scores are folded

together

1’

2’

3’

4’

5’

6’

=



MA-MCR-ALScomponent interaction

constraint

PCA =

This constraint is applied at each step of the ALS optimizationand independently and individually for each component i

XY

Z


This is analogous to a restricted Tucker3 model

Outline

• Introduction




• Conclusions

Run1Run 2

Run 3 Run 4

Daug

=>ST

mixture 1

mixture 2

C1

C2

C3

C4

D1

D2

cA cB cC cD

cE cF cG cH

mixture 3

mixture 4

D1

D2

cI cJ cK cL

cM cN cO cP

Trilinearity Constraint

(flexible for every species)

Extension of MCR-ALS to multilinear

systems

cA

cE

cI

cM


cA cE cI cM

Foldingspeciesprofile

cIzI1 zI2 zI3 zI4

PCA

1st score

1 st loading

1st scoregives thecommonshape

Loadings give therelative amounts!

Substitution of species profile cA

’ = zI1cI

cE’ = zI2cI

cI’ = = zI3cI

cM’ = = zI4cI

Refoldingspeciesprofileusing

=ST

mixture 1

mixture 2

C1

C2

C3

C4

D1

D2

cA cB cC cD

cE cF cG cH

mixture 3

mixture 4

D1

D2

cI cJ cK cL

cM cN cO cP

Bilinear Model

MCR-ALSusing trilinear

ConstraintsR.Tauler,

I.Marqués and E.Casassas. Journal of

Chemometrics, 1998; 12, 55-75

UniqueSolutions!Like in PARAFAC!

C

Z

Trilinearity constraint

Trilinear ModelCI CII CIII CIV

The profiles in the three modes are easily recovered!!!

Daug = Caug ST

zI1 zI2 zI3 zI4

zII1 zII2 zII3 zII4

zIII1 zIII2 zIII3 zIII4

zIV1 zIV2 zIV3 zIV4

Effect of application of the trilinearity constraint

Profiles withdifferentshape

Profiles withequal shape

Trilinearityconstraint

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Run 2

Run1Run 3

Run 4

Run 2

Run1Run 3

Run 4

lack of fit

I,J2

i,j i,ji,j

I,J2

i,ji,j

ˆ(d d )

lof% x100(d )

I,J

2i,j i,j

i,j2I,J

2i,j

i,j

ˆ(d d )

R 1(d )

Explained variances

Example 1 Four chromatographic runs following a trilinear model lof % R2

a) Theoretical 1.634 0.99973 (added noise)b) MA-MCR-ALS-tril 1.624 0.99974c) PARAFAC 1.613 0.99974

There is overfitting!!!

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

O PARAFAC+ MA-MCR-ALS tril- theoretical

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

O PARAFAC+ MA-MCR-ALS tril- theoretical

Run1Run 2

Run 3 Run 4

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Example 2 Four chromatographic runs not following a trilinear model

lof % R2

a) Theoretical 0.9754 0.99995 (added noise)b) MA-MCR-ALS-non-tril 0.9959 0.99990

Good MA and local rank (selectivity) conditions for total resolution without ambiguities

+ MA-MCR-ALS non tril- theoretical

+ MA-MCR-ALS non tril- theoretical

Example 2 Four chromatographic runs not following a trilinear modellof % R2

a) Theoretical 0.9754 0.9999 (added noise)b) MA-MCR-ALS-tril 17.096 0.9708

The data system is far from trilinear, and impossing trilinearity gives a much worse fit and wrong shapes of the recovered profiles

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

+ MA-MCR-ALS tril- theoretical


Example 2 Four chromatographic runs not following a trilinear modellof % R2

a) Theoretical 0.9754 0.9999 (added noise)b) PARAFAC lof (%) 14.34 0.9794

The data system is far from trilinear, and impossing trilinearity gives a much worse fit and wrong shapes of the recovered profiles

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3

3.5

O PARAFAC- theoretical

O PARAFAC- theoretical

Example 3: A hybrid bilinear-trilineal model2 components folow the trilinear model

(1st and 3rd) and 2 components (2nd and 4th) do not

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

trilinear

Non-trilinear

1

3 2

4

Run1

Run 2

Run 3

Run 4

Daug

=ST

mixture 1

mixture 2

C1

C2

C3

C4

D1

D2

cA cB cC cD

cE cF cG cH

mixture 3

mixture 4

D1

D2

cI cJ cK cL

cM cN cO cP

A hybrid bilinear-

trilinear model

cAor cC

cE orcG

cI or cK

cM or cO


cA cE cI cM

Foldingspeciesprofile

cIzI1 zI2 zI3 zI4

PCA

1st score

1 st loading

1st scoregives thecommonshape

Loadings give therelative amounts!

Substitution of species profile cA

’ = zI1cI

cE’ = zI2cI

cI’ = = zI3cI

cM’ = = zI4cI

Refoldingspeciesprofileusing

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

0.9905,0.9990,0.9928,0.99700.9989,0.9999,0.9696,0.9895

+ MA-MCR-ALS non-tril- theoretical

+ MA-MCR-ALS non-tril- theoretical

Example 3: A hybrid bilinear-trilineal modelMCR-ALS trilinearity constraint was not applied to any component

lof % R2

a) Theoretical 1.34 0.9998 (added noise)b) MA-MCR-ALS-non tril 1.35 0.9998

The fit is good but spectral shapes of 3rd and 4th notrotation ambiguity is still present!

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

0.9715,0.9426,0.9540,0.84440.9872,0.9990,0.5199,0.9584



Example 3: A hybrid bilinear-trilineal modelMCR-ALS trilinearity constraint is applied to all components

lof % R2

a) Theoretical 1.34 0.9998 (added noise)b) MA-MCR-ALS-tril 12.8 0.9936

The fit is not good and the all spectral shapes are wrong. This is the worse case!!

Assuming trilinearity for non-trilinear data is not adequate!!

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Example 3: A hyubrid bilinear-trilineal modelMCR-ALS trilinearity constraint is applied to 1st and 3rd

componentslof % R2

a) Theoretical 1.34 0.9998 (added noise)b) MA-MCR-ALS-mixt 1.36 0.9998

These are the best results obtained with the hybrid bilinear-trilineal model

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

0.9999,0.9999,0.9999,0.99980.9999,0.9999,0.9988,0.9999

+ MA-MCR-ALS partial tril- theoretical

+ MA-MCR-ALS partial tril- theoretical

Outline

• Introduction




• Conclusions

As in fish and Cd, Co and Pb in water

were not scaled; only downweigthed

As Ba Cd Co Cu Cr Fe Mn Ni Pb Zn-1

0

1

2

3

4

5mean of scaled concentrations of 11 metals

water

sedimentsfish

metals (variables)

METAL CONTAMINATION PATTERNS IN SEDIMENTS, FISH AND WATERS FROM CATALONIA RIVERS USING MULTIWAY DATA ANALYSIS METHODSEmma Peré-Trepat (UB), Antoni Ginebreda (ACA), Romà Tauler (CSIC)

17 rivers, 11 metals (As, Ba, Cd, Co, Cu, Cr, Fe,

Mn, Ni, Pb, Zn), 3 environmental

conpartments: Fish (barb’, ‘bagra comuna’,

bleak, carp and trout), Sediment and

Water samples

As Ba Cd Co Cu Cr Fe Mn Ni Pb Zn

1 2 3 4 5 6 7 8 9 10 110

2

4

Unit variance scaled concentrations boxplot

Va

lues

1 2 3 4 5 6 7 8 9 10 110

2

4

Va

lues

1 2 3 4 5 6 7 8 9 10 110

2

4

6

Va

lues

Fish

Sediment

Water

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

2

4

6

8

10

12

sample sites

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

2

4

6

8

10

12

sample sites

F S W0

0.4

0.8

compartments

F S W-0.8

-0.4

0

0.4

0.8

compartments

As Ba Cd Co Cu Cr Fe Mn Ni Pb Zn0

0.1

0.2

0.3

0.4

0.5

metals

As Ba Cd Co Cu Cr Fe Mn Ni Pb Zn-0.5

0

0.5

metals

%R2 (3-WAY)

1rst Component

2nd Component Total

64.7 11.7 76.4

67.3 13.2 80.5

MA-PCA + refolding MA-PCA

MA-PCA of scaled data and scores refolding

Little differences in samples mode!!! negative loadings

for water soluble metal ions


0.2

0.4

0.6

0.8

metals


0.2

0.4

0.6

0.8

metals1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0

5

10

15

sample sites

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

5

10

15

sample sites

F S W0

0.5

1

compartments

F S W0

0.5

1

compartments

MA-MCR-ALS of scaled data with nn constraint and scores refolding

47.0 40.7 76.9

%R2 (3-WAY)

1rst Component

2nd Component Total

MA-MCR-ALS + refoldingMA-MCR-ALS

48.2 42.8 80.5

D

contaminants

compartments

site

s

F

S

W

F

S

W

contaminants

site

ssi

tes

site

s

Xaug

YT

1

2

3

MCR-ALS

TRILINEARITY CONSTRAINT(ALS iteration step)


1

2

3

Folding

every augmentedscored wnated tofollow the trilinearmodel is refolded

MA-MCR-ALSTrilinear model constraint

PCA

Substitution ofspecies profile

Rebuilding augmented scores

1’

2’

3’



X YT

contaminants

Z

site

s


This constraintis applied at each stepof the ALS optimization

and independently for each component

individually


0.2

0.4

0.6

metals


0.2

0.4

0.6

metals

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

5

10

15

sample sites

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

5

10

15

sample sites

F S W0

0.5

1

compartments

F S W0

0.5

1

compartments

MA-MCR-ALS of scaled data with nn, with trilinearity model constraint and with scores refolding

%R2 (3-WAY)

1rst Component

2nd Component

Total

45.3 42.2 76.8

47.0 40.7 76.9

MA-MCR-ALS nn + trilinearMA-MCR-ALS nn + refolding


0.2

0.4

0.6

0.8

metals


0.2

0.4

0.6

0.8

metals

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

5

10

15

samples sites

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

5

10

15

sample sites

F S W0

0.5

1

compartments

F S W0

0.5

1

compartments

PARAFAC of scaled data

%R2 (3-WAY)

1rst Component

2nd Component Total

43.4 36.2 77.4

44.3 42.9 76.8

PARAFACMA-MCR-ALS nn + trilinear

D

=

Xaug

Y

contaminants

compartments

site

s

F

S

W

F

S

W

metals

site

ssi

tes

site

s

1

2

3

4

5

6

MCR-ALS

Folding

1 2 3 4 5 6

component interaction constraint

(ALS iteration step)

interacting augmented scores are folded

together

1’

2’

3’

4’

5’

6’

=



MA-MCR-ALScomponent interaction

constraint

PCA =

This constraint is applied at each step of the ALS optimizationand independently and individually for each component i

XY

Z



0.2

0.4

0.6

metals


0.2

0.4

0.6

metals1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0

5

10

15

sample sitesF S W

0

0.5

1

compartments

F S W0

0.5

1

compartments

MA-MCR-ALS of scaled data with nn, component interaction and scores refolding

%R2 (3-WAY)

1rst Component

2nd Component Total

45.2 41.4 75.8

45.3 42.2 76.8

MA-MCR-ALS nn + interactionMA-MCR-ALS nn

model [1 2 2]model [2 2 2]

Tucker Models with non-negativityconstraints

0 5 10 15 20 25 3064

66

68

70

72

74

76

78

80

82

84

[1 2 2] [1 2 3]

[1 3 3]

[2 2 2] [2 2 3]

[2 3 3] [3 3 3]

Explained variances (%) for each

TUCKER3 mstudied odel studied.

TUCKER3 model

Sum of Squares (%)

[1,1,1] 64.7

[1,1,2] 64.7

[1,1,3] 64.7

[1,2,1] 64.7

[1,2,2] 76.1

[1,2,3] 76.1

[1,3,1] 64.7

[1,3,2] 76.1

[1,3,3] 80.3

[2,1,1] 64.7

[2,1,2] 66.3

[2,1,3] 66.3

[2,2,1] 66.9

[2,2,2] 77.3

[2,2,3] 78.1

[2,3,1] 66.9

[2,3,2] 78.4

[2,3,3] 82.4

[3,1,1] 64.7

[3,1,2] 66.3

[3,1,3] 67.3

[3,2,1] 66.9

[3,2,2] 77.9

[3,2,3] 79.3

[3,3,1] 68.4

[3,3,2] 79.8

[3,3,3] 83.6

[3 2 3]

parsimonious model[1 2 2]

0 5 10 150

0.2

0.4

1 2 3 4 5 6 7 8 9 10110

0.5

1

1 2 30

0.5

1

1 2 3 4 5 6 7 8 9 10110

0.5

1

1 2 30

0.5

1

Tucker3-ALS of scaled data

%R2 (3-WAY)

1rst Component

2nd Component Total

50.7 35.3 76.1

43.4 36.2 77.4

TUCKER3PARAFAC

model [1 2 2]model [2 2 2]

CHEMOMETRIC METHOD

%R2 (3-WAY) %R2 (2-WAY)

1rst Component

2nd Component

Total1rst

Component2nd

ComponentTotal

MA-PCA 64.7 11.7 76.4 67.3 13.2 80.5

PARAFAC (non-negativity) 43.4 36.2 77.4 - - -

TUCKER3 (non-negativity) 50.7 35.3 76.1 - - -

MA-MCR-ALS (non-negativity) 47.0 40.7 76.9 48.2 42.8 80.5

MA-MCR-ALS (non-negativity and triliniarity) 44.3 42.9 76.8 - - -MA-MCR-ALS (non-negativity and component

interaction constraints)45.2 41.4 75.8 - - -

Summary of Results

Outline

• Introduction



• Example of application: MCR-ALS with Tuker3 constraint

• Conclusions

Conclusions

•It is possible to implement trilinearity constraints in MCRusing ALS algorithms in a flexible, adaptable, simple and fast way and it may be applied to only some of the components.

•Intermediate situations between pure bilinear and pure trilinear hybrid models can be easily implemented using MA-MCR-ALS

•Different number of components and interactions between components in different modes can be also easily implemented in hybrid MA-MCR models

•For an optimal RESOLUTION, the model should be in accordance with the 'true' data structure

Deviations from trilinearity Mild Medium Strong Array size

PARAFAC

Small PARAFAC2

Medium TUCKER

Large MCR

Guidelines for method selection(resolution purposes)

Journal of Chemometrics, 2001, 15, 749-771

Related works:

R.Tauler, A.Smilde and B.R.Kowalski. Journal of Chemometrics, 1995, 9, 31-58 (MCR-ALS extension to multiway)

R.Tauler, I.Marqués and E.Casassas. Journal of Chemometrics, 1998; 12, 55-75 (implementation of trilinearity constraint in MA-MCR-ALS)

A. de Juan and R.Tauler, Journal of Chemometrics, 2001, 15, 749-771 (comparison of MA-MCRE-ALS with PARAFAC and Tucker3)

E.Peré-Trepat, A.Ginebreda and R.Tauler, Chemometrics and Intelligent Laboratory Systems, 2006, (new implementation of the component interaction constraint in MA-MCR-ALS)

Acknowledgements

• Ana de Juan (comparison of MCR-ALS with other multiway data analysis methods)

• Emma Peré-Trepat (application of the component interaction constraint to environmental data)

• Research Grant MCYT, Spain, BQU2003-00191 • Water Catalan Agency (supply of environmental

data set)

including trilinear and restricted tucker3 models as a constraint in multivariate curve resolution...

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