Nikos Sidiropoulos - IRISA© 2001 Chemometrics Group, Rasmus Bro, [email protected] A B C Three-way data? Simply a set of two-way matrices Each mode consist of same basic entities over the

© 2001 Chemometrics Group, Rasmus Bro, [email protected]

ThreeThreeThree---way analysisway analysisway analysis

New multi-way models and algorithms for solving blind source separation problems

Rasmus BroChemometrics Group, Food TechnologyRoyal Veterinary & Agricultural University (KVL)[email protected]

Nikos SidiropoulosDept. of Electrical & Computer Engr.University of [email protected]


IntroductionModels & algorithmsApplicationsProblems

ContentContentContent

© 2001 Chemometrics Group, Rasmus Bro, [email protected] B

C

Three-way data?Simply a set of two-way matricesEach mode consist of same basic entities over the other modesE.g. same samples measured at different variables, several timesInstead of a matrix with typical elements xij, we have an array with elements xijk

Where?Sensory analysis, Process analysis, Image analysis, Experimental design, Spectroscopy, Chromatography, Environmental analysis, QSAR, Communication, Medicine, …

IntroductionIntroductionIntroduction

© 2001 Chemometrics Group, Rasmus Bro, [email protected] B

C

! Sensory analysis! Score as a function of (Food sample, Judge, Attribute)

! Process analysis! Measurement as a function of (Batch, Variable, time)! Measurement as a function of (Variable, Lag, Location)

! Image analysis! Pixelvalue as a function of (Sample, Image pixel, Variable)

! Experimental design! Response as a function of (factor 1, factor2, factor3,..)

! Spectroscopy! Intensity as a function of (Wavelength, Retention, Sample, Time, Location ,

Treatment)! Environmental analysis

! Measurement as a function of (Location, Time, Variable)

Examples from chemistryExamples from chemistryExamples from chemistry


Unfolding/Unfolding/Unfolding/matricizationmatricizationmatricization

! Traditional approach! Unfolding leading to two-way data and

analysis

! Three-way models! Natural extensions of two-way models! PCA leads to PARAFAC or Tucker3

depending on how it is extended! Rank-reduced regression, e.g. PLS leads

to multilinear PLS (N-PLS)

SampleWavelength

pH

pH Sample

Sample

Sample

Wavelength

Wavelength

Wavelength pH

pH

Unfolding/matricizationOften leads to overfitting because nature of model !nature of the data

Unfold: Wold, Geladi, Esbensen, Öhman. Principal component- and PLS-analyses generalized to multi-way (multi-order) data arrays. Copenhagen Symposium on Applied Statistics:249-277, 1986.


PARallel FACtor PARallel FACtor PARallel FACtor analysisanalysisanalysis

PCA - bilinear model,

xij = Σftifpjf + eij, i=1,..,I; j=1,..,J

PARAFAC - trilinear model,

xijk = Σfaifbjfckf + eijk , i=1,..,I; j=1,..,J; k=1,..,K

X E= + +

c2

b2

a2

c1

b1

a1

=E

+A

C

B


PARallel FACtor PARallel FACtor PARallel FACtor analysisanalysisanalysis

PCA - bilinear model,X = AB’ + E

PARAFAC - trilinear model,Xk = ADkB’ + Ek , k = 1,…,KDk = diag(C(k,:))

X E= + +

c2

b2

a2

c1

b1

a1

=E

+A

C

B


Definition of threeDefinition of threeDefinition of three---way rankway rankway rank

Rank of two-way matrixMinimum number of bilinear (PCA) components needed to reproduce matrix

Rank of three-way arrayMinimum number of trilinear (PARAFAC) components needed to reproduce array

X= +

c2

b2

a2

c1

b1

a1


Definition of threeDefinition of threeDefinition of three---way rankway rankway rank

Two-wayAny random matrix has full rank with probability oneRow-rank = column-rankSimple rules for maximal rank

Three-wayRandom 2×2×2 is rank 2 (30%) and rank 3 (70%)!Row-rank ≠ column-rank ≠ tube-rank!No rules for maximal rank!

Except simple cases such as 3×3×3 is five.

X= +

c2

b2

a2

c1

b1

a1


PARAFAC PARAFAC PARAFAC --- uniquenessuniquenessuniqueness

Emission

Excitation

S. 1 S. 2

Sample

Emission

Excitation

Three-w

ay Two-way

250 300 350 400 450240

260280

3000

500

Emission Wavelength/nm

Excitation Wavelength/nm

Inte

nsity

250 300 350 400 450240

260280

3000

200

400



Inte

nsity

PARAFAC has a unique property: It is unique!!

Practical example: FluorescenceTwo samplesMixture of three analytes

(Trp, Tyr, Phe)

PARAFAC invented in 1970 by Harshman and independently by Carroll & Chang under the name CANDECOMP. Based on a principle of parallel proportional profiles suggested in 1944 by Cattell

•R. A. Harshman. UCLA working papers in phonetics 16:1-84, 1970.•J. D. Carroll and J. Chang. Psychometrika 35:283-319, 1970.•R. B. Cattell. Psychometrika 9:267-283, 1944.



PCA orthogonal loadings PARAFAC pure spectra!

250 300 350 400 450 500-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Emission wavelength

PCA

load

ing

250 300 350 400 450 500-0.05

0

0.05

0.1

0.15

0.2

0.25

Emission wavelength

PAR

AFAC

load

ing/

Emis

sion

inte

nsity


PARAFAC for CDMA PARAFAC for CDMA PARAFAC for CDMA



Uniqueness - conditionsA PARAFAC model is unique when

kA + kB + kC " 2F + 2

F is the number of components and kA is the k-rank of loading A = maximal number of randomly chosen columns which will have full rank (#F)

J. B. Kruskal. Linear Algebra and its Applications 18:95-138, 1977.SidGiaBro, IEEE TSP, Mar 2000, N = 3, !SidBro. JChemom 14 2000, any N, "

XB

A

C

=≥ + −∑ 1

2 ( 1)Nnn

k F N



Uniqueness - what does it mean?Mixtures of analytes can be separatedConcentrations can be estimatedPure spectra and profiles can be estimated

8 × 8 × 8 array is unique for a 10 component model!!Hence with 8 snapshots, 8 symbols and 8 codes, recovery of 10 sources is possible

ExamplesOn-line purity control (Kodak) Windig, J. Magnetic Resonance 132:298-306, 1998

Determination of kinetic parameters Bijlsma, AIChE Journal 44:2713-2723, 1998

Mathematical chromatography Leurgans, Statistical Science 7:289-319, 1992

Localizing cellular phone calls Sidiropoulos & Bro. PARAFAC Techniques for Signal separation. In: Signal Processing Advances in Communications, edited by P. Stoica, G. B. Giannakis, Y. Hua, and L. Tong,Prentice-Hall, 2000,








PARAFAC PARAFAC PARAFAC --- algorithmalgorithmalgorithm

Alternating least squares algorithm

* Hadamard (elementwise product)

( ) ( ){ }

( ) ( ){ }

( ) ( ){ } ( )

1

k k1

1

k k1

1

k k

1. Initialize and

2. ' * '

2. ' ' * '

3. ' * ' diag ' , =1,..,

4. Step 2 until relative change in fit is small

K

k

K

k

diag k K

−

=

−

=

−

= =

=

∑

∑

B C

A X BD B B C C

B X AD A A C C

D B B A A A X B


PARAFAC PARAFAC PARAFAC --- algorithmalgorithmalgorithm

Speed-upCompression of data (LS or Spline-based bases)Line-search etc.

Alternative algorithmsGRAM/DTLD – direct generalized eigenvalue problem but not LSPMF3 – Gauss-Newton: fast but problematic for large arraysCOMFAC – Uses compression and separation + Gauss-Newton


Tucker3 – an extension of SVD

PCA formulated as truncated SVD – X = USV’ + E

Problem: Express X in terms of new unitary bases U and V

Solution: Regress X on U and V: S = U+XV’+ = U’XV

Hence: S equals (approximates) X in a new (truncated) coordinate system

Curiosity: S is diagonal, but that’s not mandatory

Tucker3/SVDTucker3/SVDTucker3/SVD


PCA = SVD can be written (note the off-diagonals)

=XS BT

A= [ ] [ ] T

1 2 1 21 00 2

a a b b

T T T T T T1 1 2 1 1 2 2 2 1 1 2 20 0 2 2= + + + == +a b a b a b a b a b a b

Tucker3/SVDTucker3/SVDTucker3/SVD


The Tucker3 modelThe Tucker3 modelThe Tucker3 model

For three-way data, three unitary bases, A, B, and C; one for each mode

Tucker3 is X = AG(C⊗⊗⊗⊗ B)’ + E

Loadings are truncated bases and G the representation of X in these reduced spaces

=X

G

C

B

A•L. R. Tucker. The extension of factor analysis to three-dimensional matrices. In: Contributions to Mathematical Psychology, edited by N. Frederiksen and H. Gulliksen, New York:Holt, Rinehart & Winston, 1964, p. 110-182.

•L. R. Tucker. Some mathematical notes on three-mode factor analysis.Psychometrika 31:279-311, 1966


Tucker3Tucker3Tucker3

! Important theorem! If the matricized ranks in the three

modes are found to be P, Q, and Rrespectively

! then a (P,Q,R) Tucker3 model fit the data perfect

! Practical value! If the pseudo-ranks are definitely found

to be P, Q, and R respectively! then a (P,Q,R) Tucker3 model fit the

data appropriately

pH Sample

Sample

Sample

Wavelength

Wavelength

Wavelength pH

pH

Unfold: Wold, Geladi, Esbensen, Öhman. Principal component- and PLS-analyses generalized to multi-way (multi-order) data arrays. Copenhagen Symposium on Applied Statistics:249-277, 1986.

PCA rank P

PCA rank Q

PCA rank R


Tucker3 Tucker3 Tucker3 vs vs vs PARAFACPARAFACPARAFAC

Differences from PARAFAC/PCA:The number of components can vary in A, B, and C!G is not superdiagonalTucker loadings not unique (only subspace) = rotational freedom Tucker loadings orthogonal => variance-partitioningPARAFAC best rank F model, but does not describe the part of data within that subspace!Tucker best subspace F model but not rank F !!

=X

G

C

B

A


Tucker3 Tucker3 Tucker3 --- algorithmalgorithmalgorithm

1. Initialize B and C (e.g. from SVD)2. A equals first P left singular vectors of X(I×JK)(C⊗ B)3. B equals first Q left singular vectors of X(J×IK)(C⊗ A)4. C equals first R left singular vectors of X(K×IJ)(B⊗ A)5. Go to step 2 until relative changes are small6. G = A’X(C⊗ B)

=X

G

C

B

A


Other Tucker modelsOther Tucker modelsOther Tucker models

Tucker3 has the number 3 because three modes are ’reduced’.Tucker2 and Tucker1 reduces two and one modes respectively

=X

B

A

G

=X

C

B

A

G

=X

A

G

Tucker3

Tucker1

Tucker2

Tucker2 core often called Extended

Core Array

Tucker1 is identical to PCA on

matricizes X


Multilinear PLS regressionMultilinear PLS regressionMultilinear PLS regression

N-PLS

Use a trilinear (PARAFAC-like) model of X but such that the scores are predictive of y.

X E= + +

c2

b2

t2

c1

b1

t1

T y

b

Bro. Multiway calibration. Multi-linear PLS. Journal of Chemometrics 10:47-61, 1996.



Two-way PLS principleFind w (||w||=1) and q (||q||=1) such that the one-component models of X = tw' + EX and Y = uq' + EY have maximal covariance (t'u)

Make regression model to predict u from t

Subtract the model from X and predictions from Y

Proceed with the next component from residuals


Two-way PLS: how to do it - one y.

= = = =

=

⇒

=

max cov( , )

max '

max '

max '

''

w

w

w

w

t y t Xw

t y t Xw

y Xw

z w

X ywX y


Max covariance

LS model



Three-way PLS: how to do it - one y.

= =

= = =

= = =

= =

= = = =

=

= ⇒

∑∑

∑ ∑∑

∑∑∑

∑∑

J K

J K

J K

J K

J K

J Ki ijk j k

1 1

J Ki i i ijk j k

1 1 1

J Ki ijk j k

1 1 1

J Kjk j k

1 1

J K

,

max cov( , )

max

max

max

max ( ) '

SVD on

J K

j k

I J K

i j k

I J K

i j k

J K

j k

t x w w

t y t x w w

y x w w

z w w

w w

w w

w w

w w

w w

t y

w Zw

Z


Example sensory dataExample sensory dataExample sensory data

Three-way, two-way: Does it make a difference?

5 breads (in replicates) × 11 attributes × 8 judgesData due to Magni Martens

10 breads

11 a

ttrib

utes

Judge 2 Judge 8Judge 1

11 a

ttrib

utes

11 a

ttrib

utes

X10 breads

8 Judges

11 attributes

_


Sensory dataSensory dataSensory data

PCA

Score one

Scor

e tw

o

-10 0 10 20-10

-8

-6

-4

-2

0

2

4

6

8

10

78

3

12

4

5

6

910

PARAFAC

Score one

Scor

e tw

o

-10 -5 0 5 10 15-10

-8

-6

-4

-2

0

2

4

6

8

10

34

56

78

12910

Scores from bilinear PCA and trilinear PARAFAC

Similar but note that replicates are closer for PARAFAC

Thre

e-w

ay m

ore

robu

st

beca

use

of ‘s

tron

ger’

stru

ctur

al m

odel


Loadings from bilinear PCA and trilinear PARAFAC


Off-flav

Tough Salt-t

Sweet-t

Other-t

Bread-od

Yeast-od

Colour

Moisture

Yest-t

Total

PARAFAC - Attributes PARAFAC - Judges

-0.5Loading one

0 0.5

-0.6

-0.2

0.2

0.6Lo

adin

g tw

o

0.2 0.3 0.4 0.5 0.6 0.7

0.15

0.25

0.35

0.45

0.55

Loading one

Load

ing

two

6

1

2

3

45

78

65

801014

19

21

25

30

36 4354

6374

76

851

8

11

13

22415264

67

69

77

495073

83

2

345

67

9

1215

16

17

18

20

2324

2627

28

29

31

32

33

34

35

37

38

3940

42

44

45

46

47

48

51

53

55

56

57585960

6162

66

68

7071

72

75

78

79

8182

84

86

88

PCA - unfolded

-0.2 -0.1 0 0.1 0.2

Loading one

-0.3

-0.1

0.1

0.3

Load

ing

two

PARAFAC 19 loading-elements per componentPCA 88 loading-elements per component!



Calibration - predict salt content

25% improvement!!

LV Variation explained /% RMSE

X cal. X val. Y cal. Y val. Y cal. Y val.

1 43 25 80 62 0.21 0.29

2 61 38 95 76 0.10 0.23

3 74 49 100 84 0.03 0.19

4 78 49 100 84 0.01 0.19

5 84 50 100 84 0.00 0.19

6 87 52 100 84 0.00 0.19

1 31 22 75 60 0.23 0.30

2 46 36 93 82 0.12 0.20

3 54 44 98 91 0.07 0.15

4 57 46 100 91 0.03 0.14

5 60 47 100 90 0.02 0.15

6 61 47 100 90 0.00 0.15

unfo

ld-P

LSTr

iline

ar P

LS

Three-way more predictive because less overfit


Second order calibrationSecond order calibrationSecond order calibration

Two-way classical least squares mixture model X = CS’, E.g.X the measured spectra, C the true concentrations, S the pure spectra

If pure spectra, S, known, C determined by a simple least squares regression step – C = X(S’)+

Thus, if all spectra known quantitation possible and no calibration samples are needed


How to do itHow to do itHow to do it

In three-way PARAFAC the model is Xk = ADkB’, Xk the measured spectra at occasion kA the true concentrationsB the pure spectraC (rows from Dk) pure spectra/profiles etc.

Pure spectra not necessary. They are found by decomposing with PARAFAC

Thus, without prior knowledge quantitation possible and no calibration samples are needed except for fixing the scale


Fluorescence exampleFluorescence exampleFluorescence example

250350

450260

3000

500

1000

Standard (Trp)

250350

450260

300-500

0

500

Unknown 1 (Tyr,Trp,Phe)

250350

450260

3000

200

400



Inte

nsity

Unknown 2 (Tyr,Trp,Phe)

Calibration set: One sample with oneanalyte (2.67 µM Trp)

Test set: Two samples with three analytes each (Trp, Tyr, Phe)

Three-component PARAFAC model fitted to the fluorescence data (3×201×61):

A - estimated concentrationsB - estimated emission spectraC - estimated excitation spectra

X= + +

c2

b2

a2

c1

b1

a1

c3

b3

a3


Fluorescence exampleFluorescence exampleFluorescence example

250 300 350 400 4500

1

2

3

Emis

sion

B

240 260 280 3000

1

2 C

wavelength/nm

Exci

tatio

n

2.67 0 0 2.671.52 1.30 1.29 , 1.58.86 1.12 1.06 .88

= =

A reference

Trp Tyr Phe

Spectral loadings resemble pure spectra Sample scores - estimated concentration


PredictionPredictionPrediction

How to make a prediction modelUsing sub-space regression, e.g. N-PLSUsing ‘second-order calibration’ mostly with PARAFAC

Significant differencePredictions based on ordinary regression models only work if all interferents are varying independently in the calibration data ⇒

Many samples neededOnly similar samples can be predicted

PARAFAC models work with only one calibration sample and with unknown uncalibrated interferents


Sugar processingSugar processingSugar processing

Sugar made from beets

End product sampled every 8’th hour during the three months of operationMeasured spectrofluorometrically260 samples

Thick juice

Massecuite

Standardliqour

Sugarboiling

Centrifuge

Thin juiceA little

SyrupWash syrup

Wash water

Wash juice

Water

Sugar

X7

X8

X1

X2

X3

X4

X5

X6


Sugar data modelSugar data modelSugar data model

Raw data: 268 samples from a sugar campaign - fluorescence landscapes

Concentrations

Raw data

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

160

180 Scores

Time/days

Comp. 2

Tryptophane

Tyrosine

Comp. 1

300 400 5000

1

Com

p. 1

240 260 280 300 320 3400

1

300 400 5000

1

Tyro

sine

240 260 280 300 320 3400

1

300 400 5000

1

Tryp

toph

ane

240 260 280 300 320 3400

1

300 400 5000

1

Com

p. 4

Emission /nm240 260 280 300 320 3400

1

Excitation /nm

Deconvoluted spectra

= + + +

PARAFAC model


0 33 66 99Days

Predicting CaO

0 10 20 30 40 50 60 70 80 9015

20

25

30

35

Days

PredictedReference

Predicting color

PARAFAC and fluorescence: unique combination of multivariate process control and process analytical chemistryThe process can be monitored and controlled on a chemical level

Using the modelUsing the modelUsing the model


Resolving twoResolving twoResolving two---way exponentialsway exponentialsway exponentials

Time (ms)0 100 200 300 400 500 600 700 800 900 1000

I(t) = M0 ∙ exp(-t/T2)

Two-way data but with very special structure in loadings

X = AB’ = 1 0 27 9 3 10 1 16 8 4 2

2143891627

Sample 2Sample 1


X1 = AD1B’ =9 31 0 3 18 4 20 1

00 2

43

89

1627

Sample 2Sample 1

SLAB 1 (X1)

21

43

89

Sample 2Sample 1

SLAB 2 (X2)

X2 = AD2B’ = 9 31 0 1 18 4 20 1

00 1

SLICING SLICING SLICING --- Pseudo ThreePseudo ThreePseudo Three---way Dataway Dataway Data


Example: Sensory qualityExample: Sensory qualityExample: Sensory quality

! Predicting cooked potato quality from NMR on raw data! Potatoes cooked and served hot. Ten assessors

evaluate texture profile, e.g. mealiness

! Raw (!) potato measured by NMR (CPMG pulse sequence)

0 200 400 600 800 1000 1200 14000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000Raw data

???

Inte

nsity

A. K. Thybo et. al., Food Science and Technology, 33 (2):103-111, 2000.

=

LaggingPAR

AFAC


Example: Sensory qualityExample: Sensory qualityExample: Sensory quality

0 50 100 150 200 250 300 350 400 450-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Profile s PAR AFAC decomposition

???

a.u.

2 3 4 5 6 7

2

3

4

5

6

7

Predictions mealy

Reference

Pred

icte

d

r=0.858 RMSEP=0.79

Decomposing NMR into meaningful latent variablesPARAFAC on lagged data enables meaningful decompositionFour components adequate for describing NMR data

Predictions of mealiness from NMRPredicting sensory quality of cooked potatoes from amount of latent variables (i.e. directly from NMR of raw potatoes)


Using constraintsUsing constraintsUsing constraints

Using constraints in multi-way modeling

ExampleInstead of ‘PCA’: || X - AB’||fit the model: || X - AB’||,

subject to A and B are nonnegative

Constraints are essential in two-way curve resolution because the model is unidentified

In three-way curve resolution the model is often unique but constraints are still useful


Why constraints?Why constraints?Why constraints?

Obtain sensible parametersEx.: Require chromatographic profiles to have but one peak

Obtain unique solutionEx.: Use selective channels in data to obtain uniqueness

Test hypothesesEx.: Investigate if tryptophane is present in sample

Avoiding degeneracy and numerical problemsEx.: Enabling a PARAFAC model of data otherwise inappropriate for the model

Speed up algorithmsEx.: Use truncated bases to reexpress problem by a smaller problem

Enable quantitative analysis of qualitative dataEx.: Incorporate sex and job type in a model for predicting income


Typical constraintsTypical constraintsTypical constraints

Spectroscopy Chromatography FIA Kinetics

Auto- & cross correlation !! !! !! !!

Uncertainty !! !! !! !!Nonnegativity !! !! !! !!Unimodality ((!!)) !! !!Selectivity !! !!Smoothness !! !! !! !!Known spectra !!…


FIA exampleFIA exampleFIA example

Model of a single sample with one analyte

0

50

100

250300

350400

4500

0.1

0.2

0.3

TimeWavelengthAbsorbance

250 300 350 400 4500

0.05

0.1

0.15

Wavelength/nm

0 20 40 60 80 1000

0.02

0.04

0.06

Time

Absorbance

Absorbance

Samples of 2, 3, 4-HBA (hydroxy benzaldehyde) detected by UV-VIS in FIA system with pH-gradient imposedEvery spectrum is a sum of acidic and basic spectrum. Same holds for time profiles. Only sums are measuredModel not important here. PARALIND : Xk = AHDkB’

DataConcentrations

Pure spectra


Effect of constraintsEffect of constraintsEffect of constraints

Eq : Equality of summed profilesNNLS : Non-negativity of all parametersULSR : Unimodality of FIAgrams/time profilesFix : Fixing purely acidic/basic times to only reflect acidic/basic analytes

SPECTRA 2HBAacidic

2HBAbasic

3HBAacidic

3HBAbasic

4HBAacidic

4HBAbasic

Eq 0.9893* 0.9871* 0.9689* 0.7647* 0.9106* 0.9211*

NNLS 0.9944 0.9117* 0.9952 0.9241 0.9974 0.9977

NNLS/Eq 0.9946 0.9312* 0.9953 0.9988 0.9965 0.9971

NNLS/ULSR/Eq 0.9946 0.9590* 0.9953 0.9989 0.9966 0.9943

NNLS/ULSR/Fix/Eq 0.9946 0.9989 0.9954 0.9986 0.9961 0.9977

0 20 40 60 80 100

0

0.2

Time

4HBA -a.u.

Non-negativity & equality constrained

Correlations


Problems to deal withProblems to deal withProblems to deal with

AlgorithmsTucker, N-PLS are unproblematic, even for large data setsPARAFAC problematic

Multiple local minima 2-factor degeneracy (model-problem): no solution?Solution sensitive to correct dimensionality (not sequential)Slow convergence

Other problems currently consideredStatistical measures (rank/DoF problems)Diagnostics for choosing the number of componentsHandling highly structured error covariancesHandling huge amounts of missing data


Concluding remarksConcluding remarksConcluding remarks

Uniqueness (mainly PARAFAC)Pure spectraSecond order advantage

No interferents, only one standard

Better structural modelRobustness/noise reductionSimpler modelInterpretationBetter predictions


Concluding remarksConcluding remarksConcluding remarks

ReferencesHarshman, Lundy. Comp. Stat. Data Anal., 1994, 18, 39Leurgans, Ross. Statist. Sci., 1992, 7, 289Smilde. Chemom. Intell. Lab. Syst., 1992, 5, 143Bro. J. Chemom. 1996, 10, 47Bro, Chemom.Intell.Lab.Syst., 1997, 38, 149Sidiropoulos & Bro. PARAFAC Techniques for Signal separation. In: Signal Processing Advances in Communications, edited by P. Stoica, G. B. Giannakis, Y. Hua, and L. Tong, Prentice-Hall, 2000.

Software & infohttp://www.models.kvl.dk (Free matlab code’n’course, database of papers)http://www.ece.umn.edu/users/nikos/public_html/3SPICE/3SPICEmain.htmlTRIPLE SPICE – Multi-way analysis in signal processing – matlab, presentations etc.http://www.eigenvector.com (Matlab code)http://www.fsw.leidenuniv.nl/~kroonenb (Stand alone)

Nikos Sidiropoulos - IRISA© 2001 Chemometrics Group, Rasmus Bro, [email protected] A B C Three-way data? Simply a set of two-way matrices Each mode consist of same basic entities over the

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