DESCRIPTION OF TUCKER METHOD

Historic:
Tucker's model has been found by L.R. Tucker in 1966. Articles with applications appear in the 80th/90th years. It is called in the litterature "Tucker", "Tucker3", "N-way PCA", etc...

Aim:
Tucker is an exploratory data analysis model . It is an extension of PCA for N-Way data, keeping the N-linear structure of the data, and so providing simultaneously summaries of main variations for the N modes of the data. More precisely, the aim is to find subspaces for each mode where projection of modalities are rendered faithfully.

It can be used as a multiplicative model in analysis of variance too.

A particular case of Tucker model is "Tucker 2" , where only some modes are reduced.

Criterion:

- Find subspaces Fi which satisfy:

- for N=3 (3-way data sets), and with the usual unfolded notation, that is:

- Ond set of the column-wise orthonormal matrix of size nxd.
- Tdef set of the tensors of size dxexf.
- A,B, C are called the loadings and G the core matrix.

the choice of the basis of Fi (i.e. the set of the columns of A,B,C) can be chosen with different manners: it is possible to rotate the core in order to minimise non-zero values in it.

Algorithm:
It is an iterative algorithm called Alternating Least Square in order to find N optimal subspaces. The idea is to fix N-1 loadings and to calculate the other by projection, alternatively.

Code:
Here the code employed is the one of the n-way toolbox 2.01 of Rasmus Bro and Claus Andersson: http://www.models.kvl.dk/source/nwaytoolbox/

It is freely available.

Applications:
Chromatography, environmental analysis, pharmacology research, etc....

Dataset reference:
predclim.mat (X or Y)

References:

Tucker L.R., some mathematical notes on three-mode factor analysis, Psychometrika, 31,(1966), 279.

Kroonenberg P.M., Three-mode principal component analysis. theory and applications, DSWO Press, Leiden, 1983.

Henrion R., N-way principal Component Analysis,. Theory algorithms and applications, Chemometrics Intell. Lab. Syst, 25, (1994) 1.

Franc A., Etude algébrique des multitableaux: apports de l'analyse tensorielle, PhD thesis, 1992, Université de Montpellier.

Andersson C.A., Bro R., Improving the speed of multiway algorithms, part 1: tucker3, Chemometrics Intell. Lab. Syst, Vol 42, Issues 1-2, 1998, 93-103

Andersson C.A., Bro R., The N-way Toolbox for Matlab, Chemometrics Intell. Lab. Syst., Vol 52, 2000, 1-4

Estienne F., Matthijs N., Massart D.L., Ricoux P., Leibovici D., Multivariate modelling of high dimentionnality electroencephalographic data, Chemometrics Intell. Lab. Syst., Vol 58, Issue 1, 2001, 59-72