Discriminant Analysis for Multi-way Data

International audienceIn standard multivariate data analysis, individuals × variables data table is usually considered (two-way datatable). However, from a practical view point this simple data structure appears to be somehow limitated. It isthe case for instance when individuals are charaterized by the temperature at different locations sampled overdifferent times, leading to a three-way data structure. Such multi-way structure can be viewed as a stack ofmatrices X = Xi jk1≤i≤I, 1≤j≤J, 1≤k≤K from which the I horizontal slices describe the individuals i = 1, ..., I, theJ lateral slices describe the variables (temperature) j = 1, ...,J and the K frontal slices describe the different timepoints k = 1, ...,K. Many two-way data analysis methods have been extended to the multi-way configuration. Forinstance, a multi-way formulation of Partial Least Squares Regression (N-PLS) has been proposed in [1]. N-PLSrelies on the maximization of a covariance criterion but explicitely takes into acount the multi-way structure ofthe input data. In this paper, we present a Multi-way formulation of Fisher Discriminant Analysis (MFDA) in anattempt to improve the interpretability of the resulting model compared with the results obtained with unfoldedmethods. MFDA is illustrated on a real multi-modalMagnetic Resonance Brain Imaging (MRI) datase