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