Generalized Singular Value Decomposition And Discriminant Analysis

Abstract

A fundamental and widely used approach to feature selection is the method of linear discriminant analysis. The standard formulation of the analysis requires knowledge of the total and pooled within-class scatter matrices. Estimates of these matrices are made from training sets. In spectral pattern recognition, one is usually faced with large numbers of highly correlated measurements and only a small number of representative spectra. The training sets are typically inadequate, and the resultant estimates of pooled within-class scatter matrices are ill conditioned or singular. This precludes the use of the standard formulation of discriminant analysis. By formulating the discriminant analysis in terms of generalized singular value decompositions of total and pooled within-class deviation matrices, the method can be applied in situations where the scatter matrices are ill conditioned or singular. The approach is illustrated with an example two-class problem of distinguishing between Fourier transform infrared (FTIR) spectra of aliphatic hydrocarbons and alcohols.