Regularized Gaussian Discriminant Analysis through Eigenvalue Decomposition

Friedman has proposed a regularization technique RDA of discriminant anal ysis in the Gaussian framework RDA makes use of two regularization parameters to design an intermediate classi cation rule between linear and quadratic discriminant analysis In this paper we propose an alternative approach to design classi cation rules which have also a median position between linear and quadratic discriminant analysis Our approach is based on the reparametrization of the covariance matrix k of a group Gk in terms of its eigenvalue decomposition k kDkAkD k where k speci es the volume of Gk Ak its shape and Dk its orientation Variations on constraints concerning k Ak and Dk lead to discrimination models of interest For each model we derived the maximum likelihood parameter estimates and our ap proach consists in selecting the model among the possible models by minimizing the sample based estimate of future misclassi cation risk by cross validation Nu merical experiments show favorable behavior of this approach as compared to RDA