Discriminant Analysis

Abstract

In the setting of discriminant analysis it is assumed that the so-called training data belong to certain groups. The goal is to find classification rules that allow to assign new test data to one of the groups. Different discriminant methods have been introduced, such as linear discriminant analysis (LDA), with Fisher’s method as a specific approach, and quadratic discriminant analysis (QDA). Both LDA and QDA utilize the information on prior class probabilities and heavily use the assumption of normality in ilr coordinates (normal distribution on the simplex) to represent group distributions. While for QDA individual group covariance matrices are assumed, a joint covariance matrix is computed for the case of LDA. These methods result in classification rules that allow to assign a new test set observation to one of the groups by taking the prior information on class pertinence into account. The Fisher discriminant rule aims for the same goal, but now no underlying distributions of the samples in the groups are assumed and the idea is to search for projection directions which allow for a maximum separation of the group means with respect to the spread of the projected data. As a consequence, also discriminant scores can be derived that are used to visualize relevant information for the group separation. All described procedures are invariant with respect to the choice of the orthonormal coordinates, and this also holds for the robust counterparts of the covariance-based methods if an affine equivariant location and covariance estimator (like the MCD estimator) is taken.