Linear Discrimination and Some Further Results on Best Lower Dimensional Representations

Abstract

This chapter discusses low dimensional hyperplanes, which, in some sense, best represent the populations. However, the Fisher method is not unique in providing representations. When discriminating among k p-variate normal distributions with common covariance matrix, it is well known that the optimal discrimination procedure is based on the score functions. However, for many practical reasons, it proves useful to have one-dimensional two-dimensional or three-dimensional representations of the data. Plotting the transformed observations in these lower dimensional spaces can lead to a better understanding of the relationships between populations and the detection of outliers. Such a representation is especially helpful when p is large compared to k or when the means almost lie in a low dimensional hyperplane. The p-dimensional representations, including Fisher's between-within method, can be interpreted as based on orthogonal transformations of the standardized principal components. Each representation provides a decomposition of the scores as a sum of squares.