On data depth and distribution-free discriminant analysis using separating surfaces

Abstract

A very well-known traditional approach in discriminant analysis is to use some linear (or nonlinear) combination of measurement variables which can enhance class separability. For instance, a linear (or a quadratic) classifier finds the linear projection (or the quadratic function) of the measurement variables that will maximize the separation between the classes. These techniques are very useful in obtaining good lower-dimensional views of class separability. Fisher's discriminant analysis, which is primarily motivated by the multivariate normal distribution, uses the first-and second-order moments of the training sample to build such classifiers. These estimates, however, are highly sensitive to outliers, and they are not reliable for heavy-tailed distributions. This paper investigates two distribution-free methods for linear classification, which are based on the notions of statistical depth functions. One of these classifiers is closely related to Tukey's half-space depth, while the other is based on the concept of regression depth. Both these methods can be generalized for constructing nonlinear surfaces to discriminate among competing classes. These depth-based methods assume some finite-dimensional parametric form of the discriminating surface and use the distributional geometry of the data cloud to build the classifier. We use a few simulated and real data sets to examine the performance of these discriminant analysis tools and study their asymptotic properties under appropriate regularity conditions.