Classification Using Kernel Density Estimates: Multiscale Analysis and Visualization

The use of kernel density estimates in discriminant analysis is quite well known among scientists and engineers interested in statistical pattern recognition. Using a kernel density estimate involves properly selecting the scale of smoothing, namely the bandwidth parameter. The bandwidth that is optimum for the mean integrated square error of a class density estimator may not always be good for discriminant analysis, where the main emphasis is on the minimization of misclassification rates. On the other hand, cross-validation-based methods for bandwidth selection, which try to minimize estimated misclassification rates, may require huge computation when there are several competing populations. Besides, such methods usually allow only one bandwidth for each population density estimate, whereas in a classification problem, the optimum bandwidth for a class density estimate may vary significantly, depending on its competing class densities and their prior probabilities. Therefore, in a multiclass problem, it would be more meaningful to have different bandwidths for a class density when it is compared with different competing class densities. Moreover, good choice of bandwidths should also depend on the specific observation to be classified. Consequently, instead of concentrating on a single optimum bandwidth for each population density estimate, it is more useful in practice to look at the results for different scales of smoothing for the kernel density estimates. This article presents such a multiscale approach along with a graphical device leading to a more informative discriminant analysis than the usual approach based on a single optimum scale of smoothing for each class density estimate. When there are more than two competing classes, this method splits the problem into a number of two-class problems, which allows the flexibility of using different bandwidths for different pairs of competing classes and at the same time reduces the computational burden that one faces for usual cross-validation-based bandwidth selection in the presence of several competing populations. We present some benchmark examples to illustrate the usefulness of the proposed methodology.