The Kernel Method of Density Estimation With Applications in Discrimination, Selection of Features, and Conditional and Marginal Distribution

The thesis may be thought of in three sections. The first is mainly historical and considers the continuous kernel function. The second develops the binary kernel function and extends the method to deal with various possible types of data set. The third section is concerned with various possible applications and extensions to less standard areas.;The first section introduces the continuous kernel function and discusses possible choices for the function before settling on one based on the normal distribution function. Several general problems in discrimination, parameter estimation and misclassification rate estimation are described and the approach used in the thesis is explained. An example is given of the performance of the continuous kernel function compared with the predictive normal distribution.;The binary kernel function is then introduced. Variations of it and its relationship to nearest neighbour methods and continuous kernels are discussed. Its performance in fitting probabilities is compared with loglinear models of differing orders in a simulation study. It appears that the less structure there is in the data the better the performance of the kernel Extensions of the kernel function to deal with unordered and ordered categorical data and also to mixtures of data are considered. It is shown that it is not possible to construct a kernel function to satisfy all the intuitively desirable criteria of discrete probability estimation. A kernel function to estimate probabilities in general data sets is constructed.;The applications section of the thesis first considers the performance of the multivariate binary kernel when used for discrimination. It is compared with several more standard methods on two suitable data sets. Atypicality indices are constructed for one of the data sets and there is a discussion on parameter values. The mixed data kernel is used on simulated data consisting of binary and continuous parts. Its performance in discrimination is compared with a location model in which the continuous distribution is chosen to be conditional on the binary distribution. The kernel method does reasonably well in comparison.;The second application is in the use of feature selection. A multi parameter binary kernel with parameters calculated from the marginal tables is used and a method is proposed based on ideas of information gain. A cross-validatory approach is used which, applied over the whole data set, is non-sequential. This approach is not practical for categorical data and a simpler, sequential method of minimising the misclassification rates is used.;Marginal and conditional distributions are discussed. It is shown that for the conditional probabilities to sum to one over the sample space it is necessary for computational reasons to use an approximation. An application to categorical data is given. In?a chapter of miscellaneous items at the end more applications and several possible extensions are given. If the data are continuous but non-normal it may be better to use a different kernel function. It is possible to connect the potential function method of density estimation to the kernel function method and this suggests a possible development for sequential analysis.