Variable selection for kernel methods with application to binary classification
Thesis (PhD (Statistics and Actuarial Science))—University of Stellenbosch, 2008.
The problem of variable selection in binary kernel classification is addressed in this thesis. Kernel methods are fairly recent additions to the statistical toolbox, having originated approximately two decades ago in machine learning and artificial intelligence. These methods are growing in popularity and are already frequently applied in regression and classification problems. Variable selection is an important step in many statistical applications. Thereby a better understanding of the problem being investigated is achieved, and subsequent analyses of the data frequently yield more accurate results if irrelevant variables have been eliminated. It is therefore obviously important to investigate aspects of variable selection for kernel methods. Chapter 2 of the thesis is an introduction to the main part presented in Chapters 3 to 6. In Chapter 2 some general background material on kernel methods is firstly provided, along with an introduction to variable selection. Empirical evidence is presented substantiating the claim that variable selection is a worthwhile enterprise in kernel classification problems. Several aspects which complicate variable selection in kernel methods are discussed. An important property of kernel methods is that the original data are effectively transformed before a classification algorithm is applied to it. The space in which the original data reside is called input space, while the transformed data occupy part of a feature space. In Chapter 3 we investigate whether variable selection should be performed in input space or rather in feature space. A new approach to selection, so-called feature-toinput space selection, is also proposed. This approach has the attractive property of combining information generated in feature space with easy interpretation in input space. An empirical study reveals that effective variable selection requires utilisation of at least some information from feature space. Having confirmed in Chapter 3 that variable selection should preferably be done in feature space, the focus in Chapter 4 is on two classes of selecion criteria operating in feature space: criteria which are independent of the specific kernel classification algorithm and criteria which depend on this algorithm. In this regard we concentrate on two kernel classifiers, viz. support vector machines and kernel Fisher discriminant analysis, both of which are described in some detail in Chapter 4. The chapter closes with a simulation study showing that two of the algorithm-independent criteria are very competitive with the more sophisticated algorithm-dependent ones. In Chapter 5 we incorporate a specific strategy for searching through the space of variable subsets into our investigation. Evidence in the literature strongly suggests that backward elimination is preferable to forward selection in this regard, and we therefore focus on recursive feature elimination. Zero- and first-order forms of the new selection criteria proposed earlier in the thesis are presented for use in recursive feature elimination and their properties are investigated in a numerical study. It is found that some of the simpler zeroorder criteria perform better than the more complicated first-order ones. Up to the end of Chapter 5 it is assumed that the number of variables to select is known. We do away with this restriction in Chapter 6 and propose a simple criterion which uses the data to identify this number when a support vector machine is used. The proposed criterion is investigated in a simulation study and compared to cross-validation, which can also be used for this purpose. We find that the proposed criterion performs well. The thesis concludes in Chapter 7 with a summary and several discussions for further research.