Support Vector Machines for Classification and Regression

Abstract

The problem of empirical data modelling is germane to many engineering applications.
\nIn empirical data modelling a process of induction is used to build up a model of the
\nsystem, from which it is hoped to deduce responses of the system that have yet to be observed.
\nUltimately the quantity and quality of the observations govern the performance
\nof this empirical model. By its observational nature data obtained is finite and sampled;
\ntypically this sampling is non-uniform and due to the high dimensional nature of the
\nproblem the data will form only a sparse distribution in the input space. Consequently
\nthe problem is nearly always ill posed (Poggio et al., 1985) in the sense of Hadamard
\n(Hadamard, 1923). Traditional neural network approaches have suffered difficulties with
\ngeneralisation, producing models that can overfit the data. This is a consequence of the
\noptimisation algorithms used for parameter selection and the statistical measures used
\nto select the ’best’ model. The foundations of Support Vector Machines (SVM) have
\nbeen developed by Vapnik (1995) and are gaining popularity due to many attractive
\nfeatures, and promising empirical performance. The formulation embodies the Structural
\nRisk Minimisation (SRM) principle, which has been shown to be superior, (Gunn
\net al., 1997), to traditional Empirical Risk Minimisation (ERM) principle, employed by
\nconventional neural networks. SRM minimises an upper bound on the expected risk,
\nas opposed to ERM that minimises the error on the training data. It is this difference
\nwhich equips SVM with a greater ability to generalise, which is the goal in statistical
\nlearning. SVMs were developed to solve the classification problem, but recently they
\nhave been extended to the domain of regression problems (Vapnik et al., 1997). In the
\nliterature the terminology for SVMs can be slightly confusing. The term SVM is typically
\nused to describe classification with support vector methods and support vector
\nregression is used to describe regression with support vector methods. In this report
\nthe term SVM will refer to both classification and regression methods, and the terms
\nSupport Vector Classification (SVC) and Support Vector Regression (SVR) will be used
\nfor specification. This section continues with a brief introduction to the structural risk