Support vector machines for classification and regression

The problem of empirical data modelling is germane to many engineering applications. In empirical data modelling a process of induction is used to build up a model of the system, from which it is hoped to deduce responses of the system that have yet to be observed. Ultimately the quantity and quality of the observations govern the performance of this empirical model. By its observational nature data obtained is finite and sampled; typically this sampling is non-uniform and due to the high dimensional nature of the problem the data will form only a sparse distribution in the input space. Consequently the problem is nearly always ill posed [16] in the sense of Hadamard [9]. Traditional neural network approaches have suffered difficulties with generalisation, producing models that can overfit the data. This is a consequence of the optimisation algorithms used for parameter selection and the statistical measures used to select the ‘best’ model. The foundations of Support Vector Machines (SVM) have been developed by Vapnik [21] and are gaining popularity due to many attractive features, and promising empirical performance. The formulation embodies the Structural Risk Minimisation (SRM) principle, which has been shown to be superior, [8], to traditional Empirical Risk Minimisation (ERM) principle, employed by conventional neural networks. SRM minimises an upper bound on the VC dimension ('generalisation error'), as opposed to ERM that minimises the error on the training data. It is this difference which equips SVM with a greater ability to generalise, which is the goal in statistical learning. SVM were developed to solve the classification problem, but recently they have been extended to the domain of regression problems [20]. In the literature the terminology for SVM can be slightly confusing. The term SVM is typically used to describe classification with support vector methods and support vector regression is used to describe regression with support vector methods. In this report the term SVM will refer to both classification and regression methods, and the terms Support Vector Classification (SVC) and Support Vector Regression (SVR) will be used for specification. This section continues with an introduction to the structural risk minimisation principle. In section 2 the SVM is introduced in the setting of classification, being both historical and more accessible. This leads onto mapping the input into a higher dimensional feature space by a suitable choice of kernel function. The report then considers the problem of regression. Illustrative examples are given to show the properties of the techniques.