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characteristic of the SVM optimization process that helps in determining the steps in the methodology. The SVM decision function for classifying points with respect to the hyperplane only involves dot products between points. Furthermore, the algorithm that finds a separating hyperplane in the feature space can be stated entirely in terms of vectors in the input space and dot products in the feature space. We are transforming training samples from one space into the other. But we are making computation only with scalar products of points in this new space. This product is computationally inexpensive because only a small subset of points is SVs involved in product computation. Thus, an SVM can locate a separating hyperplane in the feature space and classify points in that space without ever representing the space explicitly, simply by defining a function, called a kernel function. Kernel function K always plays the role of the dot product in the feature space:

This approach avoids the computational burden of explicitly representing all transformed source data and high-dimensional feature vectors. The two most widely used kernel functions are Polynomial Kernel

and Gaussian Kernel

The polynomial kernel is valid for all positive integers d ≥ 1. The Gaussian kernel is one of a group of kernel functions known as radial basis functions (RBFs). RBFs are kernel functions that depend only on the geometric distance between x and y, and the kernel is valid for all nonzero values of the kernel width σ. It is probably the most useful and commonly applied kernel function. The concept of a kernel mapping function is very powerful, such as in the example given in Figure 4.25. It allows SVM models to perform separations even with very complex boundaries. The relation between a kernel function and a feature space can analyze for a simplified version of quadratic kernel k(x, y) = < x, y >2 where x, y ∈ R2:

defining a 3-D feature space . Similar analysis may be performed for other kernel function. For example, through the similar process verify that for the “full” quadratic kernel (<x, y > +1)2 the feature space is 6-D.

Figure 4.25. An example of a mapping Φ to a feature space in which the data become linearly separable. (a) One-dimensional input space; (b) two-dimensional feature space.

In practical use of SVM, only the kernel function k (and not transformation function Φ) is specified. The selection of an appropriate kernel function is important, since the kernel function defines the feature space in which the training set examples will be classified. As long as the kernel function is legitimate, an SVM will operate correctly even if the designer does not know exactly what features of the training data are being used in the kernel-induced feature space. The definition of a legitimate kernel function is given by Mercer’s theorem: The function must be continuous and positive-definite.

Modified and enhanced SVM constructs an optimal separating hyperplane in the higher dimensional space. In this case, the optimization problem becomes

where K(x,y) is the kernel function performing the nonlinear mapping into the feature space, and the constraints are unchanged. Using kernel function we will perform minimization of dual Lagrangian in the feature space, and determine all margin parameter, without representing points in this new space. Consequently, everything that has been derived concerning the linear case is also applicable for a nonlinear case by using a suitable kernel K instead of the dot product.

The approach with kernel functions gives a modular SVM methodology. One module is always the same: Linear Learning Module. It will find margin for linear separation of samples. If the problem of classification is more complex, requiring nonlinear separation, then we include a new preparatory module. This module is based on kernel function, and it transforms input space into higher, feature space where the same Linear Learning Module may be applied, and the final solution is nonlinear classification model. Illustrative example is given in Figure 4.26. This combination of different kernel functions with standard SVM learning algorithm for linear separation gives the flexibility to the SVM methodology for efficient application in nonlinear cases.

Figure 4.26. SVM performs nonlinear classification by kernel-based transformations. (a) 2-D input space; (b) 3-D feature space; (c) 2-D input space.

The idea of using a hyperplane to separate the feature vectors into two groups works well when there are only two target categories, but how does SVM handle the case where the target variable has more than two categories? Several approaches have been suggested, but two are the most popular: (a) “one against many” where each category is split out and all of the other categories are merged; and (b) “one against one” where k (k − 1)/2 models are constructed and k is the number of categories.

A preparation process for SVM applications is enormously important for the final results, and it includes preprocessing of raw data and setting model parameters. SVM requires that each data sample is represented as a vector of real numbers. If there are categorical attributes, we first have to convert them into numeric data. Multi-attribute coding is recommended in this case. For example, a three-category attribute such as red, green, and blue can be represented with three separate attributes and corresponding codes such as (0,0,1), (0,1,0), and (1,0,0). This approach is appropriate only if the number of values in an attribute is not too large. Second, scaling values of all numerical attributes before applying SVM is very important in successful application of the technology. The main advantage is to avoid attributes with greater numeric ranges to dominate those in smaller ranges. Normalization for each attribute may be applied to the range [−1; +1] or [0; 1].

Selection of parameters for SVM is very important and the quality of results depends on these parameters. Two most important parameters are cost C and parameter γ for Gaussian kernel. It is not known beforehand which C and σ are the best for one problem; consequently, some kind of