3.1.5. Support Vector Machine

The SVM aims to determine a hyperplane in an N-dimensional space, where N is the number of features, which distinctly classifies the data points [40]. The SVM is a generalized version of the maximal margin classifier, with provision for more data types to be better classified. Essentially, the SVM uses the hyperplane to separate optimally two convex hulls of points (data instances), by ensuring that the hyperplane is equidistant from the two convex hulls.

In this situation, the hyperplane is a classification border. It is usually a *N* − 1 dimensional flat affine subspace, which is a line in two dimensions and a plane in three dimensions [41]. In terms of classification, the goal is to find the plane that optimizes the distance between two classes of data points. The hyperplane's size depends on the number

of features. In the SVM domain, support vectors are data points on or near the hyperplane. Using these support vectors allows us to optimize the hyperplane's margin. They are called support vectors because any change in their position affects the hyperplane.

For non-linearly separated data points, the SVM adopts the concept of kernels to classify such points. A kernel refers to a function that maps lower dimensional data into higher dimensional ones. The function takes two vectors of any dimension as input and outputs a score (a dot product) that quantifies how similar the input vectors are. If the dot product is small, then the vectors are different, whereas if they are large, the vectors are similar. The SVM can use a variety of kernel functions, but one popular kernel is the Gaussian radial basis function (RBF). The RBF Gaussian kernel *K*(*x*, *y*) is calculated as follows:

$$K(x, y) = \exp\left(\frac{||x^2 - y^2||^2}{2\sigma^2}\right) \tag{6}$$

where *x* and *y* are N-dimensional independent variables, and *σ* is assumed to be the standard deviation of *x* and *y*, with || • || being the Euclidean norm. Further formal and detailed explanation and use of the SVM can be obtained in [42,43], with the following hyperparameters fine tuned in the present study, namely, the kernel and filter type.
