In this work we present support vector machines, their mathematical derivation and practical usage. The primal problem, which we encounter while trying to separate two-class data with hyperplane, is described. We generalise given primal problem to nonlinear separable data. The theory of convex optimization is introduced which helps us to prove strong duality in the convex case. We convert the primal problem to its dual. Kernel functions are introduced into the dual problem. We prove that the polynomial and radial kernels are scalar products in some space. The problem of multiclass data is described. Methods such as cross-validation are introduced for error estimation. Different kernels are demonstrated on a simple two dimensional data set, which is a part of the IRIS data set. Support vector machines are trained on a bigger data set and an indication of a possible way of choosing final model is shown.
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