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Sensors 2021, 21, 6319 5 of 16 • binary classes. Each binary SVM finds a hyperplane that separates between every two classes, neglecting the data points of the other classes. For example, consider m = 4, say, with classes 1, 2, 3, and 4. In this case, we have six different SVMs applied to the binary classes {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, and {3, 4}. One-to-rest method: Here, the classifier can use m binary SVMs by obtaining an optimal hyperplane that separates between a class and all others at once. For example, consider m = 3, say, with classes 1, 2, and 3. In this case, we have three different SVMs applied to the binary classes {1, not 1}, {2, not 2}, and {3, not 3}. The data points classification depends on the type of the kernel function to be em- ployed, which aims to reduce the computational time and effort by transforming a non- linear decision surface to a linear equation in a higher number of dimension spaces. For any two vectors u and v, the most popular kernels, utilized with SVM algorithms, are linear, polynomial, radial basis, and sigmoid functions defined as: • Linear function: K(u, v) = u⊤v, (1) where K is the kernel function and ⊤ denotes the transpose of a vector. • Polynomial function: K(u, v) = (γu⊤v + 1)d, (2) where γ > 0 and d > 0 are the parameters of scale and degree, respectively. • Radial basis function: K(u, v) = exp(−γ||u − v||2), (3) where γ > 0 is the inverse influence radius of points selected as support vectors. • Sigmoid function: K(u, v) = tanh(γu⊤v + 1), (4) where γ > 0 is a tuning meta-parameter that defines how far the influence of a training point can reach. Let (x1, y1), . . . , (xn, yn) be the training data set, where each pair belongs to one of two classes that we name A or B, according to 1, if xk ∈ class A; yk = −1, ifxk ∈classB. Each xk , with k = 1, . . . , n, is a p-dimensional vector of real numbers. From the training data set, the maximum margin hyperplane algorithm is used to estimate the parameters of the decision function D(x). Hence, the classification of unknown patterns is predicted based on the following criteria: (i) x ∈ class A if D(x) > 0; or (ii) x ∈ class B, otherwise. In the direct space, D(x) has the form D(x) = w⊤φ(x) + b, where φ(x) is a predefined function, which separates the hyperplane in n-dimensional spaces, whereas w = (w1, . . . , wn)⊤ and b are vectors of weights and noise coefficients, respectively, with both w and b being adjustable parameters computed along with minimizing the empirical risk. The distance between the pattern x and the separated hyperplane is D(x)/||w|| (see Figure 1). Note that the axes in this figure (left) are labeled as x1 and x2. Nonetheless, the figure on the right-hand side is the same as that on the left-hand side but as an image (photography). The maximum margin hyperplane algorithm aims to find the vector w that maximizes the margin M between the training set and class boundary, which is formulated as M⋆ = max M (5) w,||w||=1 subjectto ykD(x) ≥ M, k=1,...,p,PDF Image | Predicting Cryptocurrency Returns Based on the Gold Prices
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