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Sensors 2021, 21, 6319 6 of 16 where the bound M⋆ for supporting patterns attains to satisfy the condition M⋆ = mink ykD(xk).Thedecisionfunction,D(x)namely,withmaximummarginM⋆,isshown in Figure 1. Therefore, the hyperplane with maximum margin can be obtained by solving the minimax problem established as max min ykD(xk). Usually, the product of the norm w,||w||=1 k,b of w and the margin M is scaled to be fixed, that is, M||w|| = 1, to avoid the problem of getting an infinite number of possible solutions. Therefore, maximizing M as in (5) is equivalent to minimizing ||w||. Thus, finding the hyperplane with maximum margin (M⋆ = 1/||w⋆||) reduces to solve the quadratic form given by min ||w||2, (6) w,b subjectto ykD(xk) ≥ 1, k=1,...,p. Plot of the maximum margin linear decision function D(x) = w⊤ x + b (φ(x) = x) (left) and an image of it Figure 1. (right). The gray levels encode |D(x)|, solid black corresponds to D(x) = 0, and the numbers indicate the supporting vectors (Source: [34]). Often, we may tolerate some cost errors ξ ≥ 0 by allowing for some observations to violate the margin. The cost C is defined by the number of observations beyond the margin. In this case, instead of minimizing the form stated in (6), we need to minimize where ξk ≥ 0 and ∑p ξk ≤ C. However, although the problem stated in (7) can be k=1 solved by numerical methods in direct space, these methods are impractical for high dimensionality of the dual space φ. In this dual space, the decision function is defined as p D(x)= ∑αkK(xk,x)+b, (8) k=1 where 0 ≤ αk ≤ c are adjustable parameters computed along with minimizing the empirical risk, C is the cost given by the number of observations outside the margin, and K is a predefined kernel function. The values of the kernel function stated in (8) may be calculated bytheinnerproductofφ(xi)andφ(xj),thatis,K(xi,xj) =φ(xi)◦φ(xj),wheretheoperator ◦ denotes the inner product of two vectors. We aforementioned the most popular kernels in the expressions defined in (1) to (4). The problem stated in (7) can be transformed into a dual space using a Lagrangian algorithm [31] by means of the representation formulated as 1p L(w, b, α) = 2 ||w||2 − ∑ αk (yk D(xk ) − 1), (9) 1p min ||w||2 + C ∑ ξk, (7) w,b 2 subjectto ykD(xk) ≥ 1−ξk, k=1,...,p, k=1 k=1PDF Image | Predicting Cryptocurrency Returns Based on the Gold Prices
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