LITMIR.BIZ

      The network throughput (in arbitrary units), computed by the fuzzy model, is thus 72.12.

      4.4.2 SVR

      The basic idea: Let {(x₁, y₁), . . . , (x_l, y_l)} ⊂ X × ℝ, be a given training data, where X denotes the space of the input patterns (e.g., X = ℝ^d). In ε‐SV regression, the objective is to find a function f(x) that has at most a deviation of ε from the actually obtained targets y_i for all the training data, and at the same time is as flat as possible. In other words, we do not care about errors as long as they are less than ε, but will not accept any deviation larger than this. We begin by describing the case of linear functions, f, taking the form

(4.44)

where 〈 , 〉 denotes the dot product in X. Flatness in the case of Eq. (4.44) means that one seeks a small w. One way to ensure this is to minimize the norm, that is, ‖w‖² = 〈w, w〉. We can write this problem as a convex optimization problem:

(4.45)

The tacit assumption in Eq. (4.45) was that such a function f actually exists that approximates all pairs (x_i, y_i) with ε precision, or in other words, that the convex optimization problem is feasible. Sometimes, the problem might not have a solution for the given ε, or we also may want to allow for some errors. In analogy with the “soft margin” (see Figure 2.7 of Chapter 2), one can introduce slack variables ξ_i, to cope with possibly infeasible constraints of the optimization problem: Eq. (4.45). So we have

(4.46)

      The constant C > 0 determines the trade‐off between the flatness of f and the amount up to which deviations larger than ε are tolerated. This corresponds to dealing with a so‐called ε‐insensitive loss function |ξ|_ε described by