Artificial Intelligence and Quantum Computing for Advanced Wireless Networks. Savo G. Glisic

the input domain into a lattice of fuzzy hyper‐boxes, parallel with the axes. Each of the hyper‐boxes is a Cartesian product‐space intersection of the corresponding univariate fuzzy sets. The number of rules in the conjunctive form needed to cover the entire domain is given by where p is the dimension of the input space, and N_i is the number of linguistic terms of the i‐th antecedent variable.

      By combining conjunctions, disjunctions‚ and negations, various partitions of the antecedent space can be obtained; the boundaries are, however, restricted to the rectangular grid defined by the fuzzy sets of the individual variables. As an example, consider the rule “If x₁ is not A₁₃ and x₂ is A₂₁ then …”

      The degree of fulfillment of this rule is computed using the complement and intersection operators:

      (4.42)

      The antecedent form with multivariate membership functions, Eq. (4.31), is the most general one, as there is no restriction on the shape of the fuzzy regions. The boundaries between these regions can be arbitrarily curved and opaque to the axes. Also, the number of fuzzy sets needed to cover the antecedent space may be much smaller than in the previous cases. Hence, for complex multivariable systems, this partition may provide the most effective representation.

      Defuzzification: In many applications, a crisp output y is desired. To obtain a crisp value, the output fuzzy set must be defuzzified. With the Mamdani inference scheme, the center of gravity (COG) defuzzification method is used. This method computes the y coordinate of the COG of the area under the fuzzy set B′:

(4.43)

      where F is the number of elements y_j in Y. The continuous domain Y thus must be discretized to be able to compute the COG.

      Design Example 4.5

Consider the output fuzzy set B′ = [0.2, 0.2, 0.3, 0.9, 1] from the previous example, where the output domain is Y = [0, 25, 50, 75, 100]. The defuzzified output obtained by applying Eq. (4.43) is

      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