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

High 0.0 0.0 0.3 0.9 1.0

The fuzzy relations R_i corresponding to the individual rule can now be computed by using Eq. (4.32). For rule , we have R₁ = Low × Low; for rule , we obtain R₂ = Moderate × High; and finally for rule R₃ = High × Low. The fuzzy relation R, which represents the entire rule base, is the union (element‐wise maximum) of the relations R_i :

      Similarly

      (4.35)

where from Eq. (4.33) , that is, .

      Design Example 4.3

Now consider an input fuzzy set to the model, A′ = [1, 0.6, 0.3, 0], which can be denoted as SomewhatLow traffic volume, as it is close to Low but does not equal Low. The result of max‐min composition defined by Eq. (4.34) gives

      Similarly, by using the same procedure for the input set A′ = [0, 0.2, 1, 0.2] we obtain B ′ = max (A′ ∧ R) = [0.2, 0.2, 0.3, 0.9, 1].

      Max‐min (Mamdani) inference: In the previous section, we have seen that a rule base can be represented as a fuzzy relation. The output of a rule‐based fuzzy model is then computed by the max‐min relational composition. In this section, it will be shown that the relational calculus can be bypassed. This is advantageous, as the discretization of domains and storing of the relation R can be avoided. To show this, suppose an input fuzzy value , for which the output value B′ is given by the relational composition:

      (4.36)

      After substituting for μ_R(x, y) from Eq. (4.33), the following expression is obtained:

      (4.37)