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

The fuzzy relations R icorresponding to the individual rule can now be computed by using Eq. (4.32). For rule , we have R 1= Low × Low ; for rule , we obtain R 2= Moderate × High ; and finally for rule R 3= 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, .

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)

Since the max and min operations are taken over different domains, their order can be changed as follows:

(4.38)

Denote as the degree of fulfillment of the i‐ th rule’s antecedent. The output fuzzy set of the linguistic model is thus

(4.39)

The entire algorithm, called the max ‐ min or Mamdani inference , is summarized in Algorithm 4.1and visualized in Figure 4.5.

Figure 4.5 A schematic representation of the Mamdani inference algorithm.

1. Compute the degree of fulfillment by . Note that for a singleton fuzzy set ( otherwise) the equation for βi simplifies to . 2. Derive the output fuzzy sets . 3. Aggregate the output fuzzy sets .

Let us take the input fuzzy set A ′ = [1,0.6,0.3,0] from the previous example and compute the corresponding output fuzzy set by the Mamdani inference method. Step 1 yields the following degrees of fulfillment:

In step 2, the individual consequent fuzzy sets are computed: