Data Mining - Mehmed Kantardzic [240]
where each term in T(age) is characterized by a fuzzy set of a universe of discourse X = [0, 100]. The syntactic rule refers to the way the linguistic values in the term set T(age) are generated, and the semantic rule defines the MF of each linguistic value of the term set T(age), such as the linguistic values in Figure 14.8.
14.3 EXTENSION PRINCIPLE AND FUZZY RELATIONS
As in the set theory, we can define several generic relations between two fuzzy sets, such as equality and inclusion. We say that two fuzzy sets, A and B, defined in the same space X are equal if and only if (iff) their MFs are identical:
Analogously, we shall define the notion of containment, which plays a central role in both ordinary and fuzzy sets. This definition of containment is, of course, a natural extension of the case for ordinary sets. Fuzzy set A is contained in fuzzy set B (or, equivalently, A is a subset of B) if and only if μA(x) ≤ μB(x) for all x. In symbols,
Figure 14.9 illustrates the concept of A ⊆ B.
Figure 14.9. The concept of A ⊆ B where A and B are fuzzy sets.
When the fuzzy sets A and B are defined in a finite universe X, and the requirement that for each x in X, μA(x) ≤ μB(x) is relaxed, we may define the degree of subsethood DS as
DS(A,B) provides a normalized measure of the degree to which the inequality μA(x) ≤ μB(x) is violated.
Now we have enough background to explain one of the most important concepts in formalization of a fuzzy-reasoning process. The extension principle is a basic transformation of the fuzzy-set theory that provides a general procedure for extending the crisp domains of mathematical expressions to fuzzy domains. This procedure generalizes a common point-to-point mapping of a function f between fuzzy sets. The extension principle plays a fundamental role in translating set-based concepts into their fuzzy counterparts. Essentially, the extension principle is used to transform fuzzy sets via functions. Let X and Y be two sets, and F is a mapping from X to Y:
Let A be a fuzzy set in X. The extension principle states that the image of A under this mapping is a fuzzy set B = f(A) in Y such that for each y ∈ Y:
The basic idea is illustrated in Figure 14.10. The extension principle easily generalizes to functions of many variables as follows. Let Xi, i = 1, … , n, and Y be universes of discourse, and X = X1 × X2 × … × Xn constitute the Cartesian product of the Xis. Consider fuzzy sets Ai in Xi, i = 1, … , n and a mapping y = f(x), where the input is an n-dimensional vector x = (x1, x2, … , xn) and x ∈ X. Fuzzy sets A1, A2, … , An are then transformed via f, producing the fuzzy set B = f(A1, A2, … , An) in Y such that for each y ∈ Y:
Figure 14.10. The idea of the extension principle.
subject to x ∈ X and y = f(x). Actually, in the expression above, the min operator is just a choice within a family of operators called triangular norms.
More specifically, suppose that f is a function from X to Y where X and Y are discrete universes of discourse, and A is a fuzzy set on X defined as
then the extension principle states that the image of fuzzy set A under the mapping f can be expressed as a fuzzy set B:
where yi = f(xi), i = 1, … , n. In other words, the fuzzy set B can be defined through the mapped values xi using the function f.
Let us analyze the extension principle using one example. Suppose that X = {1, 2, 3, 4} and Y = {1, 2, 3, 4, 5, 6} are two universes of discourse, and the function for transformation is y = x + 2. For a given fuzzy set A = 0.1/1 + 0.2/2 + 0.7/3 + 1.0/4 in X, it is necessary to find a corresponding fuzzy set B(y) in Y using the extension principle through function B = f(A). In this case, the process of computation is straightforward and a final, transformed fuzzy set is B = 0.1/3 + 0.2/4 + 0.7/5 + 1.0/6.
Another problem will show that the computational process is not always a one-step process. Suppose that A is given