Data Mining - Mehmed Kantardzic [239]
1. Hamming distance for p = 1,
2. Euclidean distance for p = 2, and
3. Tchebyshev distance for p = ∞.
For example, the distance between given fuzzy sets A and B, based on Euclidean measure, is
For continuous universes of discourse, summation is replaced by integration. The more similar the two fuzzy sets, the lower the distance function between them. Sometimes, it is more convenient to normalize the distance function and denote it dn(A,B), and use this version to express similarity as a straight complement, 1 − dn (A, B).
The other approach to comparing fuzzy sets is the use of possibility and necessity measures. The possibility measure of fuzzy set A with respect to fuzzy set B, denoted by Pos(A, B), is defined as
The necessity measure of A with respect to B, Nec(A, B) is defined as
For the given fuzzy sets A and B, these alternative measures for fuzzy-set comparison are
An interesting interpretation arises from these measures. The possibility measure quantifies the extent to which A and B overlap. By virtue of the definition introduced, the measure is symmetric. On the other hand, the necessity measure describes the degree to which B is included in A. As seen from the definition, the measure is asymmetrical. A visualization of these two measures is given in Figure 14.6.
Figure 14.6. Comparison of fuzzy sets representing linguistic terms A = high speed and B = speed around 80 km/h.
A number of simple yet useful operations may be performed on fuzzy sets. These are one-argument mappings, because they apply to a single MF.
1. Normalization: This operation converts a subnormal, nonempty fuzzy set into a normalized version by dividing the original MF by the height of A
2. Concentration: When fuzzy sets are concentrated, their MFs take on relatively smaller values. That is, the MF becomes more concentrated around points with higher membership grades as, for instance, being raised to power two:
3. Dilation: Dilation has the opposite effect from concentration and is produced by modifying the MF through exponential transformation, where the exponent is less than 1
The basic effects of the previous three operations are illustrated in Figure 14.7.
Figure 14.7. Simple unary fuzzy operations. (a) Normalization; (b) concentration; (c) dilation.
In practice, when the universe of discourse X is a continuous space (the real axis R or its subset), we usually partition X into several fuzzy sets whose MFs cover X in a more-or-less uniform manner. These fuzzy sets, which usually carry names that conform to adjectives appearing in our daily linguistic usage, such as “large,” “medium,” or “small,” are called linguistic values or linguistic labels. Thus, the universe of discourse X is often called the linguistic variable. Let us give some simple examples.
Suppose that X = “age.” Then we can define fuzzy sets “young,” “middle aged,” and “old” that are characterized by MFs μyoung(x), μmiddleaged(x), and μold(x), respectively. Just as a variable can assume various values, a linguistic variable “age” can assume different linguistic values, such as “young,” “middle aged,” and “old” in this case. If “age” assumes the value of “young,” then we have the expression “age is young,” and so also for the other values. Typical MFs for these linguistic values are displayed in Figure 14.8, where the universe of discourse X is totally covered by the MFs and their smooth and gradual transition from one to another. Unary fuzzy operations, concentration and dilation, may be interpreted as linguistic modifiers “very” and “more or less,” respectively.
Figure 14.8. Typical membership functions for linguistic values “young,” “middle aged,” and “old.”
A linguistic variable is characterized by a quintuple (x, T(x), X, G, M) in which x is the name of the variable; T(x) is the term set of x—the set of its linguistic values; X is the universe of discourse; G is a syntactic rule that generates the terms in T(x); and M is a semantic rule that associates with each linguistic value A its meaning M(A), where