Data Mining by Mehmed Kantardzic (good book recommendations TXT) 📕

(b) The MF has to monotonically increase the left of the core and monotonically decrease on the right. This ensures that only one peak and, therefore, only one typical value exists. The spread of the support (i.e., the nonzero area of the fuzzy set) describes the degree of imprecision expressed by the fuzzy number.

A graphical illustration of some of these basic concepts is given in Figure 14.4.

Figure 14.4. Core, support, and α-cut for fuzzy set A.

14.2 FUZZY-SET OPERATIONS

Union, intersection, and complement are the most basic operations in classic sets. Corresponding to the ordinary set operations, fuzzy sets too have operations, which were initially defined by Zadeh, the founder of the fuzzy-set theory.

The union of two fuzzy sets A and B is a fuzzy set C, written as C = A ∪ B or C = A OR B, whose MF μC(x) is related to those of A and B by

As pointed out by Zadeh, a more intuitive but equivalent definition of the union of two fuzzy sets A and B is the “smallest” fuzzy set containing both A and B. Alternatively, if D is any fuzzy set that contains both A and B, then it also contains A ∪ B.

The intersection of fuzzy sets can be defined analogously. The intersection of two fuzzy sets A and B is a fuzzy set C, written as C = A ∩ B or C = A AND B, whose MF is related to those of A and B by

As in the case of the union of sets, it is obvious that the intersection of A and B is the “largest” fuzzy set that is contained in both A and B. This reduces to the ordinary-intersection operation if both A and B are non-fuzzy.

The complement of a fuzzy set A, denoted by A′, is defined by the MF as

Figure 14.5 demonstrates these three basic operations: Figure 14.5a illustrates two fuzzy sets A and B; Figure 14.5b is the complement of A; Figure 14.5c is the union of A and B; and Figure 14.5d is the intersection of A and B.

Figure 14.5. Basic operations on fuzzy sets. (a) Fuzzy sets A and B; (b) C = A’; (c) C = A ∪ B; (d) C = A ∩ B.

Let A and B be fuzzy sets in X and Y domains, respectively. The Cartesian product of A and B, denoted by A × B, is a fuzzy set in the product space X × Y with an MF

Numeric computations based on these simple fuzzy operations are illustrated through one simple example with a discrete universe of discourse S. Let S = {1, 2, 3, 4, 5} and assume that fuzzy sets A and B are given by:

Then,

and the Cartesian product of fuzzy sets A and B is

Fuzzy sets, as defined by MF, can be compared in different ways. Although the primary intention of comparing is to express the extent to which two fuzzy numbers match, it is almost impossible to come up with a single method. Instead, we can enumerate several classes of methods available today for satisfying this objective. One class, distance measures, considers a distance function between MFs of fuzzy sets A and B and treats it as an indicator of their closeness. Comparing fuzzy sets via distance measures does not place the matching procedure in the set-theory perspective. In general, the distance between A and B, defined in the same universe of discourse X, where X ∈ R, can be defined using the Minkowski distance:

where p ≥ 1. Several specific cases are typically encountered in applications:

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;