Data Mining - Mehmed Kantardzic [238]
3. Normalization. A fuzzy set A is normal if its core is nonempty. In other words, we can always find a point x ∈ X such that μA(x) = 1.
4. Cardinality. Given a fuzzy set A in a finite universe X, its cardinality, denoted by Card(A), is defined as
Often, Card(X) is referred to as the scalar cardinality or the count of A. For example, the fuzzy set A = 0.1/1 + 0.3/2 + 0.6/3 + 1.0/4 + 0.4/5 in universe X = {1, 2, 3, 4, 5, 6} has a cardinality Card(A) = 2.4.
5. α-cut. The α-cut or α-level set of a fuzzy set A is a crisp set defined by
6. Fuzzy number. Fuzzy numbers are a special type of fuzzy sets restricting the possible types of MFs:
(a) The MF must be normalized (i.e., the core is nonempty) and singular. This results in precisely one point, which lies inside the core, modeling the typical value of the fuzzy number. This point is called the modal value.
(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