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3.4. Characteristics of a fuzzy set

As fuzzy sets are described by membership functions, we can characterize them in more detail by referring to the features being used in the characterization of the functions (Kandel, 1986; Dubois and Prade, 1993; Yager et al., 1993; Zadeh, 1975, 1978, 1979, 1981, Zimmermann, 1985). This brings up the notion of normality, height, support, convexity, and simple operations conveying modifications of the membership functions.

normality and height A fuzzy set A is normal if its membership function attains 1, that is,

If the supremum is less than 1, then A is called subnormal. The above supremum is usually referred to as the height of A, hgt(A). Hence the normality of the fuzzy set is equivalent to having the height equal to 1.

support By a support of a fuzzy set A, denoted by Supp(A), we mean all elements of X that belong to A to a nonzero degree,

core the core of a fuzzy set A is the set of all elements of X that exhibit a unit level of membership in A. More formally:

The support and core of fuzzy sets may be viewed as dual concepts. Obviously, both the support and core are sets rather than fuzzy sets.

α- cut Similarly, we define a more refined notion such as an α -cut of A

The core of A is just its 1-cut.

unimodality A is unimodal if its membership is a unimodal function, namely a function that has a single maximum.

A related concept is that of convexity.

convexity A fuzzy set A is convex if its membership function is such that

for any x1, x2X, and λ ∈ [0,1].

concavity is a dual concept of convexity, a fuzzy set A is concave if the corresponding membership function satisfies the relationship

for any x1, x2X, and λ ∈ [0,1].

Note that convex and concave fuzzy sets are unimodal, but the converse is not true.

cardinality Given a fuzzy set A in a finite universe X, its cardinality, denoted by card(A), is defined as

Frequently, Card(A) is referred to as the scalar cardinality or the sigma 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 X = {1, 2, 3, 4, 5, 6} has the cardinality equal 2.4, Card(A) = 2.4. If X is infinite, the cardinality of a defined therein is an integral of A (assuming that this operation makes sense)

We can list a collection of simple, yet useful operations on fuzzy sets. These are one-argument mappings as they apply to a single membership function

normalization. This operation converts a subnormal, nonempty fuzzy set into its normal version by dividing the original membership function by the height of A

concentration. By concentrating fuzzy sets their membership functions get relatively smaller values. That is, it gets more concentrated around points with higher membership grades as, for instance, being raised to power two,

The similar, yet even amplified effect, is obtained by using any power p > 1,

dilation. The effect of dilation is the opposite of concentration, and is produced by modifying the membership function through the transformation

or

As before one can generalize this operation by admitting any power r ∈ (0, 1.0)

contrast intensification. As the name suggests, the membership values lower than 1/2 are diminished while the grades of membership above this threshold are elevated. This intensifies contrast, once it has the effect of reducing the fuzziness of a set. The operation is defined by

or, to make the intensification more pronounced, for p > 1,

fuzzification. The effect of fuzzification is complementary to intensification, and is produced by altering the membership function as follows:

3.5. Membership function determination

Fuzzy sets are elastic constraints defining linguistic concepts. The concepts are inherently discussed and used within a certain environment. Their meaning dwells on the context. The notion of high temperature is pretty much meaningless unless we place it into a certain context (temperature in a certain building, outdoor temperature, temperature of a specific chemical process, etc.). There is a visible conceptual duality. On one hand, the linguistic terms are symbols - as such they could be exploited within any symbol-oriented environment (that is specific to AI). The numeric fabric of membership functions lies in another dimension and provides a handle necessary to calibrate the notions. That is why the membership of high outdoor temperature is very different from the membership function of the same linguistic label describing a high temperature in a gas furnace. There are a number of experimental methods aimed at the estimation of membership values (or membership functions). In what follows, we briefly elaborate on some of them including: a horizontal approach, vertical approach, pairwise comparison, and inference based on problem specification. The selection of each of them depends heavily on the specificity of an application. In particular, this refers to the way in which the uncertainty is manifested and captured during the experiment.

3.5.1. Horizontal method of membership estimation

The underlying idea is to gather information about membership values of the concept at some selected elements of the universe of discourse x1, x2, …, xn (the elements need not to be evenly distributed; on the contrary their distribution could be very uneven depending on the form of the concepts to be captured; especially we may envision a logarithmic scale governing the distribution of the points of interest). The method relies on some experimental findings collected under the following scenario. A group of testees (experts) are asked to answer the question:

Can xi be accepted as compatible with the concept A?

The admitted answers could be only either “yes” or “no” (one could eventually enlarge this repertoire by the third option of the “unknown” reply). The estimated value of the membership function at xi is taken as a ratio of the number of positive replies P(xi), to the total number N of responses,


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