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It has been shown in Ebanks (1983) that the above additive form of the functional leads to the satisfaction of (i) - (v). Moreover, since the functional “h” is symmetrical one has

The example below reveals some interesting links between the fuzzy entropy and the concept of distance between fuzzy sets.

As a brief example let us study a piecewise linear form of the functional h. Then the entropy of A computes accordingly

Denote by A1/2 the 1/2-cut set of A,

Then

and is nothing but a Hamming distance between A and its 1/2-cut multiplied by a constant (2). This, in fact, emphasizes what has been said about the entropy - the more A resembles A1/2, the higher its entropy.

In the case X is a line of reals, the entropy measure of fuzziness is modified, namely the summation is replaced by integration (obviously we assume that the resulting integral does make sense).

It should be stressed that the uncertainty effect captured by the fuzzy entropy pertains to the effect of partial rather than binary membership values conveyed by A. Undoubtedly, this is an inherently different source of uncertainty in comparison to the probabilistic (statistical) character of uncertainty represented by the previous form of uncertainty (3.1).

As fuzziness and randomness may emerge within the same problem, the uncertainty originating from these two sources could be aggregated. One of possible ways of merging can be realized in the additive form as given below,

Note that if the events (occurrence) of xi’s are equiprobable, then the second component remains whereas for sets the formula reduces to the first probability-oriented part. One should be aware that as fuzziness and probability are orthogonal concepts, the additive model as the one above could be regarded as one among feasible alternatives aimed at the aggregation. For more detailed discussion refer to Pal and Bezdek (1993).

3.10.2. Energy measure of fuzziness

An energy measure of fuzziness (energy) (De Luca, Termini, 1972; 1974), E(A), is defined as the quantity

where e:[0,1] → [0, 1] is a functional that is increasing over the entire domain with the boundary conditions e(0) = 0 and e(1) = 1.

In particular, the above functional can be regarded as an identity mapping, e(u)=u for all u’s in the unit interval. In this case the energy measure of fuzziness is referred to as a cardinality of the fuzzy set,

Some other types of the mapping e include

etc.

In general, the energy measure of fuzziness is aimed at expressing a total mass of the fuzzy set. If e(u)=up, we can regard the energy measure of fuzziness as the distance of A from the origin (empty set),

If each xi appears with probability pi, then the energy of A can include this probabilistic information by assuming the form

Essentially, (A) becomes an expected valued of the functional e(A). As in the case of the fuzzy entropies, the definition of energy can be extended into the continuous case,

3.10.3. Specificity of a fuzzy set

The introduction of this type of the characterization is motivated by the need quantifying how difficult it is to pick up a single point in the universe of discourse as a reasonable representative of the fuzzy set. In limit cases:

  if the fuzzy set is of a degenerated form, namely it is already a single element, A={x0}, there is no hesitation in the selection of x0 as an excellent (and the only one) representative of A,
  if A covers almost the entire universe of discourse and embraces a lot of elements with membership equal 1, then the choice of only one of them causes a lot of hesitation.

As seen from these considerations, in the first instance the fuzzy set is very specific whereas its specificity in the second situation is zero.

Being prompted by this problem, we can define specificity of a fuzzy set as follows (Yager, 1981, 1982). The specificity of A defined in X, Sp(A), assigns to a fuzzy set A a nonnegative number such that

  Sp(A) =1 if and only if there exists only one element of X for which A assumes 1 while the remaining membership values are equal zero
  if A(x)=0 for all elements of X then Sp(A) =0
  if A1 A2 then

In Yager (1981) the specificity measure is defined as the integral

where αmax = hgt(A).

For the finite universe of discourse (that implies a finite number of membership values), the integration is replaced by a summation

where

and α1-1 =0 whereas “n” denotes a number of the membership values of A.

As discussed by Yager (1992), the measure of specificity can be modified by introducing a linear class of such measures defined as

where aj is the jth largest membership value of A. The set of weight factors satisfies the conditions

If w2 = 1 the resulting specificity measure is the most strict and reads as

If

For all j=2, 3, …, n then the measure is equal to

3.11. Matching measures

The problem of comparing (matching) two fuzzy sets or fuzzy relations becomes of importance as its solution supports processing of linguistic quantities. As fuzzy sets are just membership functions, there is no single solution to the problem. Rather than that, the literature furnishes us with a broad spectrum of methods. Here, we review three of the approaches based upon computing possibility, necessity and compatibility measures.


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