Эволюция концепции нечеткой меры

Let be the set of fuzzy subsets on a given set, the entropy measure introduced by De Luca and Termini [1972], and the possibility or necessity measures [Higashi & Klir; 1983] are also a fuzzy p-measures.

The figure 3 shows graphically the relation between different measures and a classification of many of the given examples

Figure 3: relation between different measures and a classification of some examples

3 Some Measures On Fuzzy Sets and Fuzzy Relations

3.1 Entropy or Measures of Fuzziness

Let X be a set and let (X) be the set of fuzzy sets on X. The measures of fuzziness or entropies give a degree of fuzziness for every fuzzy set in . Some entropy measures have influence from the Shannon probabilistic entropy, which is commonly used as measures of information.

De Luca and Termini [1972] consider the entropy E of a fuzzy set of (X) as a measure that gives a value in [0, ] and satisfies the following properties:

1.E(A) = 0 if A is a crisp set.

2.E(A) is maximal if A is the constant fuzzy set A(x) = 12 for all x X.

3.E(A) ≥ E(B) if A is ‘more fuzzy’ than B by the ‘sharpen’ order.

4.E(A) = E(A’).

Note that the defined entropy measure of a fuzzy set is a fuzzy p-measure where the p preorder is the ≤S sharpen order, in which B≤SA if for any

element x in the universe of discourse when A(x) ≤ 12 then B(x) ≤ A(x)

and when A(x) ≥ 12 then B(x) ≥ A(x)

Kaufmann [1975] proposes a fuzziness index as a normal distance. Other authors as Yager [1979] and Higashi and Klir [1983] understand the entropy measures as the difference between a fuzzy set and its complementary fuzzy set.

A new proposal for entropy measure given by a t-norm, a t-conorm and a negation is as follows:

Let H: → [0, 1], H(µ) = k Sx U{T(µ(x), N(µ(x))}, where µ is a fuzzy set on U, k is a constant that is used to normalise the measure and depends on the chosen continuous t-conorm S, the continuous t-norm T and the strong negation N.

It is easy to prove that if µ is crisp then H(µ) = 0. If the t-norm T is the minimum or is in the family of the product t-norm, then H(µ) = 0 if and only if µ is a crisp set. It is proved that H(µ) = H(µc) by the symmetry property of the t-norms. When the t-norm T is the minimum or in the family of the product, then if B≤SA then H(B)≤H(A).

So the new proposed entropy verifies the properties given by De Luca and Termini [1972] when T is the minimum or in the family of the product t- norm.

If the fuzzy set is on an infinite domain, the same result can be extended to the new expression

H(µ) = k ∫ {T(µ(x), N(µ(x))}.dx. x E

3.2 Measures of Specificity

The specificity measures introduced by Yager [1990] are useful as measures of tranquillity when making a decision. Yager introduces the specific- ity-correctness trade-off principle. The output information of expert systems and other knowledge-based systems should be both specific and correct to be useful. Yager suggests the use of specificity in default reasoning, in possibility-qualified statements and data mining processes, giving several possible manifestations of this measure. Kacprzyk [1990] describes its use in a system for inductive learning. Dubois and Prade [1999] introduce the minimal specificity principle and show the role of specificity in the theory of approximate reasoning. Higashi and Klir [1983] introduce a closely related idea called non-specificity. The concept of granularity introduced by Zadeh [1978] is correlated with the concept of specificity.

Let X be a set with elements {xi} and let [0, 1]X be the class of fuzzy sets of X. A measure of specificity Sp is a function Sp: [0, 1]X →[0, 1] such that:

1.Sp(µ) = 1 if and only if µ is a singleton (µ = {x1}).

2.Sp( ) = 0

3.If µ and η are normal fuzzy sets in X and µ η, then Sp(µ)≥Sp(η).

A general expression [L. Garmendia; R. R. Yager; E. Trillas y A. Salvador, 2003] that can be used to build measures of specificity from three t-norms and negations is an application SpT: [0, 1]X→[0, 1] defined by SpT(µ) =

T1(a1, N(T2*j=2,..,n{T3(aj, wj)}) where µ is a fuzzy set in a finite set X, and ai is the membership degree of the element xi (µ(xi)=ai), the membership

degrees ai [0, 1] are totally ordered with a1 ≥ a2 ≥ ... ≥ an, N is a negation, let T1 and T3 be any t-norms, T2* a n-argument t-conorm and {wj} is a weighting vector.

For example, when N is the negation N(x) = 1-x, T1 and T2 are the Łukasiewicz t-norm defined by T1(a, b) = max{0, a+b−1}, so T2*(a1, ..., an) = min{1, a1+ ..+an}, and T3 is the product, then the previous expression gives Yager´s [1990] linear measure of specificity, defined as

Sp(µ) = a1 − ∑n wj aj. j =2

The measures of specificity are not monotone measures, because the measure of specificity of a fuzzy set is lower when some membership degrees that are not the highest degree are increased. However the measures of specificity of fuzzy sets are fuzzy p-measures, where pis a preorder that classifies the fuzzy sets by the utility of the contained information, or by a given distance to a singleton.

Let A be a fuzzy set on an infinite universe X and let Aα be its α-cut.

R.R. Yager [1998] defines a measure of specificity on a continuous do-

αmax

∫ F(M(Aα)) dα, where αmax is the maximum mem-

0

bership degree of A and F is a function F: [0, 1] → [0, 1] verifying that F(0)=1, F(1)=0, and F(x) ≤ F(y) ≤ 0 when x>y.

0

area under the fuzzy set A. So, the measure of specificity under a fuzzy set under an interval [a, b] can be interpreted as

αmax – area under A . b-a

The measures of specificity of fuzzy sets on infinite universes can be given by an expression

tion.

Yager [1991] introduced the concept of specificity of a fuzzy set under similarities using the Zadeh [1971] concept of similarity or Minindistinguishability.

The α-cut of a similarity S is a classical equivalence relation denoted Sα. Let πα be the set of equivalence classes of S for a given value α. Let µα/S

be the set of equivalence classes of πα defined in the following way: class πα(i) belongs to µα/S if there exists an element x contained in πα(i) and in

the µ´s α-cut (µα).

Yager [1991] definition of measure of specificity of a fuzzy set µ under a similarity is the following:

3.3 Measures of µ-T-Unconditionality

The µ-T-conditionality property of fuzzy relations generalises the modus ponens property when making fuzzy inference. A fuzzy relation R: E1×E2→ [0,1] is µ-T-conditional if and only if T(µ (a), R(a, b)) ≤ µ(b) for all (a, b) in E1×E2.

Some ways to measure a degree of verification of this property are discussed, which are monotonous measures on the measurable space ( , , M), where is the set of fuzzy relations R: E1×E2→[0, 1], the set of measurable subsets of and M is a measure of µ-T- unconditionality. There are two ways to define those measures [L. Garmendia; E. Trillas; A. Salvador; C. Moraga, 2005]. A first way computes a generalised distance between a fuzzy relation R and the greatest µ-T-conditional relation that is contained in R. The other way measures the difference between T(µ(a), R(a, b)) and µ(b) in all points (a, b) in which R is not µ-T-conditional.

Let JT(x, y) = Sup {z: T(x, z) ≤ y} be the residual operator of a t-norm T, let JTµ (a, b) = JT (µ(a), µ(b)), and let

mendia [2003] it is proved that JT(R(a, b), JTµ (a, b)) = JT( TRµ (a, b), µ2(a, b))

for all continuous t-norms, which shows that when a generalised distance defined from a residuated operator of the T-norm is used, both methods set the same measures of µ-T-unconditionality of fuzzy relations.

So, given any continuous t-norm T, for all (a, b) in E1×E2, the distance 1-JT between a fuzzy relation R in the point (a, b) and its µ-T-conditionalized relation JTµ (a, b), is the same than the distance 1-JT between TRµ (a, b) and

µ(b).

Some operators are frequently used to make fuzzy inference. The fuzzy operators are fuzzy relations on the universe E1×E2 = [0, 1]×[0, 1].

The following examples show the evaluation of the measures for some implication operators. For all of them, the fuzzy set µ is taken as the identity (that is, as a function µ:[0, 1]→[0, 1] such that µ(x)=x).

The chart bellow gives the measures of the most used residual implication operators, S-implications, QM-implications and conjunctions for the t- norms minimum, product and Łukasiewicz:

Table 1: measures of Id-T-unconditionality of some operators

The measures of µ-T- unconditionality of fuzzy relations are monotone measures on the measurable space ( , , M) where is the set of fuzzy relations R: E1×E2→[0, 1], is the set of measurable subsets of and M is a measure of µ-T- unconditionality.

4 Conclusions

This paper presents an overview of the evolution of the concept of fuzzy measure. Some of these are relevant to understand the process of inference, even when these are neither additive nor monotone. Proposals for nonmonotone measures on fuzzy sets (entropy and specificity) are presented. They can be considered to be measures of information on the premises or conclusions in approximate reasoning. Finally some results are discussed on a monotone measure on fuzzy relations to understand whether or with which logics is the modus ponens generalised when making fuzzy inference.

Acknowledgment

This research is partially supported by the Spanish MCyT project BFM2002-00281.

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