![]() |
|
|||
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
|
![]() |
[an error occurred while processing this directive]
3.11.1. Possibility and necessity measuresThe measures of possibility and necessity (Dubois and Prade, 1988; Zadeh, 1978) are among the most commonly used mechanisms of expressing matching between two fuzzy sets (and more generally, fuzzy relations). Let A and X be two fuzzy sets defined in the same space X. Here X is regarded as an input datum while A can be viewed as a fuzzy set of reference with respect to which X needs to be matched. The possibility measure, Poss(X, A), describes a degree of overlap (intersection) between X and A It is remarkable that the possibility measure can make different fuzzy sets (X indistinguishable with respect to A (they are equal in the sense of the derived possibility measure). The necessity measure reflects a degree of inclusion of X in A The possibility and necessity measures expand to fuzzy relations - then A and X are viewed as two fuzzy relations defined in the same space. 3.11.2. Compatibility measureThe concept behind this idea is to quantify an extent to which X is compatible with another fuzzy set defined in the same space. Let, a before, X and A be expressed in X. Formally, the compatibility of X with A, say comp(X, A), is defined as u∈ [0, 1]. Conceptually, there is an evident difference between this measure and the other approaches presented before. Firstly, the compatibility is not a single numerical quantity but a mapping between two unit intervals. In other words, we can think of comp(X, A) as a fuzzy set defined in [0, 1] - we can think of a fuzzy set of compatibility. Secondly, the compatibility is asymmetrical as it quantifies the truth of the statement and this is inherently distinct from saying that A is compatible with X. Evidently, the fuzzy set A plays a role of a certain reference (conceptual reference) with respect to which the compatibility has to be computed. For instance, one can think of compatibility of speed of about 80 km/h with the concept of high speed, Comp(about 80 km/h, high speed). The computations of the fuzzy set of compatibility are illustrated in Fig. 3.7.
For illustrative purposes, consider u0 in [0,1]. There are two arguments of A associated with this membership value, namely, x1 and x2. The supremum (maximum) taken over the corresponding membership values X(x1) and X(x2) eliminates ambiguity as the higher of the values becomes accepted as the membership of the compatibility at u0 By inspection, several basic properties can be revealed - assume that both X and A are normal,
The possibility and necessity measures are subsumed in the compatibility measure as included in the support of Comp(X, A). In particular, as shown in Fig. 3.8 where X is a set, the upper bound of the support of the compatibility is the possibility value of X with respect to A whereas the lower bound corresponds to the necessity value.
3.12. Numerical representation of fuzzy setsThe objective of transforming fuzzy sets into numerical representative(s) is to allow fuzzy sets to interact with numerical environments. It is obvious that all reasoning processes are abstract; all final decision (results of inferences to be applied) must be numeric. These mechanisms are often referred to defuzzification mechanism. The literature provides us with an abundance of methods. Let us elaborate on some of them that may be regarded as representatives in this area. mean of maxima (MoM) method. Here the values corresponding to the highest membership functions of A are selected and averaged giving rise to the result of decoding, where p is the number of modal values of A and center of gravity (CoG) method. Here the result of decoding is equal to the center of gravity of A, center of area (CoA) method. Here There are numerous modifications of these basic methods. For instance, one can reduce an impact coming from long tails of the membership function, particularly if the corresponding membership values in these regions are low. The CoG modification incorporating this modification reads as We can consider the similar modification to the CoA method, where β is the threshold level eliminating a deteriorating influence coming from low membership values. Another modification of the CoG method involves an extra weight factor δ that occurs in the expression and makes this decoding scheme more flexible. Especially, if δ=1 then this modification is the standard CoG method while for δ → 0 it performs like the MoM scheme. These methods as requiring computations of A are more demanding than the algorithms coming from the first group. Interestingly enough, we can reveal some links between the COA method and robust statistics (Barnett and Lewis, 1984). Let us assume that the universe of discourse is finite, card (X) = n. We represent A by a single number (m) such that the weighted sum of the absolute differences assumes minimum The above is an example of a robust statistics. Moreover xis are weighted by the corresponding membership functions. The update of the numeric representative is completed iteratively viz. Note that Thus and 3.13. Rough setsThe underlying idea of rough sets defined in X (Pawlak, 1991) is dominantly associated with a binary relation of indiscernibility I defined in the same space. This relation expressed for each pair (x, y) of elements in X is:
Copyright © CRC Press LLC
![]() |
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |