8.6.5. Fuzzy Statistics

Fuzzy set theory and probability theory are used to handle two different types of uncertainty. We use probability to study random phenomena. Each event itself has distinct meaning and not uncertain. However, due to the lack of sufficient condition the outcome for certain event to occur during a process cannot be determined.

In fuzzy set theory, concept or event itself does not have a clear definition. For example, "tall mean", how tall they are is not defined. Here, whether certain phenomena belong to this concept is difficult to determine. We call it fuzziness the uncertainty involved in a classification due to the imprecise concept definition. The root for fuzziness is that there exists transitions between two phenomena. Such transitions make it possible for us to label phenomena into either this or that class. Fuzzy set theory is the base for us to study membership relationships from the fuzziness of phenomena.

Fuzzy statistics is used to determine estimate the degree of membership or membership function. In order to do so we need to design a fuzzy statistic experiment. In such an experiment, similar to fuzzy statistics, there are four elements:

1. Universe of discourse W ;

2. An element w in W ;

3. An ordinary set A which is varying on the W basis. A is related to a fuzzy set  which corresponds to a fuzzy concept. Each time A is fixed, it represents a deterministic definition of the fuzzy concept as its approximation.

4. Condition S which contains all the objective and subjective factors that are related to the definition of the fuzzy concept and therefore is a constraint of the variation of A.

The purpose of fuzzy statistics is to use a deterministic approach to study the uncertainties. The requirements for a fuzzy statistical experiment is that in each experiment a deterministic decision on whether w belongs to A. Therefore, in each experiment, A is a definite ordinary set. In fuzzy statistical experiments, w is fixed while A is changing.

In n experiments, calculate the membership frequency of w belonging to fuzzy set  , denoted by f 

f =  

As n increases, f may stabilize. The stabilized membership frequency is the degree of membership for w belonging to  . We call fuzzy statistics involving more than one fuzzy concepts, multi-phase fuzzy statistics.

Definition 8. Given Pm = { 1 , ...,  m} Ai Œ F(W), i = 1, ..., m, this type of experiments is m-phase fuzzy statistical experiments, provided that in each experiment we can determine a projection such that

e : W Æ Pm .

Each fuzzy set in Pm is one phase of Pm.

The results of multi-phase fuzzy statistics enable us to obtain a fuzzy membership function for each phase on W. They have the following properties:

m 1(w) + m 2(w) + ... + m m(w) = 1

If W = {w1, w2, ... , wn} is a finite universe of discourse we have

 m 1(wj) + ... +  m m(wj) = n .