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The bound of the interval is located at the point where Ai and Ai+1, assume the same value. In other words, xi is determined by the point of intersection of these two fuzzy sets - quite an intuitively appealing finding.

3.16.4. Robustness properties of the frame of cognition

Could the fuzzy sets of the frame of cognition contribute to its increased robustness when compared with the induced Boolean partition of the same space? Could the frame of cognition tolerate or even mask some slight deviations (errors) in the input variable x not leading to the deterioration of quality any further processing carried out in terms of the membership grades of the fuzzy sets (relations) of A? Intuitively, the answer should be affirmative anticipating that the abrupt changes of the characteristic functions would amplify the changes originating from some erroneous values of the input variable processed via the frame of cognition. First, we define the notion of robustness. Let us, as in Section 3.14.3, discuss two fuzzy sets A and B (assuming that the supports of the adjacent fuzzy sets vanish between “a” and “b”).


Figure 3.21  Distribution of error for the frame of cognition (i) and its Boolean counterpart (ii)

Consider that instead of x we are furnished with x + e where e > 0 is a certain systematic bias (error). The intent is to quantify an impact this bias has on the computed membership values. Two performance indices are taken into account:

  a local performance index taken as a sum of absolute values between membership functions. Due to the character of the respective membership functions (A decreasing and B increasing) we get to, the expression

for any x in [a, b]. The value of r(x) depends on x; its upper bound equals 2. The global index of robustness r is an integral of r(x)

The set-based frame of cognition leads to the obvious results

while r =2e

Let us confine ourselves to the triangular membership functions A and B. Then

Similarly,

The global index of robustness is the same for the original frame of cognition as well as its Boolean counterpart. The distribution of r is diametrically different, though. The Boolean character of the frame of cognition leads to far higher yet concentrated error - the use of fuzzy sets reduces the maximal value of r(x) and spreads it evenly across the entire universe.

3.17. Probability and fuzzy sets

The distinction between probability and fuzzy sets is self-evident. Probability is concerned with occurrence of well-defined events (being regarded as sets). Tossing a coin, drawing a ball from an urn, accomplishing a space mission - these are examples of events described by sets. Probabilities are or can be associated with each of them. For instance, a probability of picking up randomly a red ball from an urn containing 5 red and 7 black balls is , (obviously, we make a default assumption that each ball can be picked up with equal probability 1/12). The probabilities are assessed or estimated based on a repetition of a certain series of experiment carried out in a stationary environment. The engineering approach relies heavily on the probabilities treated as limit frequencies and determined based on a finite number of experiments. Thus the probability of event A is estimated as the limit

Where nA is the number of experiments in which event A has occurred while n is the total number of experiments in the series. Hence the occurrence of an event becomes a central notion of probability: the calculus of probability deals with occurrence of events.

On the other hand, fuzzy sets deal with the problems of graduality of concepts and describe their boundaries. Therefore they have nothing to do with frequencies (repetition) of the event. This underlines the dominant difference between probabilities and fuzzy sets. To make this difference even more transparent, consider an experiment whose outcome (A) can eventually occur. Prior to the experiment one can only think of the probability of A, P(A). Once the experiment is over the probabilistic facet of uncertainty vanishes. The outcome is unambiguously known: A has happened or not. In contrast, let A be a fuzzy set - after the experiment the notion is still valid and fully retained.

The conceptual difference between these two facets of uncertainty makes that the mathematical frameworks of fuzzy sets and probability are also very distinct. The first one relies on set theory and logic while the latter hinges on the concepts of (additive) measure theory.

3.18. Hybrid fuzzy-probabilistic models of uncertainty

The previous section has strongly emphasized that the mechanism of probability and fuzziness as coping with occurrence and graduality are highly orthogonal. In spite of that there are some interesting situations that, in fact, led to the useful symbiosis of fuzzy sets and probability.

Let us consider a statement

Prob (X is A)

where A is a fuzzy set defined in XR. As X is equipped with some probability density function (p.d.f.) the quantification of the probability of this fuzzy event is computed as (Zadeh, 1968)

Formula (6.2) calculates what is known as a probability of a fuzzy event (assuming that the above integral does make sense). The underlying illustration of this idea is depicted in Fig. 3.22. The probability density function p(x) is integrated over the support of A while A(x) plays a role of a weighting function. For comparison illustrated is a situation where A is a set. The shadowed areas denote the probability of the corresponding events.


Figure 3.22  Computations of probabilities of event (i) and fuzzy event (ii)

Noticeable is that if A∴B then the entailed probabilities satisfy the inequality

implying the increase in the resulting probability. Similarly, by insisting on a certain level of probability (λ)

we determine a fuzzy event A satisfying this relationship (definitely, the choice of A is not unique). The expressions similar to those given above are examples of probabilistically - quantified linguistic statements.

The fundamental requirements about these statements can be formulated by looking at the following constraints:

  specificity constraint: the statement (linguistic label) A should be specific enough (with the predefined value of the specificity measure)
  probability constraint: the statement should be meaningful enough thus making the probability associated with it sufficiently high

The above requirements translate into the formal expressions


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