Subject: [9] How are membership values determined?

Date: 15-APR-93

Determination methods break down broadly into the following categories:

1. Subjective evaluation and elicitation

As fuzzy sets are usually intended to model people's cognitive states, they can be determined from either simple or sophisticated elicitation procedures. At they very least, subjects simply draw or otherwise specify different membership curves appropriate to a given problem. These subjects are typcially experts in the problem area. Or they are given a more constrained set of possible curves from which they choose. Under more complex methods, users can be tested using psychological methods.

2. Ad-hoc forms

While there is a vast (hugely infinite) array of possible membership function forms, most actual fuzzy control operations draw from a very small set of different curves, for example simple forms of fuzzy numbers (see [7]). This simplifies the problem, for example to choosing just the central value and the slope on either side.

3. Converted frequencies or probabilities

Sometimes information taken in the form of frequency histograms or other probability curves are used as the basis to construct a membership function. There are a variety of possible conversion methods, each with its own mathematical and methodological strengths and weaknesses. However, it should always be remembered that membership functions are NOT (necessarily) probabilities. See [10] for more information.

4. Physical measurement

Many applications of fuzzy logic use physical measurement, but almost none measure the membership grade directly. Instead, a membership function is provided by another method, and then the individual membership grades of data are calculated from it (see UZZIFICATION in [4]).

5. Learning and adaptation

For more information, see:

Roberts, D.W., "Analysis of Forest Succession with Fuzzy Graph Theory", Ecological Modeling, 45:261-274, 1989.

Turksen, I.B., "Measurement of Fuzziness: Interpretiation of the Axioms of Measure", in Proceeding of the Conference on Fuzzy Information and Knowledge Representation for Decision Analysis, pages 97-102, IFAC, Oxford, 1984.

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Subject: [10] What is the relationship between fuzzy truth values and probabilities?

Date: 21-NOV-94

This question has to be answered in two ways: first, how does fuzzy theory differ from probability theory mathematically, and second, how does it differ in interpretation and application.

At the mathematical level, fuzzy values are commonly misunderstood to be probabilities, or fuzzy logic is interpreted as some new way of handling probabilities. But this is not the case. A minimum requirement of probabilities is ADDITIVITY, that is that they must add together to one, or the integral of their density curves must be one.

But this does not hold in general with membership grades. And while membership grades can be determined with probability densities in mind (see [11]), there are other methods as well which have nothing to do with frequencies or probabilities.

Because of this, fuzzy researchers have gone to great pains to distance themselves from probability. But in so doing, many of them have lost track of another point, which is that the converse DOES hold: all probability distributions are fuzzy sets! As fuzzy sets and logic generalize Boolean sets and logic, they also generalize probability.

In fact, from a mathematical perspective, fuzzy sets and probability exist as parts of a greater Generalized Information Theory which includes many formalisms for representing uncertainty (including random sets, Demster-Shafer evidence theory, probability intervals, possibility theory, general fuzzy measures, interval analysis, etc.). Furthermore, one can also talk about random fuzzy events and fuzzy random events. This whole issue is beyond the scope of this FAQ, so please refer to the following articles, or the textbook by Klir and Folger (see [16]).

Semantically, the distinction between fuzzy logic and probability theory has to do with the difference between the notions of probability and a degree of membership. Probability statements are about the likelihoods of outcomes: an event either occurs or does not, and you can bet on it. But with fuzziness, one cannot say unequivocally whether an event occured or not, and instead you are trying to model the EXTENT to which an event occured. This issue is treated well in the swamp water example used by James Bezdek of the University of West Florida (Bezdek, James C, "Fuzzy Models --- What Are They, and Why?", IEEE Transactions on Fuzzy Systems, 1:1, pp. 1-6).

Delgado, M., and Moral, S., "On the Concept of Possibility-Probability Consistency", Fuzzy Sets and Systems 21:311-318, 1987.

Dempster, A.P., "Upper and Lower Probabilities Induced by a Multivalued Mapping", Annals of Math. Stat. 38:325-339, 1967.

Henkind, Steven J., and Harrison, Malcolm C., "Analysis of Four Uncertainty Calculi", IEEE Trans. Man Sys. Cyb. 18(5)700-714, 1988.

Kamp`e de, F'eriet J., "Interpretation of Membership Functions of Fuzzy Sets in Terms of Plausibility and Belief", in Fuzzy Information and Decision Process, M.M. Gupta and E. Sanchez (editors), pages 93-98, North-Holland, Amsterdam, 1982.

Klir, George, "Is There More to Uncertainty than Some Probability Theorists Would Have Us Believe?", Int. J. Gen. Sys. 15(4):347-378, 1989.

Klir, George, "Generalized Information Theory", Fuzzy Sets and Systems 40:127-142, 1991.

Klir, George, "Probabilistic vs. Possibilistic Conceptualization of Uncertainty", in Analysis and Management of Uncertainty, B.M. Ayyub et. al. (editors), pages 13-25, Elsevier, 1992.

Klir, George, and Parviz, Behvad, "Probability-Possibility Transformations: A Comparison", Int. J. Gen. Sys. 21(1):291-310, 1992.

Kosko, B., "Fuzziness vs. Probability", Int. J. Gen. Sys. 17(2-3):211-240, 1990.

Puri, M.L., and Ralescu, D.A., "Fuzzy Random Variables", J. Math. Analysis and Applications, 114:409-422, 1986.

Shafer, Glen, "A Mathematical Theory of Evidence", Princeton University, Princeton, 1976.