Celikyilmaz A., Türksen I.B. Modeling Uncertainty with Fuzzy Logic. With Recent Theory and Application

Springer, 2009. — 443 p.The world we live in is pervaded with uncertainty and imprecision. Is it likely to rain this afternoon? Should I take an umbrella with me? Will I be able to find parking near the campus? Should I go by bus? Such simple questions are a common occurrence in our daily lives. Less simple examples: What is the probability that the price of oil will rise sharply in the near future? Should I buy Chevron stock? What are the chances that a bailout of GM, Ford and Chrysler will not succeed? What will be the consequences? Note that the examples in question involve both uncertainty and imprecision. In the real world, this is the norm rather than exception.
There is a deep-seated tradition in science of employing probability theory, and only probability theory, to deal with uncertainty and imprecision. The monopoly of probability theory came to an end when fuzzy logic made its debut. However, this is by no means a widely accepted view. The belief persists, especially within the probability community, that probability theory is all that is needed to deal with uncertainty. To quote a prominent Bayesian, Professor Dennis Lindley, “The only satisfactory description of uncertainty is probability. By this I mean that every uncertainty statement must be in the form of a probability; that several uncertainties must be combined using the rules of probability; and that the calculus of probabilities is adequate to handle all situations involving uncertainty…probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. All other methods are inadequate…anything that can be done with fuzzy logic, belief functions, upper and lower probabilities, or any other alternative to probability can better be done with probability.” What can be said about such views is that they reflect unfamiliarity with fuzzy logic. The book “Modeling Uncertainty with Fuzzy Logic,” co-authored by Dr. A. Celikyilmaz and Professor I.B. Türksen, may be viewed as a convincing argument to the contrary. In effect, what this book documents is that in the realm of uncertainty and imprecision fuzzy logic has much to offer.
There are many misconceptions about fuzzy logic. Fuzzy logic is not fuzzy. Like traditional logical systems and probability theory, fuzzy logic is precise. However, there is an important difference. In fuzzy logic, the objects of discourse are allowed to be much more general and much more complex than the objects of discourse in traditional logical systems and probability theory. In particular, fuzzy logic provides many more tools for dealing with second-order uncertainty, that is, uncertainty about uncertainty, than those provided by probability theory. Imprecise probabilities, fuzzy sets of Type 2 and vagueness are instances of secondorder uncertainty. In many real-world settings, and especially in the context of decision analysis, the complex issue of second-order uncertainty has to be addressed. At this juncture, decision-making under second-order uncertainty is far from being well understood.
Modeling Uncertainty with Fuzzy Logic begins with an exposition of the basics of fuzzy set theory and fuzzy logic. In this part of the book as well as in other parts, there is much that is new and unconventional. Particularly worthy of note is the authors' extensive use of the formalism of so-called fuzzy functions as an alternative to the familiar formalism of fuzzy if-then rules. The formalism of fuzzy functions was introduced by Professor M. Demirci about a decade ago, and in recent years was substantially extended by the authors. The authors employ their version of the formalism to deal with fuzzy sets of Type 2, that is, fuzzy sets with fuzzy grades of membership. To understand the authors' approach, it is helpful to introduce what I call the concept of cointension. Informally, cointension is a measure of the closeness of fit of a model to the object of modeling. A model is cointensive if its degree of cointension is high. In large measure, scientific progress is driven by a quest for cointensive models of reality.
In the context of modeling with fuzzy logic, the use of fuzzy sets of Type 2 makes it possible to achieve higher levels of cointension. The price is higher computational complexity. On balance, the advantages of using fuzzy sets of Type 2 outweigh the disadvantages. For this reason, modeling with fuzzy sets of Type 2 is growing in visibility and importance.
A key problem in applications of fuzzy logic is that of construction of the membership function of a fuzzy set. There are three principal approaches. In the declarative approach, membership functions are specified by the designer of a system. This is the approach that is employed in most of the applications of fuzzy logic in the realms of industrial control and consumer products. In the computational approach, the membership function of a fuzzy set is expressed as a function of the membership functions of one or more fuzzy sets with specified membership functions. In the modelization/elicitation approach, membership functions are computed through the use of cointension-enhancement techniques. In using such techniques, successive outputs of a model are compared with desired outputs, and parameters in membership functions are adjusted to maximize cointension. For this purpose, the authors make skillful use of a wide variety of techniques centering on cluster analysis, pattern classification and evolutionary algorithms. They employ simulation to validate their results. In sum, the authors develop an effective approach to modeling of uncertainty using fuzzy sets of Type 2 employing various parameter-identification formalisms.
Modeling Uncertainty with Fuzzy Logic is an important contribution to the development of a better understanding of how to deal with second-order uncertainty. The issue of second-order uncertainty has received relatively little attention so far, but its intrinsic importance is certain to make it an object of growing attention in coming years. Through their work, the authors have opened a door to wide-ranging applications. They deserve our compliments and congratulations.Fuzzy Sets and Systems
Improved Fuzzy Clustering
Fuzzy Functions Approach
Modeling Uncertainty with Improved Fuzzy Functions
Experiments
Conclusions and Future Work