
IEEE TFS: Abstracts of Published Papers, vol. 2, no. 1
The efficacy of fuzzy representations of uncertainty
Advocates of the theory of fuzzy sets as a system for representing uncertainty have based their case on five basic arguments. These are: 1) the reality hypothesis, which holds that imprecision is an inherent property of the world external to an observer; 2) the subjectivity hypothesis, which holds that probability is an exclusively objective measure of uncertainty, and that therefore subjective uncertainty can only be represented with fuzzy sets; 3) the behaviorist hypothesis, which claims that uncertainty systems should emulate rather than prescribe human behavior in the face of uncertainty; 4) the "probability as fiction" hypothesis, which claims that probability does not comprise a field of study in its own right; and 5) the superset hypothesis, which holds that fuzzy set theory includes probability as a special case and thus provides a richer uncertainty modeling environment. We discuss and criticize all five. We then criticize the argument that fuzziness represents a type of uncertainty distinct from probability, and also the inordinate complexity of fuzzy methods. We present a method for assessing the efficacy of fuzzy representations of uncertainty and apply this method in three examples.
Fuzzy setsa convenient fiction for modeling vagueness and possibility
This paper is a reply to Laviolette and Seaman's critical discussion of fuzzy set theory. Rather than questioning the interest of the Bayesian approach to uncertainty, some reasons why Bayesian find the idea of a fuzzy set not palatable are laid bare. Some links between fuzzy sets and probability that Laviolette and Seaman seem not to be aware of are pointed out. These links suggest that, contrary to the claim sometimes found in the literature, probability theory is not a special case of fuzzy set theory. The major objection to Laviolette and Seaman is that they found their critique on as very limited view of fuzzy sets, including debatable papers, while they fail to account for significant works pertaining to axiomatic derivation of fuzzy set connectives, possibility theory, fuzzy random variables, among others.
Interpretative versus prescriptive fuzzy set theory
Both traditional fuzzy set theory and the theory of subjective probabilities postulate their formulas. The former because it does not accept a probabilistic interpretation of grades of membership, the latter because it makes no connection between subjective probabilities and relative frequencies. The interpretational theory of the TEE model ascribes a welldefined meaning to a membership value mu /sub lambda / in terms of probabilities which are limits of frequencies. mu /sub lambda / is interpreted as the estimate by the subject of the probability that a given object would be assigned the label X in an everyday situation of uncertainty. In contrast to the maxmin formulas used in many applications of fuzzy sets, the postulated "summationto1" formula of all fuzzy clustering algorithms follows from the interpretative TEE model. This agrees with the author's view that fuzzy set theory should be defined as a theory which allows partial membership values of an object in a class or cluster, not as a theory which uses specific mathematical operators. The operators must be derived from the welldefined interpretation of partial membership values.
On the alleged superiority of probabilistic representation of
uncertainty
The author identifies some points of his agreement or disagreement with the position taken by Laviolette and Seaman (1994) in their debate paper. In particular, the author argues that some statements regarding the superiority of probabilistic representation of uncertainty, which are made by Laviolette and Seaman in their paper, are untenable.
The probability monopoly
Probability is a very special case of fuzziness. It always faces two limits. First, it works with bivalent sets A. Second, probability measures need small infinities. A probability measure maps the sets in a sigmaalgebra to the unit interval (O, 1). Fuzzy theory challenges the probability monopoly. Probabilists have attacked it with gusto to keep their monopoly status, to have, the only uncertainty theory in the unit interval (O, 1). But the fuzzy math is sound. Its world view of shades of gray has a deep intuitive ring. And the new fuzzy products have come into their own in the marketplace.
Vagueness and Bayesian probability
This paper is a response to Michael Laviolette and John W. Seaman Jr.'s ( ibid. vol.2, no.1, p.4 (1994)) position paper "The efficacy of fuzzy representations of uncertainty," which criticizes fuzzy representations of uncertainty, and suggests that Bayesian probability can do better. The commenter argues that the author's make some misleading comments about Bayesian probability, and he briefly discusses the problem of giving a satisfactory interpretation of membership functions.
Comments on "The efficacy of fuzzy representations of uncertainty"
The author expresses agreement with the original paper, by M. Laviolette and J.W. Seaman (ibid., p.415), saying that probability is the only satisfactory measure of one's personal uncertainty about the world. The reason for this emphatic statement is that there exist several axiom systems that produce, as theorems, the rules of probability. In particular, the author criticises the use of fuzzy logic as an alternative.
Unity and diversity of fuzzinessfrom a probability viewpoint
Like the fields of probability and statistics, fuzzy set theory is characterized by a variety of viewpoints. Adherents of fuzzy methods, however, consistently maintain that probability is not necessarily the optimal representation of uncertainty. We rebut this view.
Comments on "Editorial: fuzzy modelswhat are they and why?"
In the editorial by J.C. Bezdek (ibid., p.1), an example is presented to demonstrate differences between fuzzy membership and probability. The authors argue that probability can be used in a way much more closely analogous to this use of fuzzy membership, weakening the argument for the latter.
The thirsty traveler visits Gamont: a rejoinder to "Comments on fuzzy setswhat are they and why?"
Replying to the comments of W.H. Woodall and R.E. Davis (ibid., p.43) on the author's editorial (ibid., p.1), the authors present two illustrations to show that they do not have a poor opinion of probability but that they believe fuzzy membership to be more useful in some cases.
Reinforcement structure/parameter learning for neuralnetworkbased fuzzy
logic control systems
This paper proposes a reinforcement neuralnetworkbased fuzzy logic neuralnetworkbased fuzzy logic controllers (NNFLC's), each of which is a connectionist model with a feedforward multilayered network developed for the realization of a fuzzy logic controller. One NNFLC performs as a fuzzy predictor, and the other as a fuzzy controller. Using the temporal difference prediction method, the fuzzy predictor can predict the external reinforcement signal and provide a more informative internal reinforcement signal to the fuzzy controller. The fuzzy controller performs a stochastic exploratory algorithm to adapt itself according to the internal reinforcement signal. During the learning process, both structure learning and parameter learning are performed simultaneously in the two NNFLC's using the fuzzy similarity measure. The proposed automatically and dynamically through a reward/penalty signal or through very simple fuzzy information feedback such as "high," "too high," "low," environment, where obtaining exact training data is expensive. It also preserves the advantages of the original NNFLC, such as the ability to find proper network structure and parameters simultaneously and dynamically and to avoid the rulematching time of the inference engine. Computer simulations were conducted to illustrate its performance and applicability.
Rulebase structure identification in an adaptivenetworkbased fuzzy
inference system
We summarize Jang's architecture of employing an adaptive network and the Kalman filtering algorithm to identify the system parameters. Given a surface structure, the adaptively adjusted inference system performs well on a number of interpolation problems. We generalize Jang's basic model so that it can be used to solve classification problems by employing parameterized tnorms. We also enhance the model to include weights of importance so that feature selection becomes a component of the modeling scheme. Next, we discuss two ways of identifying system structures based on Jang's architecture: the topdown approach, and the bottomup approach. We introduce a data structure, called a fuzzy binary boxtree, to organize rules so that the rule base can be matched against input signals with logarithmic efficiency. To preserve the advantage of parallel processing assumed in fuzzy rulebased inference systems, we give a parallel algorithm for pattern matching with a linear speedup. Moreover, as we consider the communication and storage cost of an interpolation model. We propose a rule combination mechanism to build a simplified version of the original rule base according to a given focus set. This scheme can be used in various situations of pattern representation or data compression, such as in image coding or in hierarchical pattern recognition.
A multiregion fuzzy logic controller for nonlinear process control
Although a fuzzy logic controller is generally nonlinear, a PItype fuzzy controller that uses only control error and change in control error is not able to detect the process nonlinearity and make a control move accordingly. In this paper, a multiregion fuzzy logic controller is proposed for nonlinear process control. Based on prior knowledge, the process to be controlled is divided into fuzzy regions such as highgain, lowgain, largetimeconstant, and smalltimeconstant. Then a fuzzy controller is designed based on the regional information. Using an auxiliary process variable to detect the process operating regions, the resulting multiregion fuzzy logic controller can give satisfactory performance in all regions. Rule combination and controller tuning are discussed. Application of the controller to pH control is demonstrated.
Fuzzy rulebased networks for control
The authors present a method for learning fuzzy logic membership functions and rules to approximate a numerical function from a set of examples of the function's independent variables and the resulting function value. This method uses a threestep approach to building a complete function approximation system: first, learning the membership functions and creating a cellbased rule representation; second, simplifying the cellbased rules using an informationtheoretic approach for induction of rules from discretevalued data; and, finally, constructing a computational (neural) network to compute the function value given its independent variables. This function approximation system is demonstrated with a simple control example: learning the truck and trailer backerupper control system. These abstracts are posted in order to accelerate dissemination of evolving Fuzzy Systems information. The abstracts are from papers published in the IEEE Transactions on Fuzzy Systems (TFS). 


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