Using Probability and Fuzzy Set Theory for Reliabiltiy Assessment

Uncertainties enter into a complex problem from many sources: variability, errors, and lack of knowledge. A fundamental question arises in how to characterize the various kinds of uncertainty and then to combine them within a problem such as the verification and validation of a structural dynamics computer model, reliability of a dynamic system, or a complex decision problem. Our aim is to explore how probability theory and fuzzy set theory can work together, so that uncertainty of outcomes and the uncertainty from imprecision can be treated in a unified and consistent manner. The theoretical development required for linking these two uncertainties will be summarized, and application of a linkage between the two will be presented. The application involves estimating the performance or reliability of a system and serves to illustrate how the linkage between the probability and fuzzy set theories is accomplished through the use of Bayes Theorem. INTRODUCTION: TYPES OF UNCERTAINTY In engineering and physical sciences, uncertainties are associated with phenomena such as random noise, measurement error and uncontrollable variation. Such uncertainties are known as aleatory. Probability theory has become a fundamental theory for characterizing aleatoric uncertainty and has been adopted by physicists and many engineering communities as evidenced by publications of the AIAA, SEM, ASME, IEEE, etc. With aleatoric uncertainty, the claim is that uncertainty cannot be further reduced or eliminated by additional information (data or knowledge). In communities such as those involved in eliciting knowledge from experts and computational sciences, uncertainty is interpreted as an absence of complete knowledge. Such uncertainty is called epistemic, and the claim here is that epistemic uncertainty can be reduced or eliminated by increasing the available information by increasing the sample size or further elicitation. Regrettably, some members of this community have also claimed that probability theory is inadequate for dealing with epistemic uncertainty. To those involved with complex decision problems like Probabilistic Risk Assessment (PRA), the term “uncertainty” embodies both sources: aleatoric and epistemic. To them, uncertainty is the lack knowledge, irrespective of its source. This point of view also encompasses uncertainty caused by errors, mistakes and miscalculations. The absence of a distinction between aleatoric and epistemic uncertainty is also subscribed to by modern subjective Bayesians, who also attribute all forms of uncertainty to a lack of knowledge, whatever its source. It is important to note that both the aleatoric and epistemic uncertainties pertain to an uncertainty about the outcome of an event. The event could be one that has yet to occur (so that its outcome is unknown to all) or one that has already occurred but whose outcome is unknown to the assessor of the uncertainty and yet known to others. Of late, mainly due to the work of Zadeh [1], another type of uncertainty has been brought to the forefront: the uncertainty of classification of any particular outcome. For example, suppose that a 10sided die is thrown with an outcome of 7. Is 7 classified as a medium number or a large number? Consequently, we are uncertain about the classification of 7 into the set of medium or large numbers. Thus in throwing the 10-sided die, two types of uncertainty arise: what will be the outcome and how will the outcome be classified? One may raise the question as to why bother with the uncertainty of classification? However Zadeh and his co-workers have made the case that uncertainty of classification is germane in the context of machine learning and natural language processing. Booker et al. [2] have also made the claim that uncertainty of classification arises in the context of eliciting knowledge from experts. In order to deal with the issue of a sets of medium and large numbers—sets whose boundaries are not well (crisply) defined)—Zadeh introduced the concept of fuzzy sets. While the concept previously appeared in Black [3], Zadeh introduced a calculus for operations with fuzzy sets such as unions, intersections, complements, etc. As a construct in mathematics, the theory of fuzzy sets has a legitimate role to play. However, Zadeh has also claimed that probability theory is inadequate when one is confronted with uncertainty due to vagueness and imprecision, like that defined for classification of medium numbers. Instead he has introduced an alternative to probability theory, possibility theory [4]. This theory, though embraced by many, lacks the behavioristic axiomatic foundation of probability theory. Because lack of knowledge is the cause of both uncertainties (outcome and classification), our view is that probability and its calculus should be able to work in concert with fuzzy set theory (but not possibility theory). This point of view has also been echoed by Zadeh [5], where he says “probability must be used in concert with fuzzy logic to enhance its effectiveness. In this perspective, probability theory and fuzzy logic are complementary rather than competitive.” The aim of this paper is to demonstrate how this philosophy of linkage can be coherently put to work so that uncertainty of outcome and uncertainty of classification can be treated in a unified and consistent manner. Bayes theorem and the notion of the likelihood function enable us to accomplish this linkage. The rest of this paper is organized as follows. The next section gives an overview of probability theory and the theory of fuzzy sets. Following that, is a summary of linking the two—the details are not given here. The reader is referred to Singpurwalla and Booker [6]. The paper concludes with an example from reliability showing how the linked theories can be productively put to work. PROBABILITY AND FUZZY SET THEORY Probability Theory Probability theory can be described as a calculus (an algebra) for determining the uncertainty of outcomes of an experiment or event, E. Let a universe or sample space, Ω, represent the set of all possible outcomes of E. Probability theory does not tell us how to specify Ω, but Ω may be countable and Ω = ∅. Let F denote the set of all subsets belonging to Ω. Consider subsets A and B where A, B ∈ F; then (A∪B) ∈ F and (A∩B) ∈ F. The subsets A and B are well defined (or crisp sets). There is no ambiguity in determining whether an outcome of E belongs to A or its complement, A. Because E is going to be performed, we are uncertain about an outcome of E, say. ω. Let P(A) describe our uncertainty about the outcome, where 0 ≤ P(A) ≤ 1, and in the theory of personal probability, P(A) represents our bet that ω ∈ A. P(A) is known as the probability of event A. The bet is twosided on the occurrence or non-occurrence of event A, and the bet will be unambiguously settled when E is performed