Method for Inversion of Fuzzy Relation

In this paper we investigate how the observation of symptoms which do not completely match a modeled fault can be used to find the most likely fault – and the degree to which this fault occurs. We start out by setting up fuzzy causal diagrams and then show how with the use of a proper operator the arcs of the causal diagram can be reversed. We introduce a graphical representation for fuzzy belief nets (FBN) and show how both AND and OR connected antecedents and consequents of rules can be accommodated. The paper concludes with an ill ustrative diagnostic example. Introduction While “ forward” reasoning from cause to effect is generally understood and various means exist to solve this problem using deterministic, probabili stic, or fuzzy means, solutions developed so far for “backward” reasoning from symptom to cause have various shortcomings. In the fuzzy domain Sanchez (1976) first investigated the solution to the inversion of the fuzzy relation A R B = (with fault vector A, symptom vector B, and relational matrix R) which allowed him to find a least upper bound with the help of an operator “ a ” . Mizumoto and Zimmermann (1982) approached the inversion problem by introducing several appropriate relational operators which allowed to express the inversion as a modus tollens of the form B A R x x ' ' = ° Fuzzy fault trees were used by Gmytrasiewicz et al. (1990) and Ulieru (1994). The latter manually created a diagnostic relevant network with the help of experiential knowledge and first principles. Using fuzzy modus tollens for validation, the pair with highest similarity for both methods is found as the solution. Engemann et al. proposed a methodology for decision making under uncertainty, integrating ordered weighted averaging aggregation operators and Dempster-Shafer belief structure which is used as a framework for representing information a decision maker has regarding relevant events (1996). Hisdal (1978) described conditional possibilit y extended by Dubois and Prade to a possibili stic version of Bayes' The author was with UC Berkeley at the time of this research theorem (1992). A similar solution is suggested by Kosko, who uses fuzzy supersethood (1986). This approach in turn is used by Dalton (1993) applying parsimonious covering theory (Peng and Reggia, 1990) to fuzzy logic, the former using a similarity measure assuming crisp symptoms. Shortcomings of these approaches are that they do not always provide a solution and if there is one it may by bounded by a range too wide to be useful for decision making, or inadequacies due to the nature of the operators used. In the approach introduced here, knowledge is taken from fuzzy cause-effect relationships modeled via causal diagrams. “Backward” reasoning becomes possible with the introduction of a proper fuzzy measure. We then show how this system can be understood as a step towards fuzzy belief nets. The method provides a way to incorporate the degree to which a symptom is observed into the reasoning apparatus. Background on Probabilistic Belief Nets Probabili stic belief nets and influence diagrams were developed to facilit ate automating the modeling of complex decision problems involving uncertainty using a compact graphical framework for representing the interrelationships between the variables involved in the problem under consideration (Mill er et al., 1976; Olmstead, 1984; Shachter, 1984; Agogino and A. Rege, 1987). They can be used to solve decision and probabili stic inference problems. At the topological level an influence diagram is an acyclic directed network with nodes representing variables critical to the problem and the arcs representing their interrelationships. Arcs going into nodes represent conditional influence and can be reversed through legal topological transformations on the diagram according to Bayes’ rule. Jain and Agogino developed Bayesian fuzzy probabiliti es and arithmetic operations that are consistent with Bayes’ rule and retain closure of the required properties (Jain and Agogino, 1990). Application of the arithmetic operations results in a solution in which the mean of the fuzzy function is equivalent to the point estimate obtained by using conventional Bayesian probabilit y. The resulting fuzzy function around the mean can be used for stochastic sensitivity analysis; its interpretation depends on the application. Method for Inversion of Fuzzy Relation To achieve the inversion of the cause-effect relation, a fuzzy measure is introduced which will assign a degree of similarity with each possible failure. Both sudden and gradual malfunctions can be treated using slightly different operators. We build on the notion of abduction using a fuzzy scheme. Inference in abduction looks at a general rule and a specific result. Out of a large number of hypothetical solutions one specific case is chosen to be most likely. In binary logic both rule and symptom are evaluated with respect to their truth and only when both are found to be true the rule can be hypothesized. In many valued logic, both rules and results are always true to some extent and therefore all rules can be hypothesized to some degree. Therefore, we have to come up with a way to find a method which identifies the most likely hypothesis. To begin, failure-symptom relationships are expressed in fuzzy causal diagrams as displayed in Fig. 1 (to avoid overcrowding of the graph, links with strength zero were omitted) where the fn represent the failures and the sm stand for the symptoms. This means that a fault fn causes a number of symptoms sm to occur to some extent. That is, some symptoms are produced more strongly than others. Other faults may cause the same symptoms but with a different degree of strength. 0.9 0.3 0.9 0.4 0.2 0.3 0.3 0.3 0.7 0.6 0.2 f f f f s s s s s s s 1 2 3 4 5 7 6 2 1 3 4 • • • •