Fuzzy Techniques of Pattern Recognition in Risk and Claim Classification

Introduction In 1961, Ellsberg presented the following paradox. An experiment was designed with two urns, each containing 100 balls, of which the first one was known to contain 50 red balls and 50 black balls, while no further information was given about the contents of the other urn. If asked to bet on the color of a ball drawn from one of the urns, most people were found indifferent as to which color they would choose no matter whether the ball was drawn from the first or the second urn. On the other hand, Ellsberg found that if people were asked which urn they would prefer to use for betting on either color, they consistently favored the first urn (no matter what color they were asked to bet on). What seems to be present in this experiment is the participants' perception of uncertainty. When we say "uncertainty," the usual association is with "probability." The Ellsberg paradox illustrates that some other form of uncertainty can indeed exist. Probability theory provides no basis for the outcome of the Ellsberg experiment. Klir and Folger (1988) analyze the semantic context of the term "uncertain" and arrive at the conclusion that there are two main types of uncertainty, captured by the terms "vagueness" and "ambiguity." Vagueness is associated with the difficulty of making sharp or precise distinctions among objects. "Ambiguity" is caused by situations where the choice between two or more alternatives is unspecified. The basic set of axioms of probability theory originating from Kolmogorov, rests on the assumption that the outcome of a random event can be observed and identified with precision. Any vagueness of observation is considered negligible, or not significant to the construction of the theoretical model. Yet one cannot escape the conclusion that forms of uncertainty represented by vagueness of observations, human perceptions, and interpretations, are missing from probabilistic models, which equate uncertainty with randomness (i.e., a sophisticated version of ambiguity). Several reasons may exist for wanting to search for models of a form of uncertainty other than randomness. One is that vagueness is unavoidable. Given imprecision of natural language, or human perception of the phenomena observed, vagueness becomes a major factor in any attempt to model or predict the course of events. But there is more. When the phenomena observed become so complex that exact measurement involving all features considered significant would be impossible, or longer than economically feasible for study, mathematical precision is often abandoned in favor of more workable simple, but vague, "common sense" models. Thus, complexity of the problem may be another cause of vagueness. These reasons were the driving force behind the development of the fuzzy set theory (FST). This field of applied mathematics has become a dynamic research and applications field, with success stories ranging from a fuzzy logic rice cooker to an artificial intelligence in control of Japan's Sendai subway system. The main idea of fuzzy set theory is to propose a model of uncertainty different from that given by probability, precisely because a different form of uncertainty is being modeled. Fuzzy set theory was created in Zadeh's (1965) historic article. To present this basic idea, recall that a characteristic function of a subset E of a universe of discourse U is defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] In other words, the characteristic function describes the membership of an element X in a set E. It equals one if X is a member of E, and zero otherwise. Zadeh challenged the idea that membership in all sets behaves in the manner described above. One example would be the set of "tall people." We consistently talk about the set of "tall people," yet understand that the concept used is not precise. A person who is 5'11" is tall only to a certain degree, and yet such a person is not "not tall. …