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Uncertainty in Fuzzy Logic Systems

Fuzzy logic systems expert Jerry Mendel categorizes four kinds of uncertainties that can occur in a rule-based fuzzy logic system, relates these to three general kinds of uncertainty, and explains why type-2 fuzzy logic is needed to handle them.
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Uncertainty: General Discussions

Uncertainty comes in many guises and is independent of the kind of fuzzy logic (FL), or any kind of methodology, one uses to handle it. One of the best sources for general discussions about uncertainty is the book Uncertainty-Based Information, by Klir and Wierman1. Professor Klir and his students (for example, Klir and T.A. Folger2) have been focusing on uncertainty topics since the 1980s, and this book represents an amalgamation and sharpening of the many ideas from their works. Regarding the occurrence of uncertainty, Klir and Wierman state:

    When dealing with real-world problems, we can rarely avoid uncertainty. At the empirical level, uncertainty is an inseparable companion of almost any measurement, resulting from a combination of inevitable measurement errors and resolution limits of measuring instruments. At the cognitive level, it emerges from the vagueness and ambiguity inherent in natural language. At the social level, uncertainty has even strategic uses and it is often created and maintained by people for different purposes (privacy, secrecy, propriety).

Regarding the causes of uncertainty, they state:

    Uncertainty involved in any problem-solving situation is a result of some information deficiency. Information (pertaining to the model within which the situation is conceptualized) may be incomplete, fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. In general, these various information deficiencies may result in different types of uncertainty.

Regarding the nature of uncertainty, they state:

    Three types of uncertainty are now recognized … fuzziness (or vagueness), which results from the imprecise boundaries of fuzzy sets; nonspecificity (or imprecision), which is connected with sizes (cardinalities) of relevant sets of alternatives; and strife (or discord), which expresses conflicts among the various sets of alternatives.

The word imprecision is frequently used for both fuzziness and nonspecificity. In a private correspondence with me, Klir said:

    When it [imprecision] is used for nonspecificity, it refers to information-based imprecision; here, uncertainty results from information deficiency. When it is used for fuzziness, it [imprecision] refers to linguistic imprecision. … This ambiguity can be avoided by distinguishing between information-based imprecision (equivalent to nonspecificity) and linguistic imprecision (equivalent to fuzziness).

Klir and Wierman divide the three types of uncertainty into two major classes, fuzziness and ambiguity, where ambiguity (one-to-many relationships) includes nonspecificity and strife.

Another source for some general discussions about uncertainty is H.R. Berenji3, who states, in agreement with Klir and Wierman, that "uncertainty stems from lack of complete information." He also states, "Uncertainty may also reflect incompleteness, imprecision, missing information, or randomness in data and a process."

Taken out of context, uncertainty is relatively abstract because of its many varieties; however, as we demonstrate later, taken in the context of a fuzzy logic system, uncertainty is easy to understand. First, however, we provide a high-level overview of what a rule-based fuzzy logic system is.

1. G.J. Klir and M.J. Wierman. Uncertainty-Based Information. Heidelberg, Germany: Physica-Verlag, 1998.

2. G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Englewood Cliffs, NJ: Prentice Hall, 1988.

3. H.R. Berenji. "Treatment of Uncertainty in Artificial Intelligence," in Machine Intelligence and Autonomy Aerospace Systems. Washington, D.C.: AIAA, 1988.

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