Guide Modeling Uncertainty with Fuzzy Logic: With Recent Theory and Applications

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Why We Need Type-2 Fuzzy Logic Systems

The output processor for a type-2 FLS has two components to it. First, type-2 fuzzy sets are transformed into type-1 fuzzy sets by means of type reduction. Then the type-reduced set is transformed into a number by means of defuzzifcation. Figure 1.


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Fuzzy logic system. Type-1 FLSs cannot directly handle rule uncertainties because they use type-1 fuzzy sets that are certain.


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  8. Type-2 FLSs, on the other hand, are very useful in circumstances in which it is difficult to determine an exact membership function for a fuzzy set; hence, they can be used to handle rule uncertainties and even measurement uncertainties. What is this new direction, and why is it important? To make the answers to these questions as clear as possible, let us briefly digress to review some things that are, no doubt, familiar.

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    Probability theory is used to model random uncertainty, and within that theory we begin with a probability density function pdf that embodies total information about random uncertainties. In most practical real-world applications, it is impossible to know or determine the pdf, so we fall back on the fact that a pdf is completely characterized by all of its moments if they exist. For most pdfs, an infinite number of moments are required.

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    Of course, in practice it is not possible to determine an infinite number of moments; instead, we compute as many moments as we believe are necessary to extract as much information as possible from the data. At the very least, we use two moments, the mean and the variance. In some cases, we even use higher-than-second-order moments. To use just the first-order moments would not be very useful because random uncertainty requires an understanding of dispersion about the mean, and this information is provided by the variance.


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    So, our accepted probabilistic modeling of random uncertainty focuses, to a large extent, on methods that use at least the first two moments of a pdf. For example, that is why designs based on minimizing a mean-squared error are so popular.

    Should we expect any less of an FLS for rule uncertainties or any other types of uncertainties? I do not want to get stuck in the quagmire about the equivalence between subjective probability and type-1 fuzzy sets; our "analogy" between the defuzzified output of an FLS and the mean of a pdf is meant to be just that and nothing more.

    Why We Need Type-2 Fuzzy Logic Systems | Why We Need Type-2 Fuzzy Logic Systems | InformIT

    Type-2 FL provides this measure of dispersion and seems to be as fundamental to the design of systems that include linguistic or numerical uncertainties that translate into rule or input uncertainties as variance is to the mean. Just as random uncertainties flow through a system and their effects can be evaluated using the mean and the variance, linguistic and random uncertainties flow through a type-2 FLS, and their effects can be evaluated using the defuzzified output and the type-reduced output of that system.

    Just as the variance provides a measure of dispersion about the mean and is often used in confidence intervals, the type-reduced output can be interpreted as providing a measure of dispersion about the defuzzified output.

    It can be thought of as or related to a linguistic confidence interval. Apart from the concept, two developed techniques one based on triangulated networks, one based on bitmap models are presented along with some of the operators. An overview of application fields is provided to illustrate where and how the techniques can be used. Unable to display preview. Download preview PDF. Skip to main content.

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