Fuzzy logic and probability theory are the most powerful tools to overcome the imperfection (see Fig.1). Fuzzy logic is mainly responsible for representation and processing of vague data (ill-defined, fuzzy). Probability theory is mainly responsible for representation and processing of uncertainty (randomness).
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Fig.1. Imperfection and theories to handle it.
Following table clarifies the differences between the two theories. 数据挖掘实验室
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Probability Measure 数据挖掘论坛
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Membership Function 数据挖掘交友
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Calculates the probability that an
ill-known variable x ranging on U hits the well-known set A
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Calculates the membership of a
well-known variable x ranging on U hits the ill-known set A
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Before an event happens 数据挖掘交友
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After it happened 数据挖掘论坛
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Measure Theory 数据挖掘实验室
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Set Theory
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Domain is 2U (Boolean Algebra)
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Domain is [0,1]U (Cannot be a Boolean Algebra)
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A Bridge
Consider the following statements:
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It is probable that it will rain a lot tomorrow.
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It is probable that the image will be very dark. 数据挖掘论坛
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It is probable that her new friend is handsome. 数据挖掘论坛
In such cases (which are very usual in pattern recognition, for instance), we are interested in probability of an event that cannot be defined exactly. Therefore, the only sophisticated way is to calculate the probability of a fuzzy event represented by a fuzzy set:
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Probability space
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Membership function 数据挖掘实验室
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The probability of the fuzzy event F 数据挖掘实验室
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For more details see following papers/books:
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Probability measures of fuzzy events, L.A.Zadeh, Journal Math. Anal. Appl., vol 23, pp. 421-427, 1968
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Fuzzy Sets as a basis for a theory of possibility, L.A. Zadeh, Fuzzy Sets and Systems, vol. 1, pp. 3-28, 1978
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Possibility Theory, D.Dubois, H. Prade, Plenum Press, 1988 数据挖掘论坛
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Fuzzy sets and probability : Misunderstandings, bridges and gaps, D.Dubois, H. Prade, Proc. of the Second IEEE Inter. Conf. on Fuzzy Systems, volume 2, pp. 1059-1068, 1993 数据挖掘工具
