<noscript id="y8mss"><tr id="y8mss"></tr></noscript><xmp id="y8mss">
<xmp id="y8mss"><menu id="y8mss"></menu>
  • <xmp id="y8mss"><menu id="y8mss"></menu><menu id="y8mss"><nav id="y8mss"></nav></menu>
    <xmp id="y8mss"><nav id="y8mss"></nav>
  • <nav id="y8mss"></nav>
    <xmp id="y8mss"><nav id="y8mss"></nav>
  • <nav id="y8mss"></nav>

    Fuzzy Logic & Probability Theory: 模糊邏輯與概率論
    Clarification towards Building a Bridge

    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).


    Fig.1. Imperfection and theories to handle it.

    Following table clarifies the differences between the two theories.

    Probability Measure

    Membership Function

    Calculates the probability that an
    ill-known variable x ranging on U hits the well-known set A

    Calculates the membership of a
    well-known variable x ranging on U hits the ill-known set A

    Before an event happens

    After it happened

    Measure Theory

    Set Theory

    Domain is 2U (Boolean Algebra)

    Domain is [0,1]U (Cannot be a Boolean Algebra)

    A Bridge

    Consider the following statements:

    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:

    Probability space

    Membership function

    The probability of the fuzzy event F

    For more details see following papers/books:

    1. Probability measures of fuzzy events, L.A.Zadeh, Journal Math. Anal. Appl., vol 23, pp. 421-427, 1968

    2. Fuzzy Sets as a basis for a theory of possibility, L.A. Zadeh, Fuzzy Sets and Systems, vol. 1, pp. 3-28, 1978

    3. Possibility Theory, D.Dubois, H. Prade, Plenum Press, 1988

    4. 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

    www.dd244.com