Fuzzy Knowledge Representation Using Probability Measures of Fuzzy Events

The concepts of Soft Computing introduced by Lotfi A. Zadeh in 1991 has integrated different methodologies and approaches, as: fuzzy set theory, fuzzy logic, approximate reasoning, linguistic expression of knowledge, probabilistic reasoning, and others for solving problems of complex systems in the way similar to human perception, recognition and solving problem methods. Linguistic fuzzy modelling gives the formal, mathematical instruments for expressing human knowledge described in natural language. Probability of fuzzy meanings of linguistic variables determines a frequency of the occurrence the imprecisely expressed events. This work presents the methods of applications linguistic modelling and probability measures of fuzzy events for creating models compatible to the features of real systems, and more flexible than traditional rule-based models derived from linguistic knowledge. In Section 2. we remind the notions of a linguistic variable and a probability of fuzzy events, formulated by Zadeh, which have become fundamental for the development of fuzzy systems. We define probability distributions of a linguistic variable and a linguistic vector as well as a mean fuzzy value (a mean fuzzy set) of the linguistic variable. The conditional probability of fuzzy events will be the base for the inference procedure. Section 3. shows an exemplary probabilistic modelling for the characteristics representing features of particles in a certain population, formulated in fuzzy categories. The created knowledge representation states a collection of weighted rules (Section 4.) Weights of rules represent probabilities of fuzzy events of input and output system variables. Construction of fuzzy models is presented for different stochastic systems. The weights are involved in the inference procedures.