Plenary lecture VII: probability measures of fuzzy events and linguistic fuzzy modelling - forms expressing randomness and imprecision
Anna Walaszek-Babiszewska · International Conference on Software Engineering · 2008
Statistical methods have a long history of applications to data analysis and modelling in many fields of human activities, as financial markets, business, optimal control. Zadeh's theory of fuzzy systems, soft computing and computing with words gave possibilities of utilising many approaches to modelling, taking into account different types of uncertainties, especially, human knowledge and perception expressed imprecisely in linguistic categories. The sets of numerical data, collected in many real systems, and expert's experiences would be very useful to modelling both randomness and imprecision. Starting with the reminding the basic notions of probability of fuzzy events, 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. We formulate also a stochastic process with fuzzy states. The next, we try to modelling randomness and imprecision, using linguistic fuzzy models with weights of rules. A structure of the fuzzy model is predefined at the beginning of the task. Probability of fuzzy events has been used to formulate probabilities of the occurrence linguistic values of input and output variables in a product-space. Marginal probabilities of linguistic values of input variables are the weights of file rules. Conditional probabilities of linguistic values of the output variable are the weights in the consequence of elementary rules. The inference procedures are proposed and exemplary calculations are presented.