Full Description
"Statistical Modeling, Analysis and Management of Fuzzy Data," or SMFD for short, is an important contribution to a better understanding of a basic issue -an issue which has been controversial, and still is though to a lesser degree. In substance, the issue is: are fuzziness and randomness distinct or coextensive facets of uncertainty? Are the theories of fuzziness and random ness competitive or complementary? In SMFD, these and related issues are addressed with rigor, authority and insight by prominent contributors drawn, in the main, from probability theory, fuzzy set theory and data analysis com munities. First, a historical perspective. The almost simultaneous births -close to half a century ago-of statistically-based information theory and cybernetics were two major events which marked the beginning of the steep ascent of probability theory and statistics in visibility, influence and importance. I was a student when information theory and cybernetics were born, and what is etched in my memory are the fascinating lectures by Shannon and Wiener in which they sketched their visions of the coming era of machine intelligence and automation of reasoning and decision processes. What I heard in those lectures inspired one of my first papers (1950) "An Extension of Wiener's Theory of Prediction," and led to my life-long interest in probability theory and its applications to information processing, decision analysis and control.
Contents
1. Fuzziness and Randomness.- Fuzziness and randomness.- 2. Fuzzy-Valued Random Elements.- On the variance of random fuzzy variables.- f-inequality indices for fuzzy random variables.- Traditional techniques to prove some limit theorems for fuzzy random variables.- Convergence in graph for fuzzy valued martingales and smartingales.- Remarks on Korovkin-type approximation of fuzzy random variables.- Several notions of differentiability for fuzzy set-valued mappings.- 3. Possibility, Probability and Fuzzy Measures.- Average level of a fuzzy set.- Second order possibility measure induced by a fuzzy random variable.- Measure extension from meet-systems and falling measures representation.- The structure of fuzzy measure families induced by upper and lower probabilities.- Statistical classes and fuzzy set theoretical classification of probability distributions.- 4. Statistics and Fuzzy Data Analysis.- Statistics with one-dimensional fuzzy data.- Testing fuzzy hypotheses with vague data.- Possibilistic interpretation of fuzzy statistical tests.- Possibilistic regression analysis.- Linear regression in a fuzzy context. The least square method.- Linear regression with random fuzzy observations.
