Statistical Modeling, Analysis and Management of Fuzzy Data

1. Front Matter

   Pages I-XIV

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2. Fuzziness and Randomness

   1. Front Matter

      Pages 1-1

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   2. Fuzziness and randomness

      • Irwin R. Goodman, Hung T. Nguyen

      Pages 3-21

3. Fuzzy-Valued Random Elements

   1. Front Matter

      Pages 23-23

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   2. On the variance of random fuzzy variables

      • Ralf Körner, Wolfgang Näther

      Pages 25-42

   3. f-inequality indices for fuzzy random variables

      • María Asunción Lubiano, María Ángeles Gil

      Pages 43-63

   4. Traditional techniques to prove some limit theorems for fuzzy random variables

      • Ana Colubi

      Pages 64-71

   5. Convergence in graph for fuzzy valued martingales and smartingales

      • Shoumei Li, Yukio Ogura

      Pages 72-89

   6. Remarks on Korovkin-type approximation of fuzzy random variables

      • Pedro Terán, Miguel López-Díaz

      Pages 90-103

   7. Several notions of differentiability for fuzzy set-valued mappings

      • Luis J. Rodríguez-Muñiz

      Pages 104-116

4. Possibility, Probability and Fuzzy Measures

   1. Front Matter

      Pages 117-117

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   2. Average level of a fuzzy set

      • Dan A. Ralescu

      Pages 119-126

   3. Second order possibility measure induced by a fuzzy random variable

      • Inés Couso, Susana Montes, Pedro Gil

      Pages 127-144

   4. Measure extension from meet-systems and falling measures representation

      • P. Z. Wang, Y. C. Chen, B. T. Low

      Pages 145-159

   5. The structure of fuzzy measure families induced by upper and lower probabilities

      • Andrew G. Bronevich, Alexander N. Karkishchenko

      Pages 160-172

   6. Statistical classes and fuzzy set theoretical classification of probability distributions

      • Andrew G. Bronevich, Alexander N. Karkishchenko

      Pages 173-195

5. Statistics and Fuzzy Data Analysis

   1. Front Matter

      Pages 197-197

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   2. Statistics with one-dimensional fuzzy data

      • Reinhard Viertl

      Pages 199-212

   3. Testing fuzzy hypotheses with vague data

      • Przemysław Grzegorzewski

      Pages 213-225

   4. Possibilistic interpretation of fuzzy statistical tests

      • Olgierd Hryniewicz

      Pages 226-238

"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.

• Extension
• Probability distribution
• Probability theory
• Random variable
• Variance
• artificial intelligence
• calculus
• data analysis
• fuzziness
• fuzzy set
• linear regression
• modeling
• perception
• set theory
• statistics

• Dipartimento di Informatica e Sistemistica, Università degli Studi di Pavia, Pavia, Italy

  Carlo Bertoluzza

• Departamento de Estadística e I.O. y D.M., Universidad de Oviedo, Oviedo, Spain

  María-Ángeles Gil

• Department of Mathematical Sciences, University of Cincinnati, Cincinnati, USA

  Dan A. Ralescu

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