Towards an (even more) natural probabilistic interpretation of fuzzy transforms (and of fuzzy modeling)

  • Authors:

  • Irina Perfilieva;Vladik Kreinovich

  • Affiliations:

  • Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Ostrava, Czech Republic;Department of Computer Science, University of Texas at El Paso, El Paso, TX

  • Venue:
  • Advances in Fuzzy Systems - Special issue on Fuzzy Functions, Relations, and Fuzzy Transforms: Theoretical Aspects and Applications to Fuzzy Systems
  • Year:
  • 2011

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Abstract

In many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have functions A1(x) ≥ 0, ...,An ≥ 0, with Σi=1n Ai(x) = 1, we can then represent each function f (x) by the coefficients Fi = (∫ f (x) ċAi(x)dx)/(∫ Ai(x)dx). Once we know the coefficients Fi, we can (approximately) reconstruct the original function f(x) as Σi=1n Fi ċ Ai(x). The original motivation for this transformation came from fuzzy modeling, but the transformation itself is a purely mathematical transformation. Thus, the empirical successes of this transformation suggest that this transformation can be also interpreted in more traditional (nonfuzzy) mathematics as well. Such an interpretation is presented in this paper. Specifically, we show that the 2002 probabilistic interpretation of fuzzy modeling by Sánchez et al. can be modified into a natural probabilistic explanation of fuzzy transform formulas.