Probabilistic Interpretation of Fuzzy Transforms and Fuzzy Control

Authors

Irina Perfilieva
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

6-2009

Comments

Technical Report: UTEP-CS-09-17

Abstract

In many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have non-negative functions A₁(x), ..., A_n(x), with A₁(x) + ... + A_n(x) = 1, we can then represent each function f(x) by the coefficients F_i which are defined as the ratio of two integrals: of f(x) * A_i(x) and of A_i(x). Once we know the coefficients F_i, we can (approximately) reconstruct the original function f(x) as F₁ * A₁(x) + ... + F_n * A_n(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 (non-fuzzy) mathematics as well.

Such an interpretation is presented in this paper. Specifically, we show that fuzzy transform has a natural probabilistic interpretation -- related to the known interpretation of fuzzy sets as equivalence classes of random sets. We also show that a similar interpretation is possible for fuzzy control techniques.