#### Publication Date

6-2009

#### Abstract

In many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have non-negative functions A_{1}(x), ..., A_{n}(x), with A_{1}(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 F_{1} * A_{1}(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.

## Comments

Technical Report: UTEP-CS-09-17