Abstract
In many practical applications, it turns out to be useful to use the notion offuzzy transform:once we have functionsA1(x)≥0,...,An≥0,with∑i=1nAi(x)=1, we can then represent each functionf(x)by the coefficientsFi=(∫f(x)⋅Ai(x)dx)/(∫Ai(x)dx).Once we know the coefficientsFi, we can (approximately) reconstruct the original functionf(x)as∑i=1nFi⋅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.
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