In many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have functions ??1( ?? ) = 0 , . . . , ????= 0 , with??? ?? = 1????\r\n( ?? ) = 1, we can then represent each function ?? ( ?? ) by the coefficients ????? = ( ?? ( ?? ) �· ????? ?? ( ?? ) ?? ?? ) / (??( ?? ) ?? ?? ) . Once we know the coefficients \r\n????, we can (approximately) reconstruct the original function ?? ( ?? ) as ??? ?? = 1????�· ????( ?? )\r\n. 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 S�¡nchez et al. can be modified into a natural probabilistic explanation of fuzzy transform formulas.
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