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Consider a sequence {Z,}={Z" t= 1, 2,...} of identically distributed random column vectors. Suppose we are interested in the relationship between some elements of Y, of Z, and the remaining elements X,. We write Z,=(Y;, X;)', where a prime superscript denotes vector transposition. For example, in classification or pattern recognition problems Y, is a binary or multinomial variable designating class membership and X, is a set of variables influencing the classification. In forecasting problems, Y, is the set of variables (digital or analog) that we wish to forecast on the basis of variables X" which may itself contain past values of Y,. In image enhancement problems, Y, encodes a target image, and X, encodes a degraded version of the image. In pattern completion problems, Y, is the missing part of a pattern, and X, is the supplied part of the pattern (we suppose the same part is always to be supplied for this application). Regardless of whether a deterministic or stochastic relationship exists between Y, and X" a natural object of interest in such situations is the conditional expectation of Y, given X" written E (Y, IX,).(See White, 1989a.) This can be represented as a regression function,() o (X,)= E (Y, IX,).