A GENERALIZED MELLIN TRANSFORM FOR THE ANALYSIS OF LEARNING MACHINES.
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OHIO SCHOOL OF ENGINEERING
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This work introduces a generalization of the Mellin integral transform, and then applies it as a new technique for the analysis of statistical learning machines. Specifically, this transform technique is demonstrated for Bayes estimation and binary detection of communication signals in the framework of statistical decision theory. It is shown that the use of the Mellin transform markedly simplifies the analysis of sequential estimation of an unknown parameter in a statistical population, whether or not an optimum admissible decision rule is used for processing the observations. The loss function, which need not be error-squared, or the likelihood ratio is treated as a random variable, and is described by a transform expression derived from the basic Bayes learning or estimation equation. This transform expression for the loss is the Mellin transform pair to the probability density of the loss, and contains, as unspecified parameters, the number of observations, the a priori assumptions about the unknown parameter that is to be estimated, and the decision data processing rule. The transform expression for the loss is shown to be a convenient and appealing measure of the performance of the learning machine. Convergence, bias, and the rate of convergence of the estimation procedure are readily displayed through the transform associated with the loss function. The average cost of errors for a binary detector, previously adapted through the use of sequential estimation, is derived through the convolution integral and Parsevals relationships for the Mellin transform. Author