On the Applicability of Wiener's Canonical Expansions

Previous papers have proposed the application of Wiener's Hermite-Laguerre expansion procedure to the multiple-alternative, discrete-decision problem with learning, characteristic of many waveform or stochastic-process pattern-recognition problems. Both sequential and nonsequential procedures were formulated; the resulting models are functionally analogous to a generalized Bayes'-net type of pattern recognizer or decision maker for stochastic processes, differing from usual Bayes' nets in their actual mathematical or circuit configurations and size-determining factors. It is to be noted that for ergodic processes (or approximations thereto), the procedure, if it can be applied, is nonparametric, i.e., not dependent upon prior or explicit knowledge of the form of the probability distribution governing the behavior of the stochastic process. The applicability of the resulting system to problems in cybernetics, intelligence, and learning was discussed previously. The present paper summarizes the results of a subsequent analytical investigation which included a digital simulation of the procedure. Emphasis is on the aspects of realizability, convergence, and applicability of the method with regard to 1) the classes of stochastic inputs for which the procedure is valid, and 2) the parameters of those processes. A Wiener or Wiener-derived white-noise process is used as the bench mark process here. Based on the results of the analysis, the introduction of certain preprocessors to extend the applicability of the procedure are suggested.