Wiener Expansion Research Articles

The Volterra Representation and the Wiener Expansion: Validity and Pitfalls

Volterra and Wiener series provide a general representation for a wide class of nonlinear systems. In this paper we derive rigorous results concerning (a) the conditions under which a nonlinear functional admits a Volterra-like integral representation, (b) the class of systems that admit a Wiener representation and the meaning of such a representation, (c) some sufficient conditions providing a connection between the Volterra-like and the Wiener representations, (d) the mathematical validity of the method of Lee and Schetzen for identifying a nonlinear system.

Wiener analysis of nonlinear systems using Poisson-Charlier crosscorrelation.

Nonlinear systems with event-sequence input, such as are often encountered in neurophysiology, may be experimentally tested with all possible input sequences by stimulation with a Poisson process eventsequence. A complete predictive model of the system's response may be constructed from this data with the Wiener expansion based on the Poisson-Charlier polynomials. Here it is shown how this formulation leads to an efficient method for the evaluation of unknown systems by crosscorrelation, generalizing previous methods. The basic statistical properties of the procedure are demonstrated and the length of experiment required for accurate estimation of the model is computed. The procedure is translated into digital algorithms and the analogous procedures for white noise analysis are presented.

Comments on Wiener's model of certain nonlinear systems

The letter presents the Wiener expansion of the output of certain nonlinear systems in a new way. This is used to condense the restrictions on the input and the form of the model to two simple relationships, and also to show that certain accepted difficulties of Wiener's method can be overcome.

Pattern Recognition and Self-Organization using Wiener's Canonical Forms

Brick and Zames showed how Wiener's expansion procedure could be used to synthesize decision functions for stochastic inputs. In this chapter, it is accomplished by adopting Ashby's approach to intelligence. Thus creativity, planning, induction, and hypothesis generation are features, which will not specifically be a concern for the present model of the automaton. However, generalization of existing hypotheses arid selection, recognition and learning with respect to these are major factors in the present chapter. The model of the recognition system derived here is, in fact, a form of conditional probability computer, which operates on stochastic processes. Of major interest is the fact that this procedure appears to avoid the tendency toward exponential growth normally considered to be characteristic of conditional probability computers. The narrowing-down of the artificial intelligence problem to the recognition, learning, and prediction phases, however one proceeds to justify it, is not without precedence.