On Building Nonlinear Dynamical Stochastic Process Models from Empirical Data
Allen Jay Goldberg
Doctoral Dissertation, Year: 1986, Advisor: John S. Baras
Wiener series is revisited as an approach to developing approximate models for arbitrary, scalar stochastic processes with continuous time parameter. The usual barrier to applying Wiener series to this problem is the difficulty of resolving the Wiener kernals from statistics of the given data. There are, however, known methods for determining Wiener kernals for Wiener series expansions ofa white noise funtional when one has access to both the functional output and the white noise excitiation. One part of this paper is a unified discussion of Wiener series, Wiener series kernal identification procedures, and methods for approximating causal Wiener series by finite-dimensional causal bilinear systems driven by white Gaussian noise.
Attention is then directed toward using a construction due to Wiener and Nisio, built upon a typical path of the process data, which could serve as a starting cadidate white noise functional. The given scalar process must be stationalary, ergodic, and continuous in probability. The sequence of Wiener-Nisio functionals, when driven by the flow of the white noise excitation gives rise to output processes which converge in finite-order distributions to the given process. The functional admits a causal Wiener series expansion which, when truncated, is realizable as a finite-order causal bilinear dynamical system driven by white noise excitation which also approximates the given process in finite distributions.
The paper then concludes with an analysis of thsi approach for two imporatnt classes of process known to admit finite-order Wiener series expanisions: Gaussian processes and Rayleigh power processes.