Stochastic catastrophe models and multimodal distributions

Abstract

Nonlinear models such as have been appearing in the applied catastrophe theory literature are almost universally deterministic, as opposed to stochastic (probabilistic). The purpose of this article is to show how to convert a deterministic catastrophe model into a stochastic model with the aid of several reasonable assumptions, and how to calculate explicitly the resulting multimodal equilibrium probability density. Examples of such models from epidemiology, psychology, sociology, and demography are presented. Lastly, a new statistical technique is presented, with which the parameters of empirical multimodal frequency distributions may be estimated.KEY WORDS: catastrophe theory, stochastic models, epidemiology, parameter estimation, multimodal distributions. Fu PIRICALfrequency distributions E which unmistakably possess more than one mode, or relative maximum, arise from time to time in all the sciences. Unfortunately, none of the commonly used theoretical probability densities, normal, exponential, gamma, etc., are bimodal, and so the phenomenon is usually ignored altogether. It is not generally recognized that bimodality is strong evidence for the existence of a fundamentally nonlinear underlying stochastic process, and that to each such process there corresponds a possibly multimodal probability density.By fundamentally nonlinear we mean that there is more than one stable equilibrium position available to the system described by the variable in question. By stochastic we mean the system is continuously perturbed by random influences. It wanders about the general neighborhood of one of the stable equilibria, occasionally crossing over into the neighborhood of an adjoining stable equilibrium. A random sample drawn from an ensemble of such systems, each possessing an identical dynamic, would yield a multimodal frequency distribution.Nonlinear dynamic systems can and do exhibit quite exotic behavior, even when they are unperturbed in the stochastic sense. In this paper we shall restrict our attention to a class of nonlinear systems that are particularly well behaved in many ways: those that are topologically equivalent to the canonical catastrophes of Thom. Although this class is not large compared to the universe of nonlinear systems, a strong argument can be made that it encompasses within its scope almost all of the fundamentally nonlinear models that are likely to be used in the behavioral sciences for some time to come. In the first section of this paper we present a stochastic form of these nonlinear systems which is seen to generate equilibrium probability densities that bear a very close relationship to the potential functions of the canonical catastrophes. Several varieties of the stochastic cusp are discussed in the next section. In the third we shift our attention to the problems of the statistical analysis of data using catastrophe models. In the last section we present a new technique for analyzing multimodal distributions with a worked example.'

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