Before catastrophe models can achieve scientific legitimacy, they must be subjected to empirical tests with real data. This research report provides the details of a recently developed statistical procedure for the estimation of the parameters of the canonical cusp catastrophe model. The procedure is computationally no more complicated than multiple regression, and is based on the method of moments. As an example of the use of this estimation technique, it is shown that extroversion can be used as a splitting factor in the prediction of driving speed after the ingestion of alcohol. This is an example in which catastrophe theory supplies a nontrivial and successful description of the effect of a higher-level system characteristic (extroversion-introversion) on a lower-level measure of system performance (driving speed). Of course, catastrophe models can be used in and between all system levels. m y WORDS: statistical catastrophe theory, bifurcation of behavior, nonlinear system dynamics, cusp model, cross-level control. GugHE ELEMENTARY catastrophe models of T Thom (1975) and Zeeman (1977) have attracted the attention of researchers and theorists throughout the behavioral sciences. Indeed, Behavioral Science recently (September, 1978) devoted an entire special issue to applications of catastrophe theory. A persistent problem with these applications, however, has been the absence of proven statistical procedures for detecting the presence of a catastrophe in any given body of data. This lack has resulted in some severe criticism of catastrophe models for being, among other things, too speculative (Sussmann & Zahler, 1978). Thus catastrophe models have unfortunately become associated in many minds with reckless speculation and intellectual irresponsibility. This paper presents the needed statistical procedure, in a form useful at all levels of systems science for which quantitative measurements are available. It is particularly useful for hypotheses which suggest that there are multiple stable states that a system can occupy when the control variables are in certain configurations. Catastrophe models come in both dynamic and static forms, the static forms being simply the equilibria (stable and un. stable) of the dynamic forms. The capacity for multiple stable equilibria is inherent in catastrophe models; this is the principal feature which distinguishes them from the standard models used in linear and polynomial regression. In effect, the control factors of a catastrophe model correspond to the independent variables of a statistical model, and the behavioral variable of a. catastrophe model corresponds to the dependent variable of a statistical model. When the control factors are such that the behavioral variable is in a multistable situation, then each stable equilibrium value is a predicted value of the behavioral variable; thus there is more than one predicted value. In addition, the unstable equilibria which separate the stable equilibria are also predictions of a sort: They are the values that we predict the ...

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and Lp bounds on the ensemble then give Lp convergence.

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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