Stochastic realization and factor analysis

Summary

A general problem of science and engineering is to represent observations by a model. R.E. Kalman [1982,1983] has been voicing a critique of modeling in econometrics. His suggestion is to formulate this modeling problem as a realization problem of system theory.

For the representation of an observed random vector several models have been proposed. Some of these models are: the regression model, the errors-in-variables model, the factor analysis model and the confluence analysis model. From a system theoretic view point only the factor analysis model, or equivalently the confluence analysis model, is acceptable. The key property of this model is the con-ditional independence of the components of the observation vector given the factor.

In this paper the problem will be considered of representing an observation vector in a nondynamic context. The questions in this problem are the existence of a factor model that represents the given observations and the classification of all minimal such models. Only the strong stochastic realization problem for a vector of small dimension will be discussed in some detail.

In a dynamic context where an observed stochastic process has to be modeled, the criticism of econometrics leads to a stochastic realintion problem for a Gaussian process. The novel aspect here is to select a class of stochastic dynamic systems such that the inherent causality relation of the obser-vation process is made explicit. A class of stochastic dynamic systems with which this may be done is proposed and the resulting stochastic realization problem discussed.