Analysis and Optimization of Systems pp 440-440 | Cite as

# 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.

## References

- 1.R.E. Kalman (1982). Identification from real data. M. Hazewinkel, A.H.G. Rinnooy Kan (eds.).
*Current developments in the interface: Economics, Econometrics, Mathematics*, D. Reidel Fubl. Co., Dordrecht, 161–196.Google Scholar - 2.R.E. Kalman (1983). Identifiability and modeling in econometrics. P.R. Krishnaiah (ed.).
*Developments in statistics, volume**4*, Academic Press, New York, 97–136.Google Scholar - 3.G. Picci, S. Pinzoni (1986).
*Dynamic factor analysis models for stationary processes*preprint.Google Scholar - 4.J.H. van Schuppen (1985).
*Stochastic realization problems motivated by econometric modeling*, Report 0S - R8507, Centre for Mathematics and Computer Science, Amsterdam.Google Scholar