Robust Estimation and Prediction Using Arma Models: A Nonstatistical Approach

Abstract

This paper presents an approach to robust estimation and prediction using ARMA models which include unknown but bounded uncertainties. In contrast to existing literature, we not only include the usual additive disturbance but also include disturbances in the input matrix. The set theoretic approach in this paper and others can be viewed as an alternative to classical least squares in the following sense: It is not assumed that the input and output disturbances are realizations of a stochastic process with partially known statistics. Instead, only a priori bounds are assumed for the disturbances. Within this setting, the set of all robust estimates (system parameters and/or predictions of the output) is defined. Roughly speaking, a robust estimate must be compatible with the measured input-output data and all possible disturbances within the given bounds. As a specific estimate in the set of robust estimates We consider the so-called “maximally robust estimate” Furthermore, it is shown that this estimate has a certain error minimizati on property and is computable via linear programming. After characterizing the set of robust. estimates, an example showing that the “true” system parameters generating the data may lie outside the set of robust solutions is provided; similarly, the least squares solution may also exhibit this pathology. To illustrate the application of the methods presented herein, an example is considered using real data involving the well known Canadian Lynx Trappings Series.