Robust Models for Time Series

Abstract

This paper introduces the concept of robust models in time series. A model is said to be robust if the accuracies of the inferences derived from it such as 1 step ahead or k step ahead forecasts are not affected even if the given process y does not exactly obey the given model. Specifically consider a stationary process y (.) about which we know only its autocorrelations of order k, k⩽m. Suppose we want to construct a model y(.) for obtaining inferences such as k step ahead forecastes of y (t) based on the past values of y (.). The criterion for the choice of the model in that optimal k-step ahead forecasts derived from the model are robust, ie, the mean square error of forecast computed from the model does not alter appreciably even if the given process y(.) does not obey the model. We show that the robust model is the mth order autoregressive model whose coefficients are chosen so that its autocorrelations of order k⩽m equal the corresponding specified values.

Next we consider the robust modeling of a process x which cannot be observed directly, but a related process y(.) can be observed where y(t) is a sum of the stochastic signal x (t) and a white noise η (t). We obtain robust models for y and x given the first m + 1 autocorrelations of the process y and the variance of the noise η(t). In all of this development, we do not need any assumption regarding the probability distribution of the process.

Indexing terms:

• Robust models
• Time Series