Filtering and Prediction

Abstract

Stochastic systems can be applied for forecasting purposes. The classical solution for filtering, smoothing and prediction of linear systems was proposed by Wiener and Kolmogorov in terms of spectral representations. The Kalman filter is a much more efficient, recursive solution in terms of state space models. 7.1 The Filtering Problem For a deterministic input-output system the future outputs are exactly known once the future inputs have been chosen. For stochastic systems, however, the future disturbances are unknown, and therefore the future outputs can only be predicted with some error. The objective is to construct predictions that minimize the prediction error in some sense. Forecasting is one of the major applications of stochastic systems, in economics, engineering and many other disciplines. The filtering problem is formalized as follows. Suppose that two jointly stationary processes, y and z, are mutually correlated and that the covariances (or the spectrum) of the joint process are completely known, but that only y is observed and z is not. As an example, you may think of z as the state in a model of the type (6.19), and y as the output. The aim is to form an optimal reconstructionˆz reconstructionˆ reconstructionˆz of the unobserved process z on the basis of the observed process y, via some function f of (possibly only some of) the values {y(s); s ∈ Z}, ie., f {y(s); s ∈ Z}. So the problem will be to determine this function f. If for the reconstructionˆzreconstructionˆ reconstructionˆz of z(t) only the past and current values of y, i.e., {y(s); s ≤ t}, can be used, this is called filtering. If only {y(s); s ≤ t − m} for some m > 0 can be used, this is the m-step ahead prediction problem, and if m < 0, this is called smoothing. The case where m = −∞ is called unrestricted smoothing. The one-step ahead prediction problem is often called the filtering problem, and we will pay special attention to this case. For this case, as objective we consider