Variable Prediction Steps and Longer Term Prediction

It is usual to model the dynamics of an observed time series by building a model F (x t) = x t+1 + e t of a map. Generally F is constructed so that P e 2 t is minimal (often subject to some constraint on F which will avoid over-tting). One may obtain a prediction further into the future by iterating F. In this paper we propose alternative formulations to build a function G(x t ;) = x t+ + e t;;. We propose two distinct method of obtaining G: (i) by modelling G directly (in the same manner as F was constructed), and (ii) by applying a nonlinear model to iterates of F so as to \reene" them. We demonstrate that method (i) is computationally intensive and useful only for relatively short term predictions, or when only very limited data is available. Method (ii) appears to be generally applicable, requires only moderate computational eeort, and may be used to make relatively long term predictions. We demonstrate the applicability of these methods with traditional \test systems" and a variety of experimental data sets. We show that method (ii) may be applied with reasonable success in many situations, and that it is even possible to make useful estimates of the error of our predictions.