Gaussian Process priors with Uncertain Inputs : Multiple - Step - Ahead Prediction ∗

We consider the problem of multi-step ahead prediction in time series analysis using the non-parametric Gaussian process model. k-step ahead forecasting of a discrete-time nonlinear dynamic system can be performed by doing repeated one-step ahead predictions. For a state-space model of the form yt = f(yt−1, . . . , yt−L), the prediction of y at time t + k is based on the estimates ŷt+k−1, . . . , ŷt+k−L of the previous outputs. We show how, using an analytical Gaussian approximation, we can formally incorporate the uncertainty about intermediate regressor values, thus updating the uncertainty on the current prediction. In this framework, the problem is that of predicting responses at a random input and we compare the Gaussian approximation to the Monte-Carlo numerical approximation of the predictive distribution. The approach is illustrated on a simulated non-linear dynamic example, as well as on a simple one-dimensional static example. ∗Technical Report TR-2002-119, Department of Computing Science, University of Glasgow, October, 2002.