Prediction of non-stationary response functions using a Bayesian composite Gaussian process

The modeling and prediction of functions that can exhibit non-stationarity characteristics is important in many applications; for example, this is often the case for simulator output. One approach to predict a function with unknown stationarity properties is to model it as a draw from a flexible stochastic process that can produce stationary or non-stationary realizations. One such model is the composite Gaussian process (CGP) which expresses the large-scale (global) trends of the output and the small-scale (local) adjustments to the global trend as independent Gaussian processes; an extension of the CGP model can produce realizations with non-constant variance by allowing the variance of the local process to vary over the input space. A new, Bayesian extension of a global-trend plus local-trend model is proposed that also allows measurement errors. In contrast to the original CGP model, the new Bayesian CGP model introduces a weight function to allow the total process variability to be apportioned between the large- and small-scale processes. The proposed prior distributions ensure that the fitted global mean is smoother than the local deviations, a feature built into the CGP model. The log of the process variance for the Bayesian CGP is modeled as a Gaussian process to provide a flexible mechanism for handling variance functions that vary across the input space. A Markov chain Monte Carlo algorithm is proposed that provides posterior estimates of the parameters for the Bayesian CGP. It also yields predictions of the output and quantifies uncertainty about the predictions. The method is illustrated using both analytic and real-data examples.