A simulation approach to Bayesian emulation of complex dynamic computer models

Abstract

In recent times, complex computer models have received wide attention in scientific research. However, in order to make conventional statistical statements regarding the scientific research, many expensive runs of the computer model are usually needed. New statistical theories, making their appearances, hold promise to alleviate the technical challenges. However, in cases where the underlying complex system is evolving with time, an effective theory for statistical analyses is lacking. In this paper, we propose a novel Bayesian methodology that extends the existing methodologies to the case of complex dynamic systems. The approach described in the paper exploits the recursive nature of dynamic simulation models to give a more efficient and accurate emulator.

The motivating example, although not a real model for any physical process, may be thought of as a proxy for a model representing climate change, where it is of interest to predict, over time $t$, the four-dimensional proxy time series $\mathbb{y}_t=\mbox{(temperature},$ $\mbox{ice melting rate}$, $\mbox{barren land, $\mbox{CO}_2$ emission)}$. Also available are proxy observations on deforestation, recorded over time; hence treated as known. The latter is known as forcing input, denoted by $z_t$. The computer model is treated as a black box.

Typically, Gaussian processes are used to model unknown computer models, which we adopt in our article. In order to exploit the recursive nature of dynamic computer models, we introduce a grid within the range of the unknown function where the entire dynamic sequence is expected to lie. This grid essentially defines a look-up table. %Instead of the full high-dimensional posterior %predictive distribution for the future time series, Our proposed method then assumes that conditional on the response surface on the grid, and the available training data, the future responses are approximately independent. Exploiting the properties of Gaussian process, we justify our proposal theoretically and with ample simulation studies. We also apply our proposed methodology to the motivating example.