Emulating computer models with step-discontinuous outputs using Gaussian processes

In many real-world applications, we are interested in approximating functions that are analytically unknown. An emulator provides a "fast" approximation of such functions relying on a limited number of evaluations. Gaussian processes (GPs) are commonplace emulators due to their properties such as the ability to quantify uncertainty. GPs are essentially developed to emulate smooth, continuous functions. However, the assumptions of continuity and smoothness is unwarranted in many situations. For example, in computer models where bifurcation, tipping points occur in their systems of equations, the outputs can be discontinuous. This paper examines the capacity of GPs for emulating step-discontinuous functions using two approaches. The first approach is based on choosing covariance functions/kernels, namely neural network and Gibbs, that are most appropriate for modelling discontinuities. The predictive performance of these two kernels is illustrated using several examples. The results show that they have superior performance to standard covariance functions, such as the Matern family, in capturing sharp jumps. The second approach is to transform the input space such that in the new space a GP with a standard kernel is able to predict the function well. A parametric transformation function is used whose parameters are estimated by maximum likelihood.