The confluence of Gaussian process emulation and wavelets

We discuss two thriving research areas, emulation (in the statistical sense) and wavelet analysis, and explore ways in which the two areas can complement each other to tackle problems that both areas face. The Gaussian process, which is the popular choice in emulation, is used due to its ability to be a surrogate for a function when we are only able to make a limited number of observations from the function. The Gaussian process, however, does not perform well when the underlying function contains a discontinuity. Wavelet analysis, on the other hand, is known for its ability to model and analyse functions that contain discontinuities. Wavelet analysis tends to require a large number of datapoints to be able to model functions accurately, tending to struggle when the amount of data is limited.
As it appears that one area’s strength is the other area’s weakness, this thesis is aimed at exploring the possible overlaps between the two methods, and the ways in which they could benefit each other. Particular attention in the thesis is paid to the challenges that are faced when the function that we are attempting to model contains discontinuities, or, areas of space in which there is a sharp increase/decrease in the value of our observations. We develop methods to select the location of additional design points after we have observed the function at our original design points with the objective of better defining the location of the discontinuity. We also develop novel methods to model the unknown function that we believe contains discontinuities, and look to accurately find our uncertainty in this function.