Gaussian process regression for non-Gaussian data

Abstract

Gaussian process regression (GPR) is a kernel-based non-linear non-parametric model which is widely used in many research fields. Functional data analysis handles data which are in the form of functions, shapes or more general objects, such as a stochastic process. A challenge in GPR is how to deal with non-Gaussian distributed data which has been increasingly recorded in scientific and industrial fields. The thesis contains two main contributions: (i) extensions of GPR for non-Gaussian data, and (ii) the development of a Gaussian process functional regression (GPFR) model for manifold-valued data. First, we review the GPR model and the GPFR model with their properties and inference methods, followed by a summary of background mathematics for Riemannian manifolds, some special probability distributions and some useful algorithms. After the literature review, we introduce a truncated Gaussian process regression (TGPR) model which is an extension of Gaussian process regression for data with a truncated normal distribution. We also study a normal distribution with multi-truncation. In addition, Gaussian process regression is extended to model Gamma-distributed data and this model is denoted as GPRG. Afterwards, we generalise the GPRG model to high-dimensional outputs. Simulation studies are used to assess the performance of the TGPR and GPRG models have good prediction accuracy. We then move on to consider more complex manifold-valued data. We introduce a novel regression model for such data within a probabilistic framework, called wrapped Gaussian process functional regression. We describe an algorithm in which mean structure and covariance structure are estimated simultaneously and then updated iteratively. Various simulation experiments are presented, and these show our model outperforms some other models. The proposed method is applied to some flight trajectory data and provides accurate predictions for missing parts of the trajectories.