Deep Statistical Models with Application to Environmental Data

When analyzing environmental data, constructing a realistic statistical model is important, not only to fully characterize the physical phenomena, but also to provide valid and useful predictions. Gaussian process models are amongst the most popular tools used for this purpose. However, many assumptions are usually made when using Gaussian processes, such as stationarity of the covariance function. There are several approaches to construct nonstationary spatial and spatio-temporal Gaussian processes, including the deformation approach. In the deformation approach, the geographical domain is warped into a new domain, on which the Gaussian process is modeled to be stationary. One of the main challenges with this approach is how to construct a deformation function that is complicated enough to adequately capture the nonstationarity in the process, but simple enough to facilitate statistical inference and prediction. In this thesis, by using ideas from deep learning, we construct deformation functions that are compositions of simple warping units. In particular, deformation functions that are composed of aligning functions and warping functions are introduced to model nonstationary and asymmetric multivariate spatial processes, while spatial and temporal warping functions are used to model nonstationary spatio-temporal processes. Similarly to the traditional deformation approach, familiar stationary models are used on the warped domain. It is shown that this new approach to model nonstationarity is computationally efficient, and that it can lead to predictions that are superior to those from stationary models. We show the utility of these models on both simulated data and real-world environmental data: ocean temperatures and surface-ice elevation. The developed warped nonstationary processes can also be used for emulation. We show that a warped, gradient-enhanced Gaussian process surrogate model can be embedded in algorithms such as importance sampling and delayed-acceptance Markov chain Monte Carlo. Our surrogate models can provide more accurate emulation than other traditional surrogate models, and can help speed up Bayesian inference in problems with exponential-family likelihoods with intractable normalizing constants, for example when analyzing satellite images using the Potts model.