Nonstationary Gaussian processes for regression and spatial modelling.

摘要

Recent work in the areas of nonparametric regression and spatial smoothing has focused on modelling inhomogeneity in the smoothness of the function of interest. In the regression literature, important progress has been made in fitting free knot spline models in a Bayesian context, with knots automatically being placed more densely in regions of the covariate space in which the function varies more quickly. In the spatial statistics literature, attention has focused on using nonstationary covariance structures to account for inhomogeneity of the spatial field.; In this dissertation, I use nonstationary covariance functions in Gaussian process (GP) prior distributions over functions to perform both nonparametric regression and spatial smoothing in a Bayesian fashion. I extend the kernel convolution method of Higdon et al. (1999) to create a general class of nonstationary covariance functions. I prove that the nonstationary covariance functions retain the smoothness properties of the stationary correlation functions on which they are based, provided there is sufficient smoothness in the underlying kernel structure used to generate the nonstationarity. The stationary Matern correlation function has desirable smoothness properties; the generalized kernel convolution method developed here provides a Matern-based nonstationary correlation function.; I develop a generalized nonparametric regression model and assess difficulties in identifiability and fitting of the model using Markov Chain Monte Carlo (MCMC) algorithms. Of particular note, I show how to improve MCMC performance for non-Gaussian data based on an approximate conditional posterior mean. The modelling approach produces a flexible response surface that responds to inhomogeneity while naturally controlling overfitting. On test datasets in one dimension, the GP model performs well, but not as well as the free-knot spline method. However, in two and three dimensions, the nonstationary GP model seems to outperform both free-knot spline models and a stationary GP model. Unfortunately, the method is not feasible for datasets with more than a few hundred observations because of the computational difficulties involved in fitting the model.; The nonstationary covariance model can also be embedded in a spatial model. In particular, I analyze spatiotemporal climate data, using a nonstationary covariance matrix to model the spatial structure of the residuals. I demonstrate that the nonstationary model fits the covariance structure of the data better than a stationary model, but any improvement in point predictions relative to a stationary model or to the maximum likelihood estimates is minimal, presumably because the data are very smooth to begin with. My comparison of various correlation models for the residuals highlights the difficulty in fitting high-dimensional covariance structures.