Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere. (Less)

https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9868.2011.00777.x

An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach

With scientific data available at geocoded locations, investigators are increasingly turning to spatial process models for carrying out statistical inference. Over the last decade, hierarchical models implemented through Markov chain Monte Carlo methods have become especially popular for spatial modelling, given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics. However, fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations, rendering such models infeasible for large spatial data sets. This computational burden is exacerbated in multivariate settings with several spatially dependent response variables. It is also aggravated when data are collected at frequent time points and spatiotemporal process models are used. With regard to this challenge, our contribution is to work with what we call predictive process models for spatial and spatiotemporal data. Every spatial (or spatiotemporal) process induces a predictive process model (in fact, arbitrarily many of them). The latter models project process realizations of the former to a lower dimensional subspace, thereby reducing the computational burden. Hence, we achieve the flexibility to accommodate non-stationary, non-Gaussian, possibly multivariate, possibly spatiotemporal processes in the context of large data sets. We discuss attractive theoretical properties of these predictive processes. We also provide a computational template encompassing these diverse settings. Finally, we illustrate the approach with simulated and real data sets.

/pdf/gaussian-predictive-process-models-for-large-spatial-data-25ay8hc7gs.pdf

Gaussian predictive process models for large spatial data sets

We provide a new approach to approximate emulation of large computer experiments. By focusing expressly on desirable properties of the predictive equations, we derive a family of local sequential design schemes that dynamically define the support of a Gaussian process predictor based on a local subset of the data. We further derive expressions for fast sequential updating of all needed quantities as the local designs are built up iteratively. Then we show how independent application of our local design strategy across the elements of a vast predictive grid facilitates a trivially parallel implementation. The end result is a global predictor able to take advantage of modern multicore architectures, providing a nonstationary modeling feature as a bonus. We demonstrate our method on two examples using designs with thousands of data points, and compare to the method of compactly supported covariances. Supplementary materials for this article are available online.

Local Gaussian Process Approximation for Large Computer Experiments

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Matern, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

Strictly and non-strictly positive definite functions on spheres

Summary. Gaussian process models have been widely used in spatial statistics but face tremendous computational challenges for very large data sets. The model fitting and spatial prediction of such models typically require O(n3) operations for a data set of size n. Various approximations of the covariance functions have been introduced to reduce the computational cost. However, most existing approximations cannot simultaneously capture both the large- and the small-scale spatial dependence. A new approximation scheme is developed to provide a high quality approximation to the covariance function at both the large and the small spatial scales. The new approximation is the summation of two parts: a reduced rank covariance and a compactly supported covariance obtained by tapering the covariance of the residual of the reduced rank approximation. Whereas the former part mainly captures the large-scale spatial variation, the latter part captures the small-scale, local variation that is unexplained by the former part. By combining the reduced rank representation and sparse matrix techniques, our approach allows for efficient computation for maximum likelihood estimation, spatial prediction and Bayesian inference. We illustrate the new approach with simulated and real data sets.

A full scale approximation of covariance functions for large spatial data sets

We introduce a new class of nonstationary covariance functions for spatial modelling. Nonstationary covariance functions allow the model to adapt to spatial surfaces whose variability changes with location. The class includes a nonstationary version of the Matern stationary covariance, in which the differentiability of the spatial surface is controlled by a parameter, freeing one from fixing the differentiability in advance. The class allows one to knit together local covariance parameters into a valid global nonstationary covariance, regardless of how the local covariance structure is estimated. We employ this new nonstationary covariance in a fully Bayesian model in which the unknown spatial process has a Gaussian process (GP) prior distribution with a nonstationary covariance function from the class. We model the nonstationary structure in a computationally efficient way that creates nearly stationary local behavior and for which stationarity is a special case. We also suggest non-Bayesian approaches to nonstationary kriging.To assess the method, we use real climate data to compare the Bayesian nonstationary GP model with a Bayesian stationary GP model, various standard spatial smoothing approaches, and nonstationary models that can adapt to function heterogeneity. The GP models outperform the competitors, but while the nonstationary GP gives qualitatively more sensible results, it shows little advantage over the stationary GP on held-out data, illustrating the difficulty in fitting complicated spatial data.

/pdf/spatial-modelling-using-a-new-class-of-nonstationary-3yiil78dr4.pdf

Mark J. Schervish

Papers

Spatial Modelling Using a New Class of Nonstationary Covariance Functions.