Spatial processes and statistics thereof

Kriging

The spatial statistics name for Gaussian process regression. Many complications arise in the spatial context as seen in, e.g. Bayesian inverse problems. Various cunning tricks are needed to make spatial GPs practical A recent one well-adapted to this context is Fixed Rank Kriging (Zammit-Mangion and Cressie 2021), which summarises a few approaches.

Fixed rank kriging (FRK) is a spatial/spatio-temporal modeling and prediction framework that is scaleable, works well with large datasets, and can deal easily with data that have different spatial supports. FRK hinges on the use of a spatial random effects (SRE) model, in which a spatially correlated mean-zero random process is decomposed using a linear combination of spatial basis functions with random coefficients plus a term that captures the random process’ fine-scale variation. Dimensionality reduction through a relatively small number of basis functions ensures computationally efficient prediction, while the reconstructed spatial process is, in general, non-stationary. The SRE model has a spatial covariance function that is always nonnegative-definite and, because any (possibly non-orthogonal) basis functions can be used, it can be constructed so as to approximate standard families of covariance functions (Kang and Cressie 2011). For a detailed treatment of FRK, see Cressie and Johannesson (2008); Cressie, Shi, and Kang (2010); Nychka et al. (2015).[…]

A few variants of FRK have been developed to date, and the one that comes closest to the present software is LatticeKrig (Nychka et al. 2015). LatticeKrig implements what we call a LatticeKrig (LTK) model, which is made up of Wendland basis functions (that have compact support) decompos- ing a spatially correlated process. LatticeKrig models use a Markov assumption to construct a precision matrix to describe the dependence between the coefficients of these basis functions. This, in turn, results in efficient computations and the potential use of a large number (> 10,000) of basis functions. LatticeKrig models do not cater for what we term fine-scale-process variation and, instead, the finest scale of the process is limited to the finest resolution of the basis functions used. The package INLA (Lindgren and Rue 2015) is a general-purpose package for model fitting and prediction. One advantage of INLA is that it contains functionality for fitting Gaussian processes that have covariance functions from the Matérn class (see Lindgren and Rue (2015), for details on the software interface) by approximating a stochastic partial differential equation (SPDE) using a Gaussian Markov random field (GMRF). Specifically, the process is decomposed using basis functions that are triangular ‘tent’ functions, and the coefficients of these basis functions are normally distributed with a sparse precision matrix. Thus, these models, which we term SPDE-GMRF models, share many of the features of LatticeKrig models. A key advantage of INLA over LatticeKrig is that once the spatial or spatio-temporal model is constructed, one has access to all the approximate-inference machinery and likelihood models available within the package.

Kang and Cressie (2011) develop Bayesian FRK; they keep the spatial basis functions fixed and put a prior distribution on K. The predictive-process approach of Banerjee et al. (2008) can also be seen as a type of Bayesian FRK, where the basis functions are constructed from the postulated covariance function of the spatial random effects and hence depend on parameters (see Katzfuss and Hammerling (2017), for an equivalence argument). An R package that implements predictive processes is spBayes (Finley, Banerjee, and Gelfand 2015; Finley, Banerjee, and Carlin 2007). It allows for multivariate spatial or spatio-temporal processes, and Bayesian inference is carried out using Markov chain Monte Carlo (MCMC), thus allowing for a variety of likelihood models.

Spatial processes are unlikely to be truly Gaussian; we often justify the approximation with a Gaussian process by Laplace approximation.

Spatial point processes

A particular sub-case combining point processes with spatial statistics, now with its own notebook

Tools

All recommendations made to me and passed on here are offered unreviewed and unendorsed.

References