Gaussian Markov random fields (GMRFs) are frequently used as computationally efficient models in spatial statistics. Unfortunately, it has traditionally been difficult to link GMRFs with the more traditional Gaussian random field models as the Markov property is difficult to deploy in continuous space. Following the pioneering work of Lindgren et al. (2011), we expound on the link between Markovian Gaussian random fields and GMRFs. In particular, we discuss the theoretical and practical aspects of fast computation with continuously specified Markovian Gaussian random fields, as well as the clear advantages they offer in terms of clear, parsimonious and interpretable models of anisotropy and non-stationarity.
Deep Dive into Think continuous: Markovian Gaussian models in spatial statistics.
Gaussian Markov random fields (GMRFs) are frequently used as computationally efficient models in spatial statistics. Unfortunately, it has traditionally been difficult to link GMRFs with the more traditional Gaussian random field models as the Markov property is difficult to deploy in continuous space. Following the pioneering work of Lindgren et al. (2011), we expound on the link between Markovian Gaussian random fields and GMRFs. In particular, we discuss the theoretical and practical aspects of fast computation with continuously specified Markovian Gaussian random fields, as well as the clear advantages they offer in terms of clear, parsimonious and interpretable models of anisotropy and non-stationarity.
From a practical viewpoint, the primary difficulty with spatial Gaussian models in applied statistics is dimension, which typically scales with the number of observations. Computationally speaking, this is a disaster! It is, however, not a disaster unique to spatial statistics. Time series models, for example, can suffer from the same problems. In the temporal case, the ballooning dimensionality is typically tamed by adding a conditional independence, or Markovian, structure to the model. The key advantage of the Markov property for time series models is that the computational burden then grows only linearly (rather than cubically) in the dimension, which makes inference on these models feasible for long time series.
Despite its success in time series modelling, the Markov property has had a less exalted role in spatial statistics. Almost all instances where the Markov property has been used in spatial modelling has been in the form of Markov random fields defined over a set of discrete locations connected by a graph. The most common Markov random field models are Gaussian Markov random fields (GMRFs), in which the value of the random field at the nodes is jointly Gaussian (Rue and Held, 2005). GMRFs are typically written as
x ∼ N (µ, Q -1 ),
where Q is the precision matrix and the Markov property is equivalent to requiring that Q is sparse, that is Q ij = 0 iff x i and x j are conditionally independent (Rue and Held, 2005).
As problems in spatial statistics are usually concerned with inferring a spatially continuous effect over a domain of interest, it is difficult to directly apply the fundamentally discrete GMRFs. For this reason, it is commonly stated that there are two essential fields in spatial statistics: the one that uses GMRFs and the one that uses continuously indexed Gaussian random fields. In a recent read paper, Lindgren et al. (2011) showed that these two approaches are not distinct. By carefully utilising the continuous space Markov property, it is possible to construct Gaussian random fields for which all quantities of interest can be computed using GMRFs!
The most exciting aspect of the Markovian models of Lindgren et al. (2011) is their flexibility. There is no barrier-conceptual or computational-to extending them to construct non-stationary, anisotropic Gaussian random fields. Furthermore, it is even possible to construct them on the sphere and other manifolds. In fact, Simpson et al. (2011a) showed that there is essentially no computational difference between inferring a log-Gaussian Cox process on a rectangular observation window and inferring one on a non-convex, multiply connected region on the sphere! This type of flexibility is not found in any other method for constructing Gaussian random field models.
In this paper we carefully review the connections between GMRFs, Gaussian random fields, the spatial Markov property and deterministic approximation theory. It is hoped that this will give the interested reader some insight into the theory and practice of Markovian Gaussian random fields. In Section 2 we briefly review the practical computational properties of GMRFs. Section 3 we take a detailed tour of the theory of Markovian Gaussian random fields. We begin with a discussion of the spatial Markov property and show how it naturally leads to differential operators. We then present a practical method for approximating Markovian Gaussian random fields and discuss what is mean by a continuous approximation. In particular, we show that deterministic approximation theory can provide essential insights into the behaviour of these approximations. We then discuss some practical issues with choosing sets of basis functions before discussing extensions of the models. Finally we mention some further extensions of the method.
Gaussian Markov random fields possess two pleasant properties that make them useful for spatial problems: they facilitate fast computation for large problems, and they are quite stable with respect to conditioning. In this section we will explore these two properties in the context of spatial statistics.
As in the temporal setting, the Markovian property allows for fast computation of samples, likelihoods and other quantities of interest (Rue and Held, 2005). This allows the investigation of much larger models than would be available using general multivariate Gaussian models. The situation is not, however, as good as it is in the one dimensional case, where all of these quantities can be computed using O(n) operations, where n is the dimension of the GMRF. Instead, for the two dimensional spatial models, samples and likelihoods can be computed in O(n 3/2 ) operations, which is still a significant saving on the O(n 3 ) operations required for a general Gaussian model. A quick order calculation shows that computing a sample from an ñ-dimensional Gaussian random vector without any special structure takes the same amount of time as computing a sample from GMRF of dime
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