Bayesian reference analysis for Gaussian Markov random fields

Gaussian Markov random fields (GMRF) are important families of distributions for the modeling of spatial data and have been extensively used in different areas of spatial statistics such as disease mapping, image analysis and remote sensing. GMRFs have been used for the modeling of spatial data, both as models for the sampling distribution of the observed data and as models for the prior of latent processes/random effects; we consider mainly the former use of GMRFs. We study a large class of GMRF models that includes several models previously proposed in the literature. An objective Bayesian analysis is presented for the parameters of the above class of GMRFs, where explicit expressions for the Jeffreys (two versions) and reference priors are derived, and for each of these priors results on posterior propriety of the model parameters are established. We describe a simple MCMC algorithm for sampling from the posterior distribution of the model parameters, and study frequentist properties of the Bayesian inferences resulting from the use of these automatic priors. Finally, we illustrate the use of the proposed GMRF model and reference prior for studying the spatial variability of lip cancer cases in the districts of Scotland over the period 1975-1980.     

Gaussian and non-Gaussian random fields associated with Markov processes

To every Markov process with a symmetric transition density, there correspond two random fields over the state space: a Gaussian field (the free field) φ and the occupation field T which describes amount of time the particle spends at each state. A relation between these two random fields is established which is useful both for the field theory and theory of Markov processes.     

Markov random fields and gibbs random fields

Spitzer has shown that every Markov random field (MRF) is a Gibbs random field (GRF) and vice versa when (i) both are translation invariant, (ii) the MRF is of first order, and (iii) the GRF is defined by a binary, nearest neighbor potential. In both cases, the field (iv) is defined onZ ^v, and (v) at anyxεZ^v, takes on one of two states. The current paper shows that a MRF is a GRF and vice versa even when (i)−(v) are relaxed, i.e., even if one relaxes translation invariance, replaces first order bykth order, allows for many states and replaces finite domains of Z^v by arbitrary finite sets. This is achieved at the expense of using a many body rather than a pair potential, which turns out to be natural even in the classical (nearest neighbor) case when Z^v is replaced by a triangular lattice. The contents of this paper were presented in August, 1971, at a seminar of the Battelle Rencontre in Statistical Mechanics and also at a pair of seminars in December, 1971, at the Weizmann Institute of Science. Partially supported by NSF GP 7469 and a Weizmann Institute senior fellowship while on sabbatical leave from Indiana University.     

Gaussian random fields

Nonanticipative representations of Gaussian random fields equivalent to the two-parameter Wiener process are defined, and necessary and sufficient conditions for their existence derived. When such representations exist they provide examples of canonical representations of multiplicity one. In contrast to the one-parameter case, examples are given where nonanticipative representations do not exist. Nonanticipative representations along increasing paths are also studied.University of Minnesota and University of North Carolina. Presented at theWorkshop on Stochastic Processes in Infinite Dimensional Spaces and Random Fields, held at UCLA in April 1979.     

Gaussian random fields,infinite dimensional Ornstein-Uhlenbeck processes,and symmetric Markov processes

A general treatment of infinite dimensional Ornstein-Uhlenbeck processes (OUPs) is presented. Emphasis is put on their connection with ordinary Gaussian random fields, and OUPs as symmetric Markov processes. We also discuss the relation to second quantisation and Gaussian Markov random fields.Supported in part by the Swedish Natural Science Research Council, NFR.     

Markov property of random fields

The survey is devoted to an exposition of general results characterizing Markov random fields and their properties.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 14, pp. 3–70, 1979.     

Singularities in Gaussian random fields

This paper studies a class of Gaussian random fields defined on lattices that arise in pattern analysis. Phase transitions are shown to exist at a critical temperature for these Gaussian random fields. These are established by showing discontinuous behavior for certain field random variables as the lattice size increases to infinity. The discontinuities in the statistical behavior of these random variables occur because the growth rates of the eigenvalues of the inverse of the variance-covariance matrix at the critical temperature are different from the growth rates at noncritical temperatures. It is also shown that the limiting specific heat has a phase transition with a power law behavior. The critical temperature occurs at the end point of the available values of temperature. Thus, although the critical behavior is not extreme, caution should be exercised when using such models near critical temperatures.Research supported by AFOSR Grant No. 91-0048 and by USARO Grant No. DAAL03-90-G-0103.     

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

Finitary coding of Markov random fields

Summary There exists a finitary code from any stationary ergodic Markov random field to any i.i.d. random field of strictly lower entropy.     

Dynamically evolving Gaussian spatial fields

We discuss general non-stationary spatio-temporal surfaces that involve dynamics governed by velocity fields. The approach formalizes and expands previously used models in analysis of satellite data of significant wave heights. We start with homogeneous spatial fields. By applying an extension of the standard moving average construction we obtain models which are stationary in time. The resulting surface changes with time but is dynamically inactive since its velocities, when sampled across the field, have distributions centered at zero. We introduce a dynamical evolution to such a field by composing it with a dynamical flow governed by a given velocity field. This leads to non-stationary models. The models are extensions of the earlier discretized autoregressive models which account for a local velocity of traveling surface. We demonstrate that for such a surface its dynamics is a combination of dynamics introduced by the flow and the dynamics resulting from the covariance structure of the underlying stochastic field. We extend this approach to fields that are only locally stationary and have their parameters varying over a larger spatio-temporal horizon.     

Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

Thermodynamics and statistical analysis of Gaussian random fields

Summary Gaussian fields are considered as Gibbsian fields. Thermodynamic functions are calculated for them and the variational principle is proved. As an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for Gaussian Markov fields.     

Markov approximation of homogeneous lattice random fields

We refine some well-known approximation theorems in the theory of homogeneous lattice random fields. In particular, we prove that every translation invariant Borel probability measure on the space X of finite-alphabet configurations on ^d, d1, can be weakly approximated by Markov measures _n with supp(_n)=X and with the entropies h(_n)h(). The proof is based on some facts of Thermodynamic Formalism; we also present an elementary constructive proof of a weaker version of this theorem.Mathematics Subject Classifications (2000): Primary 28D20, 37C85, 60G60; secondary 82B20Dedicated to Professor A. I. Vorobyov, member of the Russian Academy of Sciences and Director of the Hematology Research Center of the Russian Academy of Medical Sciences, on the occasion of his 75th birthday     

Representation of Gaussian isotropic spin random fields

We develop a technique for the construction of random fields on algebraic structures. We deal with two general situations: random fields on homogeneous spaces of a compact group and in the spin line bundles of the 2$2$-sphere. In particular, every complex Gaussian isotropic spin random field can be represented in this way. Our construction extends P. Lévy’s original idea for the spherical Brownian motion.     

Path properties of l_p-valued Gaussian random fields

General limit theorems are established for l~p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l~p-valued Gaussian random fields under(?)explicit conditions.