Composite likelihood estimation for a Gaussian process under fixed domain asymptotics

Bachoc, François; Bevilacqua, Moreno; Velandia, Daira

doi:10.1016/j.jmva.2019.104534

Cited by 11 publications

(5 citation statements)

References 53 publications

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“…Further work that is of interest following these findings includes deriv-ing the sandwich covariance matrix for weighted versions of the composite likelihood functions. Our developments can be extended to allow for general weights, but the sandwich covariance matrix and subsequent efficiency expressions will likely be less interpretable except under specific weight configurations such as binary weights (Bevilacqua and Gaetan, 2015) or optimal weights (Bachoc, Bevilacqua and Velandia, 2019;Pace, Salvan and Sartori, 2019). One particular set of weights that would be of interest is the composite full conditional likelihood with alternating binary weights, in line with the work of Besag (1974).…”

Section: Discussionmentioning

confidence: 99%

On the Efficiency of Composite Likelihood Estimation for Gaussian Spatial Processes

Chua¹,

Hui²,

Welsh³

2024

STAT SINICA

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Maximum composite likelihood estimation is an attractive and commonly used alternative to standard maximum likelihood estimation that typically involves sacrificing statistical efficiency for computational efficiency. This statistical efficiency can be quantified by evaluating the sandwich information matrix of the maximum composite likelihood estimator, and then comparing it to the analogous Fisher information matrix for the maximum likelihood estimator. In this paper, we derive new closed-form expressions for the asymptotic relative efficiency of various maximum composite likelihood estimators for a one-dimensional exponential covariance Gaussian process. These expressions are based on a sampling scheme that allows for analysis under three commonly considered spatial asymptotic frameworks: expanding domain, infill and hybrid. Our results demonstrate how the choice of composite likelihood affects estimation efficiency and consistency, particularly with respect to the infill and hybrid frameworks.

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Section: Discussionmentioning

confidence: 99%

On the Efficiency of Composite Likelihood Estimation for Gaussian Spatial Processes

Chua¹,

Hui²,

Welsh³

2024

STAT SINICA

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“…Perhaps the most successful approximation is Vecchia's method [163], which has attracted a remarkable amount of attention in recent times [inc. 147,48,49,67,47]. The Vecchia approximation can be used with any correlation model and its basic idea is is to replace (13) with a product of Gaussian conditional distributions, in which each conditional distribution involves only a small subset of the data. This approximation requires that the data are ordered and the number m of 'previous' data on which to condition is to be specified.…”

Section: Approximate Likelihood and The Matérn Modelmentioning

confidence: 99%

“…See the recent review of Katzfuss and Guinness [83] for further detail. The Vecchia likelihood can be viewed as a specific instance of a more general class of estimation methods called quasi-or composite likelihood [103,162] that have been widely used for the estimation of Gaussian fields with the Matérn model [52,26,13].…”

Section: Approximate Likelihood and The Matérn Modelmentioning

confidence: 99%

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The Matérn Model: A Journey through Statistics, Numerical Analysis and Machine Learning

Porcu¹,

Bevilacqua²,

Schaback³

et al. 2023

Preprint

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The Matérn model has been a cornerstone of spatial statistics for more than half a century. More recently, the Matérn model has been central to disciplines as diverse as numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. In this article we take a Matérn-based journey across these disciplines. First, we reflect on the importance of the Matérn model for estimation and prediction in spatial statistics, establishing also connections to other disciplines in which the Matérn model has been influential. Then, we position the Matérn model within the literature on big data and scalable computation: the SPDE approach, the Vecchia likelihood approximation, and recent applications in Bayesian computation are all discussed. Finally, we review recent devlopments, including flexible alternatives to the Matérn model, whose performance we compare in terms of estimation, prediction, screening effect, computation, and Sobolev regularity properties.

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“…Our work has important relations to several research topics. First, our theory can be viewed as the Bayesian counterpart of the frequentist fixed-domain asymptotic theory on the maximum likelihood estimator in Ying [80], Ying [81], Zhang [82], Chen et al [14], Loh [46], Du et al [21], Wang and Loh [75], Kaufman and Shaby [38], Chang et al [12], Velandia et al [73], Bachoc et al [3], Bachoc and Lagnoux [4], etc. Second, our posterior asymptotic efficiency result is a counterpart of Stein's work in the Bayesian setup and guarantees the optimal estimation of prediction MSE.…”

Section: Our Contributionsmentioning

confidence: 99%

Bayesian Fixed-domain Asymptotics: Bernstein-von Mises Theorem for Covariance Parameters in a Gaussian Process Model

Li¹

2020

Preprint

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Gaussian process models typically contain finite dimensional parameters in the covariance functions that need to be estimated from the data. We study the Bayesian fixed-domain asymptotic properties of the covariance parameters in a Gaussian process with an isotropic Matérn covariance function, which has many applications in spatial statistics. Under fixeddomain asymptotics, it is well known that when the dimension of data is less than or equal to three, the microergodic parameter can be consistently estimated with asymptotic normality while the variance parameter and the range (or length-scale) parameter cannot. Motivated by the frequentist theory, we prove a Bernstein-von Mises theorem for the covariance parameters in isotropic Matérn covariance functions. We show that under fixed-domain asymptotics, the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors as the sample size goes to infinity. The posterior of the microergodic parameter converges in total variation norm to a normal distribution with shrinking variance, while the posterior distribution of the range parameter does not necessarily converge to any degenerate distribution in general. Our theory allows an unbounded prior support for the range parameter on the whole positive real line. Furthermore, we propose a new property called the posterior asymptotic efficiency in linear prediction, and show that the Bayesian kriging predictor at a new spatial location with covariance parameters randomly drawn from their posterior distribution has the same prediction mean squared error as if the true parameters were known. In the special case of one-dimensional Ornstein-Uhlenbeck process, we derive an explicit form for the limiting posterior distribution of the range parameter and an explicit posterior convergence rate for the posterior asymptotic efficiency in linear prediction. We verify these asymptotic results in numerical examples.

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Composite likelihood estimation for a Gaussian process under fixed domain asymptotics

Cited by 11 publications

References 53 publications

On the Efficiency of Composite Likelihood Estimation for Gaussian Spatial Processes

On the Efficiency of Composite Likelihood Estimation for Gaussian Spatial Processes

The Matérn Model: A Journey through Statistics, Numerical Analysis and Machine Learning

Bayesian Fixed-domain Asymptotics: Bernstein-von Mises Theorem for Covariance Parameters in a Gaussian Process Model

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