Statistics for Spatial Data.

A survey of recent developments in the theory and application of com- posite likelihood is provided, building on the review paper of Varin(2008). A range of application areas, including geostatistics, spatial extremes, and space-time mod- els, as well as clustered and longitudinal data and time series are considered. The important area of applications to statistical genetics is omitted, in light ofLarribe and Fearnhead(2011). Emphasis is given to the development of the theory, and the current state of knowledge on e!ciency and robustness of composite likelihood inference.

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An overview of composite likelihood methods

It is shown that in model-based geostatistics, not all parameters in the Matern class can be estimated consistently if data are observed in an increasing density in a fixed domain, regardless of the estimation methods used. Nevertheless, one quantity can be estimated consistently by the maximum likelihood method, and this quantity is more important to spatial interpolation. The results are established by using the properties of equivalence and orthogonality of probability measures. Some sufficient conditions are provided for both Gaussian and non-Gaussian equivalent measures, and necessary conditions are provided for Gaussian equivalent measures. Two simulation studies are presented that show that the fixed-domain asymptotic properties can explain some finite-sample behavior of both interpolation and estimation when the sample size is moderately large.

https://www.stat.purdue.edu/~zhanghao/Paper/JASA2004.pdf

Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics

Geostatistical space–time models are used increasingly for addressing environmental problems, such as monitoring acid deposition or global warming, and forecasting precipitation or stream flow. Each discipline approaches the problem of joint space–time modeling from its own perspective, a fact leading to a significant amount of overlapping models and, possibly, confusion. This paper attempts an annotated survey of models proposed in the literature, stating contributions and pinpointing shortcomings. Stochastic models that extend spatial statistics (geostatistics) to include the additional time dimension are presented with a common notation to facilitate comparison. Two conceptual viewpoints are distinguished: (1) approaches involving a single spatiotemporal random function model, and (2) approaches involving vectors of space random functions or vectors of time series. Links between these two viewpoints are then revealed; advantages and shortcomings are highlighted. Inference from space–time data is revisited, and assessment of joint space–time uncertainty via stochastic imaging is suggested.

Geostatistical Space–Time Models: A Review

For likelihood-based inference involving distributions in which high-dimensional dependencies are present it may be useful to use approximate likelihoods based, for example, on the univariate or bivariate marginal distributions. The asymptotic properties of formal maximum likelihood estimators in such cases are outlined. In particular, applications in which only a single q x I vector of observations is observed are examined. Conditions under which consistent estimators of parameters result from the approximate likelihood using only pairwise joint distributions are studied. Some examples are analysed in detail.

/pdf/a-note-on-pseudolikelihood-constructed-from-marginal-25l8farxto.pdf

A note on pseudolikelihood constructed from marginal densities

https://aisberg.unibg.it/bitstream/10446/49178/1/Graspa19_VarinHost.pdf

Pairwise likelihood inference in spatial generalized linear mixed models

Separate modeling of the spatial mean field, the spatial variance field, and the space-time residual fields can give a more detailed and possibly more accurate representation of spatial interpolation errors when we have repeated observations on a fixed monitoring network. This article gives expressions for the spatial interpolation errors in terms of the statistics of the component fields, which enable us to assess the relative importance of different kinds of uncertainty. This modeling approach is applied to data of sulfur dioxide concentrations in Europe, and a comparison with neighborhood kriging is done by means of cross-validation.

Spatial Interpolation Errors for Monitoring Data

In aquatic studies, spatial interactions may be both easier to interpret and to quantify by using water distance than by using geographic distance. The water distance is the shortest path between those two sites that may be traversed entirely over water. One problem is that water distances may be non-Euclidean, and thus covariance and variogram functions are not necessarily valid when using the water distance as a distance metric. Another problem is that the computation of water distances for a large set of spatial locations is computationally expensive. Our alternative is a computationally efficient method for calculation of a Euclidean approximation to water distances. The first step of the method is to define a triangular grid covering the complex domain of interest. Using this triangular grid, we pre-compute approximate water distances using a graph search algorithm. These water distances are then approximated by multidimensional scaling, giving a Euclidean space. Finally, we use linear interpolation to move the data locations into the new Euclidean space. By using this method, subsequent computations of water distances between any locations can be done very fast and the method leads to a theoretically valid spatial covariance model. We apply our method to herring data from the Vestfjord system in Northern Norway. Copyright © 2003 John Wiley & Sons, Ltd.

Spatial covariance modelling in a complex coastal domain by multidimensional scaling

We present a statistical framework for model calibration and uncertainty estimation for complex deterministic models. A Bayesian approach is used to combine data from observations, the deterministic model, and prior parameter distributions to obtain forecast distributions. A case study is presented in which the statistical framework is applied using the hydrogeochemical model (MAGIC) for an assessment of recovery from acidification of soils and surface waters at a long-term study site in Norway under different future acid deposition conditions. The water quality parameters are coupled with a simple dose-response model for trout population health. Uncertainties in model output parameters are estimated and forecast results are presented as probability distributions for future water chemistry and as probability distributions of future healthy trout populations. The forecast results are examined for three different scenarios of future acid deposition corresponding to three different emissions control strategies for Europe. Despite the explicit consideration of uncertainties propagated into the future forecasts, there are clear differences among the scenarios. The case study illustrates how inclusion of uncertainties in model predictions can strengthen the inferences drawn from model results in support of decision making and assessments.

Gudmund Høst

Papers

Pairwise likelihood inference in spatial generalized linear mixed models

Spatial Interpolation Errors for Monitoring Data

Spatial covariance modelling in a complex coastal domain by multidimensional scaling

Forecasting acidification effects using a Bayesian calibration and uncertainty propagation approach.

Kriging by local polynomials