Mean squared prediction error in the spatial linear model with estimated covariance parameters

The problem considered is that of predicting the value of a linear functional of a random field when the parameter vector of the covariance function (or generalized covariance function) is unknown. The customary predictor when is unknown, which we call the EBLUP, is obtained by substituting an estimator j for in the expression for the best linear unbiased predictor (BLUP). Similarly, the customary estimator of the mean squared prediction error (MSPE) of the EBLUP is obtained by substituting j for in the expression f for the BLUP's MSPE; we call this the EMSPE. In this article, the appropriateness of the EMSPE as an estimator of the EBLUP's MSPE is examined, and alternative estimators of the EBLUP's MSPE for use when the EMSPE is inappropriate are suggested. Several illustrative examples show that the performance of the EMSPE depends on the strength of spatial correlation; the EMSPE is at its best when the spatial correlation is strong.This research was partially supported by a University of Iowa Old Gold Fellowship (Zimmerman) and by the NSF under grant DMS-8703083 (Cressie).     

Bayesian source detection and parameter estimation of a plume model based on sensor network measurements

We consider a network of sensors that measure the intensities of a complex plume composed of multiple absorption–diffusion source components. We address the problem of estimating the plume parameters, including the spatial and temporal source origins and the parameters of the diffusion model for each source, based on a sequence of sensor measurements. The approach not only leads to multiple‐source detection, but also the characterization and prediction of the combined plume in space and time. The parameter estimation is formulated as a Bayesian inference problem, and the solution is obtained using a Markov chain Monte Carlo algorithm. The approach is applied to a simulation study, which shows that an accurate parameter estimation is achievable. Copyright © 2010 John Wiley & Sons, Ltd.     

A Loss function approach to identifying environmental exceedances

A frequent problem in environmental science is the prediction of extrema and exceedances. It is well known that Bayesian and empirical-Bayesian predictors based on integrated squared error loss (ISEL) tend to overshrink predictions of extrema toward the mean. In this paper, we consider a geostatistical extension of the weighted rank squared error loss function (WRSEL) of Wright et al. (2003), which we call the integrated weighted quantile squared error loss (IWQSEL), as the basis for prediction of exceedances and their spatial location. The loss function is based on an ordering of the underlying spatial process using a spatially averaged cumulative distribution function. We illustrate this methodology with a Bayesian analysis of surface-nitrogen concentrations in the Chesapeake Bay and compare the new IWQSEL predictor with a standard ISEL predictor. We also give a comparison to predicted extrema obtained from a “plug-in” goestatistical analysis. AMS 2000 Subject Classification Primary—62M30; Secondary—62H11     

Testing for the equality of two binomial proportions

There are two statistics one might choose when testing whether two binomial probabilities are the same. This note provides a large sample answer to Robbins' question of which is preferable.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.     

Statistical properties of the state obtained by solving a nonlinear multivariate inverse problem

This article takes a statistical approach to solving a multivariate state‐space problem where many data are nonlinearly related to a state vector. The state is unknown and to be predicted, but the problem can be ill posed. A state‐space model quantifies the variability of the physical process (state equation) and of the measurements related to the process (measurement equation). The resulting posterior distribution is then maximized, yielding the predicted state vector. Statistical properties of the predicted state vector, in particular its first two moments with respect to the joint distribution, are approximated using the delta method. These are then applied to the problem of retrieving, from satellite data, a profile of CO₂ values in a column of the atmosphere. Copyright © 2012 John Wiley & Sons, Ltd.     

Spatial Point Process Models of Defensive Strategies: Detecting Changes

The study of stochastic processes can take many forms. Theoretical properties are important to ensure consistent model definition. Statistical inference on unknown parameters is equally important but can be difficult. This is principally because many of the standard assumptions for proving consistency and asymptotic normality of estimators involve independence and homogeneity. In the case where inference is concerned with detecting change in a spatial process from one time point to another, a statistical-computing approach can be rewarding. Regardless of the complexity of the stochastic process, if simulating from it is relatively easy, then detecting change is possible using a Monte Carlo approach. The methodology is applied in a military scenario, where a country’s defensive posture changes as a function of its perceived threat. For tactical-decision purposes, it is extremely important to know whether the country’s perceived threat level has changed.     

The Construction of Multivariate Distributions from Markov Random Fields

We address the problem of constructing and identifying a valid joint probability density function from a set of specified conditional densities. The approach taken is based on the development of relations between the joint and the conditional densities using Markov random fields (MRFs). We give a necessary and sufficient condition on the support sets of the random variables to allow these relations to be developed. This condition, which we call the Markov random field support condition, supercedes a common assumption known generally as the positivity condition. We show how these relations may be used in reverse order to construct a valid model from specification of conditional densities alone. The constructive process and the role of conditions needed for its application are illustrated with several examples, including MRFs with multiway dependence and a spatial beta process.