Bayesian Spatial Prediction of Random Space-Time Fields With Application to Mapping PM_2.5 Exposure

B. M. Golam Kibria B. M. Golam Kibria is Assistant Professor, Department of Statistics, Florida International University, University Park Campus, Miami, FL 33199 . Li Sun is Research Associate, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada . James V. Zidek is Professor and Head, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada . Nhu D. Le is Adjunct Professor, University of British Columbia, and Senior Scientist, BC Cancer Agency, Vancouver, BC, V5Z 4E6, Canada . Much of this article was written while Golam Kibria was working as an Assistant Professor in the Department of Statistics, University of British Columbia. The authors are indebted to Luther Smith of the ManTech Environmental Technology, Inc. for valuable suggestions that improved the earlier version of the article. The authors are also grateful to the editor, associate editor, and referees for their valuable comments and suggestions, which led to an improved presentation of these materials. Although the research described in this article has been funded wholly or in part by the Environmental Protection Agency contract 68-D5-0049 to ManTech Environmental Technology, Inc., it has not been subject to the Agency's review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred.

Li Sun B. M. Golam Kibria is Assistant Professor, Department of Statistics, Florida International University, University Park Campus, Miami, FL 33199 . Li Sun is Research Associate, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada . James V. Zidek is Professor and Head, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada . Nhu D. Le is Adjunct Professor, University of British Columbia, and Senior Scientist, BC Cancer Agency, Vancouver, BC, V5Z 4E6, Canada . Much of this article was written while Golam Kibria was working as an Assistant Professor in the Department of Statistics, University of British Columbia. The authors are indebted to Luther Smith of the ManTech Environmental Technology, Inc. for valuable suggestions that improved the earlier version of the article. The authors are also grateful to the editor, associate editor, and referees for their valuable comments and suggestions, which led to an improved presentation of these materials. Although the research described in this article has been funded wholly or in part by the Environmental Protection Agency contract 68-D5-0049 to ManTech Environmental Technology, Inc., it has not been subject to the Agency's review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred.

James V Zidek B. M. Golam Kibria is Assistant Professor, Department of Statistics, Florida International University, University Park Campus, Miami, FL 33199 . Li Sun is Research Associate, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada . James V. Zidek is Professor and Head, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada . Nhu D. Le is Adjunct Professor, University of British Columbia, and Senior Scientist, BC Cancer Agency, Vancouver, BC, V5Z 4E6, Canada . Much of this article was written while Golam Kibria was working as an Assistant Professor in the Department of Statistics, University of British Columbia. The authors are indebted to Luther Smith of the ManTech Environmental Technology, Inc. for valuable suggestions that improved the earlier version of the article. The authors are also grateful to the editor, associate editor, and referees for their valuable comments and suggestions, which led to an improved presentation of these materials. Although the research described in this article has been funded wholly or in part by the Environmental Protection Agency contract 68-D5-0049 to ManTech Environmental Technology, Inc., it has not been subject to the Agency's review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred.

Nhu D Le B. M. Golam Kibria is Assistant Professor, Department of Statistics, Florida International University, University Park Campus, Miami, FL 33199 . Li Sun is Research Associate, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada . James V. Zidek is Professor and Head, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada . Nhu D. Le is Adjunct Professor, University of British Columbia, and Senior Scientist, BC Cancer Agency, Vancouver, BC, V5Z 4E6, Canada . Much of this article was written while Golam Kibria was working as an Assistant Professor in the Department of Statistics, University of British Columbia. The authors are indebted to Luther Smith of the ManTech Environmental Technology, Inc. for valuable suggestions that improved the earlier version of the article. The authors are also grateful to the editor, associate editor, and referees for their valuable comments and suggestions, which led to an improved presentation of these materials. Although the research described in this article has been funded wholly or in part by the Environmental Protection Agency contract 68-D5-0049 to ManTech Environmental Technology, Inc., it has not been subject to the Agency's review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred.

Abstract

This article presents a multivariate spatial prediction methodology in a Bayesian framework. The method is especially suited for use in environmetrics, where vector-valued responses are observed at a small set of ambient monitoring stations “(gauged sites)” at successive time points. However, the stations may have varying start-up times so that the data have a “staircase” pattern (“monotone” pattern in the terminology of Rubin and Shaffer). The lowest step corresponds to the newest station in the monitoring network. We base our approach on a hierarchical Bayes prior involving a Gaussian generalized inverted Wishart model. For given hyperparameters, we derive the predictive distribution for currently gauged sites at times before their start-up when no measurements were taken. The resulting predictive distribution is a matric t distribution with appropriate covariance parameters and degrees of freedom. We estimate the hyperparameters using the method of moments (MOM) as an easy-to-implement alternative to the more complex EM algorithm. The MOM in particular gives exact parameter estimates and involves less cumbersome calculations than the EM algorithm. Finally, we obtain the predictive distribution for unmeasured responses at “ungauged” sites. The results obtained here allow us to pool the data from different sites that measure different pollutants and also to treat cases where the observed data monitoring stations have a monotonic “staircaserldquo; structure. We demonstrate the use of this methodology by mapping PM_2.5 fields for Philadelphia during the period of May 1992 to September 1993. Large amounts of data missing by design make this application particularly challenging. We give empirical evidence that the method performs well.

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Bayesian Spatial Prediction of Random Space-Time Fields With Application to Mapping PM_2.5 Exposure

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Bayesian Spatial Prediction of Random Space-Time Fields With Application to Mapping PM_2.5 Exposure