Bayesian Analysis

Multivariate spatiotemporal CDFs with random effects and measurement error

Bradley P. Carlin and Margaret B. Short

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Spatial cumulative distributions (SCDFs) are useful in environmental applications -- for example, by helping assess the fraction of a region exposed to harmful pollutants. Data sets containing the requisite spatial information often contain temporal data as well. We therefore extend the notion of an SCDF to a spatiotemporal cumulative distribution function (STCDF), with the goal of increasing precision by making use of repeated measurements. Ours is a hierarchical Bayesian approach, with estimation carried out by Markov chain Monte Carlo (MCMC) methods. We develop linear algebra results and corresponding computational techniques to handle the difficulties in evaluating the likelihood wrought by the large data sets (due to the added temporal component), the inclusion of spatial and temporal random effects, the need to account for measurement error, and the handling of missing data. We illustrate the concepts in a univariate setting with an Atlanta ozone data set, and in a bivariate (two pollutant) setting with a California NO/NO$_2$ data set.

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Bayesian Anal. Volume 1, Number 3 (2006), 595-624.

First available in Project Euclid: 22 June 2012

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Air pollution Bayesian methods Change of support problem Kriging Markov chain Monte Carlo (MCMC) methods


Short, Margaret B.; Carlin, Bradley P. Multivariate spatiotemporal CDFs with random effects and measurement error. Bayesian Anal. 1 (2006), no. 3, 595--624. doi:10.1214/06-BA120.

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  • Banerjee, S. (2005). "On geodetic distance computations in spatial modelling." Biometrics, 61: 617–625.
  • Banerjee, S., Carlin, B., and Gelfand, A. (2004). Hierarchical Modeling and Analysis for Spatial Data. Boca Raton, Florida: Chapman and Hall/CRC Press.
  • Banerjee, S. and Gelfand, A. (2002). "Prediction, interpolation and regression for spatially misaligned data." Sankhya, 64: 227–245.
  • Celeux, G., Forbes, F., Robert, C., and Titterington, D. (2005). "Deviance Information Criteria for missing data models." Bayesian Analysis. (To appear, this issue.).
  • Cressie, N. (1993). Statistics for Spatial Data. New York: Wiley, second edition.
  • Fuentes, M. (2003). "Testing for separability of spatial-temporal covariance functions." Mimeo Series Report #2545, Department of Statistics, North Carolina State University.
  • Gelfand, A., Zhu, L., and Carlin, B.P. (2001). "On the change of support problem for spatio-temporal data." Biostatistics, 2: 31–45.
  • Gelman, A., Roberts, G., and Gilks, W. (1996). "Efficient Metropolis jumping rules." In Bernardo, J., Berger, J., Dawid, A., and Smith, A. (eds.), Bayesian Statistics 5, 599–607. Oxford: Oxford University Press.
  • Golub, G. and Van Loan, C. (1996). Matrix Computations. Baltimore, MD: Johns Hopkins University Press, third edition.
  • Handcock, M. (1999). (Comment on “Prediction of spatial cumulative distribution functions using subsampling.”).
  • Higdon, D., Holloman, C., and Lee, H. (2003). "Markov chain Monte Carlo-based approaches for inference in computationally intensive inverse problems (with discussion)." In Bernardo, J., Berger, J., Dawid, A., and Smith, A. (eds.), Bayesian Statistics 7, 181–197. Oxford: Oxford University Press.
  • Overton, W. (1989). "Effects of measurements and other extraneous errors on estimated distribution functions in the National Surface Water Surveys." Technical Report 129, Department of Statistics, Oregon State University.
  • Schott, J. (1997). Matrix Analysis for Statistics. New York: Wiley.
  • Short, M., Carlin, B., and Gelfand, A. (2005). "Bivariate spatial process modeling for constructing indicator or intensity weighted spatial CDFs." Journal of Agricultural, Biological, and Environmental Statistics, 10: 259–275.
  • Spiegelhalter, D., Best, N., Carlin, B., and van der Linde, A. (2002). "Bayesian measures of model complexity and fit (with discussion)." J. Roy. Statist. Soc., Ser. B, 64: 583–639.
  • Zhu, L., Carlin, B., and Gelfand, A. (2003). "Hierarchical regression with misaligned spatial data: Relating ambient ozone and pediatric asthma ER visits in Atlanta." Environmetrics, 14: 537–557.