The Annals of Applied Statistics

A spatial analysis of multivariate output from regional climate models

Stephan R. Sain, Reinhard Furrer, and Noel Cressie

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Climate models have become an important tool in the study of climate and climate change, and ensemble experiments consisting of multiple climate-model runs are used in studying and quantifying the uncertainty in climate-model output. However, there are often only a limited number of model runs available for a particular experiment, and one of the statistical challenges is to characterize the distribution of the model output. To that end, we have developed a multivariate hierarchical approach, at the heart of which is a new representation of a multivariate Markov random field. This approach allows for flexible modeling of the multivariate spatial dependencies, including the cross-dependencies between variables. We demonstrate this statistical model on an ensemble arising from a regional-climate-model experiment over the western United States, and we focus on the projected change in seasonal temperature and precipitation over the next 50 years.

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Ann. Appl. Stat., Volume 5, Number 1 (2011), 150-175.

First available in Project Euclid: 21 March 2011

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Lattice data Markov random field (MRF) conditional autoregressive (CAR) model Bayesian hierarchical model climate change


Sain, Stephan R.; Furrer, Reinhard; Cressie, Noel. A spatial analysis of multivariate output from regional climate models. Ann. Appl. Stat. 5 (2011), no. 1, 150--175. doi:10.1214/10-AOAS369.

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  • Apanasovich, T. V. and Genton, M. G. (2010). Cross-covariance functions for multivariate random fields based on latent dimensions. Biometrika 97 15–30.
  • Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC Press, Bacon Raton, FL.
  • Berliner, L. M. and Kim, Y. (2008). Bayesian design and analysis for superensemble based climate forecasting. Journal of Climate 21 1891–1910.
  • Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). J. Roy. Statist. Soc. Ser. B 35 192–236.
  • Billheimer, D., Cardoso, T., Freeman, E., Guttorp, P., Ko, H. and Silkey, M. (1997). Natural variability of benthic species composition in the Delaware Bay. Environ. Ecol. Stat. 4 95–115.
  • Carlin, B. P. and Banerjee, S. (2003). Hierarchical multivariate CAR models for spatiotemporally correlated data. In Bayesian Statistics 7 45–63. Oxford University Press, Oxford.
  • Cooley, D. and Sain, S. R. (2010). Spatial hierarchical modeling of precipitation extremes from a regional climate model. J. Agric. Biol. Environ. Stat. 15 381–402.
  • Cressie, N. A. C. (1993). Statistics for Spatial Data, rev. ed. Wiley, New York.
  • Daniels, M. J., Zhou, Z. and Zou, H. (2006). Conditionally specified space–time models for multivariate processes. J. Comput. Graph. Statist. 15 157–177.
  • Davis, T. A. (2006). Direct Methods for Sparse Linear Systems. SIAM, Philadelphia.
  • Furrer, R. (2008). spam: SPArse Matrix. R package version 0.14-1.
  • Furrer, R. and Sain, S. R. (2010). spam: A sparse matrix r package with emphasis on mcmc methods for Gaussian Markov random fields. Journal of Statistical Software 36 1–25.
  • Furrer, R., Knutti, R., Sain, S. R., Nychka, D. and Meehl, G. A. (2007a). Spatial patterns of probabilistic temperature change projections from a multivariate Bayesian analysis. Geophysical Research Letters 34 L06711. DOI: 10.1029/2006GL027754.
  • Furrer, R., Sain, S. R., Nychka, D. and Meehl, G. A. (2007b). Multivariate Bayesian analysis of atmosphere–ocean general circulation models. Environ. Ecol. Stat. 14 249–266.
  • Gelfand, A. E., Hills, S. E., Racine-Poon, A. and Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling. J. Amer. Statist. Assoc. 85 972–985.
  • Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398–409.
  • Gelfand, A. E. and Vounatsou, P. (2003). Proper multivariate conditional autoregressive models for spatial data analysis. Biostatistics 4 11–15.
  • Gelman, A. (1996). Inference and monitoring convergence. In Markov Chain Monte Carlo in Practice 131–144. Chapman & Hall, London.
  • Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6 721–741.
  • Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1996). Introducing Markov chain Monte Carlo. In Markov Chain Monte Carlo in Practice 1–19. Chapman & Hall, London.
  • Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
  • Houghton, J. T., Ding, Y., Griggs, D. J., Noguer, M., van der Linden, P. J., Dai, X., Maskell, K. and Johnson, C. A., eds. (2001). Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge Univ. Press, Cambridge.
  • Jin, X., Banerjee, S. and Carlin, B. P. (2007). Order-free coregionalized lattice models with application to multiple disease mapping. J. Roy. Statist. Soc. Ser. B 69 817–838.
  • Jin, X., Carlin, B. P. and Banerjee, S. (2005). Generalized hierarchical multivariate CAR models for areal data. Biometrics 61 950–961.
  • Kang, E. L., Cressie, N. and Sain, S. (2010). Combining outputs from the NARCCAP regional climate models using a Bayesian hierarchical model. Technical Report 837, Dept. Statistics, Ohio State Univ., Columbus, OH.
  • Kaufman, C. G. and Sain, S. R. (2010). Bayesian functional ANOVA modeling using Gaussian process prior distributions. Bayesian Anal. 5 123–150.
  • Kim, H., Sun, D. and Tsutakawa, R. K. (2001). A bivariate Bayes method for improving the estimates of mortality rates with a twofold conditional autoregressive model. J. Amer. Statist. Assoc. 96 1506–1521.
  • Leung, L., Qian, Y., Bian, X., Washington, W. M., Han, J. and Roads, J. O. (2004). Mid-century ensemble regional climate change scenarios for the western United States. Climatic Change 62 75–113.
  • Mardia, K. V. (1988). Multidimensional multivariate Gaussian Markov random fields with applications to image processing. J. Multivariate Anal. 24 265–284.
  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics 21 1087–1091.
  • Pettitt, A. N., Weir, I. S. and Hart, A. G. (2002). A conditional autoregressive Gaussian process for irregularly spaced multivariate data with application to modeling large sets of binary data. Stat. Comput. 12 353–367.
  • R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
  • Royle, A. M. and Berliner, L. M. (1999). A hierarchical approach to multivariate spatial modeling and prediction. J. Agric. Biol. Environ. Stat. 4 29–56.
  • Rue, H. and Held, L. (2005). Gaussian Markov Random Fields: Theory and Application. Chapman & Hall/CRC Press, Boca Raton, FL.
  • Sain, S. R. and Cressie, N. (2007). A spatial model for multivariate lattice data. J. Econometrics 140 226–259.
  • Sain, S. R., Furrer, R. and Cressie, N. (2007). Combining regional climate model output via a multivariate Markov random field model. In Bulletin of the International Statistical Institute LXII 1375–1382. Instituto Nacional de Estatística, Lisboa, Portugal.
  • Sansó, B. and Guenni, L. (2004). A Bayesian approach to compare observed rainfall data to deterministic simulations. Environmetrics 15 597–612.
  • Schabenberger, O. and Gotway, C. A. (2005). Statistical Methods for Spatial Data Analysis. Chapman & Hall/CRC Press, Boca Raton, FL.
  • Schliep, E. M., Cooley, D., Sain, S. R. and Hoeting, J. A. (2010). A comparison study of extreme precipitation from six different regional climate models via spatial hierarchical modeling. Extremes 13 219–239.
  • Smith, R. L., Tebaldi, C., Nychka, D. and Mearns, L. O. (2009). Bayesian modeling of uncertainty in ensembles of climate models. J. Amer. Statist. Assoc. 104 97–116.
  • Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K. B., Tignor, M. and Miller, H. L., eds. (2007). Climate Change 2007: The Physical Science Basis: Working Group I Contribution to the Fourth Assessment Report of the IPCC. Cambridge Univ. Press, Cambridge, UK and New York, NY, USA.
  • Tebaldi, C. and Sansó, B. (2009). Joint projections of temperature and precipitation change from multiple climate models: A hierarchical Bayes approach. J. Roy. Statist. Soc. Ser. A 172 83–106.
  • Tebaldi, C., Smith, R. L., Nychka, D. and Mearns, L. O. (2005). Quantifying uncertainty in projections of regional climate change: A Bayesian approach to the analysis of multimodel ensembles. Journal of Climate 18 1524–1540.
  • Ver Hoef, J. M., Cressie, N. and Barry, R. M. (2004). Flexible spatial models for kriging and cokriging using moving averages and the Fast Fourier Transform (FFT). J. Comput. Graph. Statist. 13 265–282.
  • Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley, New York.