Bayesian Analysis

Bayesian inference for directional conditionally autoregressive models

Sujit K. Ghosh and Minjung Kyung

Full-text: Open access


Counts or averages over arbitrary regions are often analyzed using conditionally autoregressive (CAR) models. The neighborhoods within CAR models are generally determined using only the inter-distances or boundaries between the sub-regions. To accommodate spatial variations that may depend on directions, a new class of models is developed using different weights given to neighbors in different directions. By accounting for such spatial anisotropy, the proposed model generalizes the usual CAR model that assigns equal weight to all directions. Within a fully hierarchical Bayesian framework, the posterior distributions of the parameters are derived using conjugate and non-informative priors. Efficient Markov chain Monte Carlo (MCMC) sampling algorithms are provided to generate samples from the marginal posterior distribution of the parameters. Simulation studies are presented to evaluate the performance of the estimators and are used to compare results with traditional CAR models. Finally the method is illustrated using data sets on local crime frequencies in Columbus, OH and on the elevated blood lead levels of children under the age of 72 months observed in Virginia counties for the year of 2000.

Article information

Bayesian Anal. Volume 4, Number 4 (2009), 675-706.

First available in Project Euclid: 22 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Anisotropy Bayesian estimation Conditionally autoregressive models Lattice data Spatial analysis


Kyung, Minjung; Ghosh, Sujit K. Bayesian inference for directional conditionally autoregressive models. Bayesian Anal. 4 (2009), no. 4, 675--706. doi:10.1214/09-BA425.

Export citation


  • Anselin, L., 1988. Spatial Econometrics: Methods and Models. Kluwer Academic Publishers.
  • Besag, J., 1974. Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society, Series B. 36, 192-236 (with discussion).
  • Besag, J., 1975. Spatial analysis of non-lattice data. The Statistician 24, 179-195.
  • Besag, J and Kooperberg, C., 1995. On Conditional and Intrinsic Autoregression. Biometrika 82, 733-746.
  • Breslow, N. E. and Clayton, D. G., 1993. Approximate inference in generalized linear mixed models. Journal of the American Statistical Association. 88, 9-25.
  • Brook, D., 1964, On the distinction between the conditional probability and the joint probability approaches in the specification of nearest-neighbour systems. Biometrika. 51, 481-483.
  • Clayton, D. and Kaldor, J., 1987, Empirical Bayes Estimates of Age-Standardized Relative Risks for Use in Disease Mapping. Biometrics. 43, 671-681.
  • Cliff, A. D. and Ord, J. K., 1981. Spatial Processes: Models & Applications. Pion Limited.
  • Cressie, N., 1993. Statistics for Spatial Data. John Wiley & Sons, Inc.
  • Cressie, N. and Chan, N. H., 1989. Spatial modeling of regional variables. Journal of the American Statistical Association. 84, 393-401.
  • Freeman, M. F. and Tukey, J. W., 1950. Transformations related to the angular and the square root. Annals of Mathematical Statistics. 21, 607-611.
  • Fuentes, M., 2002. Spectral methods for nonstationary spatial processes. Biometrika. 89, 197-210.
  • Fuentes, M., 2005. A formal test for nonstationarity of spatial stochastic processes. Journal of Multivariate Analysis. 96, 30-54.
  • Fuentes, M and Smith, R., 2001. A new class of nonstationary spatial models. Technical report, North Carolina State University, Department of Statistics.
  • Griffith, D. A. and Csillag, F., 1993. Exploring Relationships Between Semi-Variogram and Spatial Autoregressive Models. Papers in Regional Science. 72, 283-295.
  • Higdon, D., 1998. A process-convolution approach to modelling temperatures in the North Atlantic Ocean. Journal of Environmental and Ecological Statistics. 5, 173-190.
  • Higdon, D., Swall, J. and Kern, J., 1999. Non-stationary spatial modeling. In Bayesian Statistics 6, eds. J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith. Oxford: Oxford University Press, 761-768.
  • Hrafnkelsson, B. and Cressie, N., 2003. Hierarchical modeling of count data with application to nuclear fall-out. Journal of Environmental and Ecological Statistics. 10, 179-200.
  • Hughes-Oliver, J. M., Heo, T. Y. and Ghosh, S. K., 2009. An Autoregressive Point Source Model for Spatial Processes. Environmetrics. 20, 575-594.
  • Hyndman, R. J., 1996. Computing and Graphing Highest Density Regions. The American Statistician. 50, 120-126.
  • Journel, A. G. and Huijbregts, C. J., 1978. Mining geostatistics. London:Academic.
  • Kyung, M., 2006. Generalized Conditionally Autoregressive Models. Ph. D. Thesis, North Carolina State University, Department of Statistics.
  • Mardia, K. V. and Marshall, R. J., 1984. Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika. 71, 135-146.
  • McCullagh, P. and Nelder, J. A., 1989. Generalized Linear Models. Chapman and Hall, London.
  • Miller, H. J., 2004. Tobler's first law and spatial analysis. Annals of the Association of American Geographers. 94, 284-295.
  • Ord, K., 1975. Estimation methods for models of spatial interaction. Journal of the American Statistical Association. 70, 120-126.
  • Ortega, J. M., 1987. Matrix Theory. New York:Plenum Press.
  • Paciorek, C. J. and Schervish, M. J., 2006. Spatial modelling using a new class of nonstationary covariance functions. Environmetics. 17, 483-506.
  • Reich, B. J., Hodges, J. S. and Carlin, B. P., 2007. Spatial analyses of periodontal data using conditionally autoregressive priors having two classes of neighbor relations. Journal of the American Statistical Association. 102, 44-55.
  • Rue, H. and Tjelmeland, H., 2002. Fitting Gaussian Markov Random Fields to Gaussian Fields. Scandinavian Journal of Statistics. 29, 31-49.
  • Schabenberger, O. and Gotway, C. A., 2005. Statistical Methods for Spatial Data Analysis. Chapman & Hall/CRC.
  • Song, H. R., Fuentes, M. and Ghosh, S., 2008. A comparative study of Gaussian geostatistical models and Gaussian Markov random field models. Journal of Multivariate Analysis. 99, 1681-1697.
  • Spiegelhalter, D. J., Best, N. J., Carlin, B. P. and van der Linde, A., 2002. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B. 64, 583-639 (with discussion).
  • Sun, D., Tsutakawa, R. K. and Speckman, P. L., 1999. Posterior distribution of hierarchical models using CAR(1) distributions. Biometrika. 86, 341-350.
  • van der Linde, A., Witzko, K.-H. and Jöckel, K.-H., 1995. Spatio-temporal analysis of mortality using splines. Biometrics. 4, 1352-1360.
  • Wahba, G., 1977. Practical approximate solutions to linear operator equations when the data are noisy. SIAM Journal on Numerical Analysis. 14, 651-667.
  • White, G. and Ghosh, S. K., 2008. A Stochastic neighborhood Conditional Auto-Regressive Model for Spatial Data. Computational Statistics and Data Analysis. 53, 3033-3046.