Smoothness Properties and Gradient Analysis Under Spatial Dirichlet Process Models

  • Michele Guindani
  • Alan E. Gelfand


When analyzing point-referenced spatial data, interest will be in the first order or global behavior of associated surfaces. However, in order to better understand these surfaces, we may also be interested in second order or local behavior, e.g., in the rate of change of a spatial surface at a given location in a given direction. In a Bayesian parametric setting, such smoothness analysis has been pursued by Banerjee and Gelfand (2003) and Banerjee et al. (2003). We study continuity and differentiability of random surfaces in the Bayesian nonparametric setting proposed by Gelfand et al. (2005), which is based on the formulation of a spatial Dirichlet process (SDP). We provide conditions under which the random surfaces sampled from a SDP are smooth. We also obtain complete distributional theory for the directional finite difference and derivative processes associated with those random surfaces. We present inference under a Bayesian framework and illustrate our methodology with a simulated dataset.


Bayesian nonparametrics Directional derivatives Dirichlet process mixture models Finite differences Matérn correlation function, nonstationarity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions, Dover: New York, 1965.Google Scholar
  2. R. J. Adler, and J. E. Taylor, “Random fields and their geometry,” Unpublished, 2003.Google Scholar
  3. T. W. Anderson, An Introduction to Multivariate Statistical Analysis, third edition, Wiley: Hoboken, NJ, 2003.Google Scholar
  4. S. Banerjee, and A. E. Gelfand, “On smoothness properties of spatial processes,” Journal of Multivariate Analysis vol. 84 pp. 85–100, 2003.MATHMathSciNetCrossRefGoogle Scholar
  5. S. Banerjee, and A.E. Gelfand, “Curvilinear boundary analysis under spatial process models,” Journal of the American Statistical Association, 2005. (in press).Google Scholar
  6. S. Banerjee, A. E. Gelfand, and C. F. Sirmans, “Directional rates of change under spatial process models,” Journal of the American Statistical Association vol. 98 pp. 946–954, 2003.MATHMathSciNetCrossRefGoogle Scholar
  7. C. A. Bush, and S. N. MacEachern, “A semiparametric Bayesian model for randomised block designs,” Biometrika vol. 83 pp. 275–285, 1996.MATHCrossRefGoogle Scholar
  8. D. Damian, P. Sampson, and P. Guttorp, “Bayesian estimation of semi-parametric non-stationary spatial covariance structures,” Environmetrics vol. 12 pp. 161–178, 2001.CrossRefGoogle Scholar
  9. V. De Oliveira, B. Kedem, and D. A. Short. “Bayesian prediction of transformed Gaussian random fields” Journal of the American Statistical Association vol. 96 pp. 1361–1374, 1997.Google Scholar
  10. P. J. Diggle, J. A. Tawn, and R. A. Moyeed, “Model-based geostatistics (with discussion),” Applied Statistics vol. 47 pp. 299–350, 1998.MATHMathSciNetGoogle Scholar
  11. J. A. Duan, M. Guindani, and A. E. Gelfand. Generalized Spatial Dirichlet Process Models. Submitted, 2005.Google Scholar
  12. M. D. Escobar, “Estimating normal means with a Dirichlet process prior,” Journal of the American Statistical Association vol. 89 pp. 268–277, 1994.MATHMathSciNetCrossRefGoogle Scholar
  13. M. D. Escobar, and M. West, “Bayesian density estimation and inference using mixtures,” Journal of the American Statistical Association vol. 90 pp. 577–588, 1995.MATHMathSciNetCrossRefGoogle Scholar
  14. T. S. Ferguson, “A Bayesian analysis of some nonparametric problems,” The Annals of Statistics vol. 1 pp. 209–230, 1973.MATHMathSciNetGoogle Scholar
  15. M. Fuentes, and R. L. Smith, “A new class of nonstationary spatial models,” Technical report, Department of Statistics, North Carolina State University, 2001.Google Scholar
  16. A. E. Gelfand, and A. F. M. Smith, “Sampling-based approaches to calculating marginal densities,” Journal of the American Statistical Association vol. 85 pp. 398–409, 1990.MATHMathSciNetCrossRefGoogle Scholar
  17. A. E. Gelfand, A. Kottas, and S. N. MacEachern, “Bayesian nonparametric spatial modeling with Dirichlet processes mixing,” Journal of the American Statistical Association vol. 100 pp. 1021–1035, 2005.MathSciNetCrossRefGoogle Scholar
  18. S. Ghosal, J. K. Ghosh, and R. V. Ramamoorthi, “Posterior consistency of Dirichlet mixture in density estimation,” The Annals of Statistics vol. 27 pp. 143–158, 1999.MATHMathSciNetCrossRefGoogle Scholar
  19. M. S. Handcock, and M. Stein, “A Baesian analysis of kriging,” Technometrics vol. 35(4) pp. 403–410, 1993.CrossRefGoogle Scholar
  20. D. A. Harville, Matrix Algebra from a Statistician’s Viewpoint, Springer Series in Statistics, Springer-Verlag: New York, 1997.Google Scholar
  21. D. Higdon, J. Swall, and J. Kern, “Non-stationary spatial modeling.” In J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith (eds.), Bayesian Statistics 6, Oxford University Press: Oxford, 1999.Google Scholar
  22. H. Ishwaran, and M. Zarepour, “Exact and approximate sum-representations for the Dirichlet process,” Canadian Journal of Statistics vol. 30 pp. 269–283, 2002.MATHMathSciNetGoogle Scholar
  23. J. T. Kent, “Continuity properties of random fields,” Annals of Probability vol. 17 pp. 1432–1440, 1989.MATHMathSciNetGoogle Scholar
  24. S. N. MacEachern, Dependent Dirichlet processes. Technical report, Department of Statistics, The Ohio State University, 2000.Google Scholar
  25. S. N. MacEachern, and P. Müller, “Estimating mixture of Dirichlet process models,” Journal of Computational and Graphical Statistics vol. 7 pp. 223–238, 1998.CrossRefGoogle Scholar
  26. A. Majumdar, H. Munneke, S. Banerjee, A.E. Gelfand, and F. Sirmans, “Gradients in spatial response surfaces, land value gradients, and endogenous central business districts,” Submitted to The Journal of Business and Economic Studies, 2004.Google Scholar
  27. B. Matern, Spatial Variation, 2nd edition, Springer Verlag: Berlin, 1986.MATHGoogle Scholar
  28. M. B. Palacios, and M. F. J. Steel, “Non-gaussian Bayesian geostatistical modeling,” Warwick Statistics Research report 426, University of Warwick, 2004.Google Scholar
  29. P. D. Sampson, and P. Guttorp, “Nonparametric estimation of nonstationary spatial covariance structure,” Journal of the American Statistical Association vol. 87(417) pp. 108–119, 1992.CrossRefGoogle Scholar
  30. A. M. Schmidt, and A. O’Hagan, “Bayesian inference for non-stationary spatial covariance structure via spatial deformations,” Journal of the Royal Statistical Society. Series B, Statistical methodology vol. 65(3) pp. 743–758, 2003.MATHMathSciNetCrossRefGoogle Scholar
  31. J. Sethuraman, “A constructive definition of Dirichlet priors,” Statistica Sinica vol. 4 pp. 639–650, 1994.MATHMathSciNetGoogle Scholar
  32. M. L. Stein, Interpolation of Spatial DataSome theory of Kriging, Springer: New York, 1999.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of Biostatistics and Applied MathematicsUT MD Anderson Cancer Care CenterHoustonUSA
  2. 2.Institute of Statistics and Decision SciencesDuke UniversityDurhamUSA

Personalised recommendations