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Smoothness Properties and Gradient Analysis Under Spatial Dirichlet Process Models

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  • Michele Guindani

    (UT MD Anderson Cancer Care Center)

  • Alan E. Gelfand

    (Duke University)

Abstract

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.

Suggested Citation

  • Michele Guindani & Alan E. Gelfand, 2006. "Smoothness Properties and Gradient Analysis Under Spatial Dirichlet Process Models," Methodology and Computing in Applied Probability, Springer, vol. 8(2), pages 159-189, June.
    • Handle: RePEc:spr:metcap:v:8:y:2006:i:2:d:10.1007_s11009-006-8547-8
    DOI: 10.1007/s11009-006-8547-8
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    References listed on IDEAS

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    1. Alexandra M. Schmidt & Anthony O'Hagan, 2003. "Bayesian inference for non‐stationary spatial covariance structure via spatial deformations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(3), pages 743-758, August.
    2. Gelfand, Alan E. & Kottas, Athanasios & MacEachern, Steven N., 2005. "Bayesian Nonparametric Spatial Modeling With Dirichlet Process Mixing," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1021-1035, September.
    3. Banerjee, S. & Gelfand, A. E., 2003. "On smoothness properties of spatial processes," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 85-100, January.
    4. Banerjee S. & Gelfand A.E. & Sirmans C.F., 2003. "Directional Rates of Change Under Spatial Process Models," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 946-954, January.
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    Cited by:

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