Smoothness Properties and Gradient Analysis Under Spatial Dirichlet Process Models

Michele Guindani () and Alan E. Gelfand ()
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Michele Guindani: UT MD Anderson Cancer Care Center
Alan E. Gelfand: Duke University

Methodology and Computing in Applied Probability, 2006, vol. 8, issue 2, 159-189

Abstract: 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.

Keywords: Bayesian nonparametrics; Directional derivatives; Dirichlet process mixture models; Finite differences; Matérn correlation function; nonstationarity (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1007/s11009-006-8547-8

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