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Biometrika Advance Access published online on December 3, 2007
Biometrika, doi:10.1093/biomet/asm071
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© 2007 Biometrika Trust

Articles

Generalized Spatial Dirichlet Process Models

Jason A. Duan

School of Management, Yale University, New Haven, Connecticut 06520-8200, U.S.A. jd522{at}som.yale.edu

Michele Guindani

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, U.S.A. michele{at}stat.unm.edu

Alan E. Gelfand

Institute of Statistics and Decision Sciences, Duke University, Durham, North Carolina 27708-0251, U.S.A. alan{at}stat.duke.edu

Received for publication 1 December 2005. Revision received 1 April 2007.
  Abstract

Many models for the study of point-referenced data explicitly introduce spatial random effects to capture residual spatial association. These spatial effects are customarily modelled as a zero-mean stationary Gaussian process. The spatial Dirichlet process introduced by Gelfand et al. (2005) produces a random spatial process which is neither Gaussian nor stationary. Rather, it varies about a process that is assumed to be stationary and Gaussian. The spatial Dirichlet process arises as a probability-weighted collection of random surfaces. This can be limiting for modelling and inferential purposes since it insists that a process realization must be one of these surfaces. We introduce a random distribution for the spatial effects that allows different surface selection at different sites. Moreover, we can specify the model so that the marginal distribution of the effect at each site still comes from a Dirichlet process. The development is offered constructively, providing a multivariate extension of the stick-breaking representation of the weights. We then introduce mixing using this generalized spatial Dirichlet process. We illustrate with a simulated dataset of independent replications and note that we can embed the generalized process within a dynamic model specification to eliminate the independence assumption.

Key Words: Dirichlet process mixing • Dynamic model • Latent process • Non-Gaussian • Nonstationary • Stick breaking


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