Generalized spatial dirichlet process models

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. Copyright 2007, Oxford University Press.

[1]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[2]  T. Ferguson.  Prior Distributions on Spaces of Probability Measures , 1974 .

[3]  John T. Kent,et al.  Continuity Properties for Random Fields , 1989 .

[4]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[5]  J. Sethuraman.  A CONSTRUCTIVE DEFINITION OF DIRICHLET PRIORS , 1991 .

[6]  P. Guttorp,et al.  Nonparametric Estimation of Nonstationary Spatial Covariance Structure , 1992 .

[7]  S. Chib,et al.  Bayesian analysis of binary and polychotomous response data , 1993 .

[8]  S. MacEachern.  Estimating normal means with a conjugate style dirichlet process prior , 1994 .

[9]  M. Escobar,et al.  Bayesian Density Estimation and Inference Using Mixtures , 1995 .

[10]  S. MacEachern,et al.  A semiparametric Bayesian model for randomised block designs , 1996 .

[11]  S. MacEachern,et al.  Estimating mixture of dirichlet process models , 1998 .

[12]  David Higdon,et al.  Non-Stationary Spatial Modeling , 2022, 2212.08043.

[13]  Nils Lid Hjort,et al.  Bayesian analysis for a generalised Dirichlet process prior , 2000 .

[14]  Lancelot F. James,et al.  Gibbs Sampling Methods for Stick-Breaking Priors , 2001 .

[15]  P. Guttorp,et al.  Bayesian estimation of semi‐parametric non‐stationary spatial covariance structures , 2001 .

[16]  Montserrat Fuentes,et al.  A New Class of Nonstationary Spatial Models , 2001 .

[17]  H. Ishwaran,et al.  Exact and approximate sum representations for the Dirichlet process , 2002 .

[18]  P. Green,et al.  Modelling spatially correlated data via mixtures: a Bayesian approach , 2002 .

[19]  H. Ishwaran,et al.  DIRICHLET PRIOR SIEVES IN FINITE NORMAL MIXTURES , 2002 .

[20]  C. F. Sirmans,et al.  Directional Rates of Change Under Spatial Process Models , 2003 .

[21]  P. Diggle,et al.  A Non‐Gaussian Spatial Process Model for Opacity of Flocculated Paper , 2003 .

[22]  A. O'Hagan,et al.  Bayesian inference for non‐stationary spatial covariance structure via spatial deformations , 2003 .

[23]  A. Gelfand,et al.  Spatial modeling and prediction under stationary non-geometric range anisotropy , 2003, Environmental and Ecological Statistics.

[24]  S. MacEachern,et al.  An ANOVA Model for Dependent Random Measures , 2004 .

[25]  S. MacEachern,et al.  Bayesian Nonparametric Spatial Modeling With Dirichlet Process Mixing , 2005 .

[26]  Deepak K. Agarwal,et al.  Slice sampling for simulation based fitting of spatial data models , 2005, Stat. Comput..

[27]  J. E. Griffin,et al.  Order-Based Dependent Dirichlet Processes , 2006 .

[28]  Mark F. J. Steel,et al.  Non-Gaussian Bayesian Geostatistical Modeling , 2006 .

[29]  D. Dey,et al.  Semiparametric Bayesian estimation of random coefficients discrete choice models , 2007 .

[30]  Donato Michele Cifarelli,et al.  Nonparametric statistical problems under partial exchangeability . The role of associative means . Translated from Problemi statistici non parametrici in condizioni di scambiabilità parziale : impiego di medie associative , 2008 .