Generalized Spatial Dirichlet Process Models

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Presentation transcript:

Generalized Spatial Dirichlet Process Models Jason A. Duan, Michele Guindani and Alan E. Gelfand Presenter: Lu Ren ECE@Duke Oct 23, 2008

Outline Introduction Spatial Dirichlet process (SDP) Generalized spatial Dirichlet process (GSDP) The spatially varying probabilities model Simulation-based model fitting Simulation example

Introduction Distributional modelling for point-referenced spatial data e.g. stationary Gaussian process, spatially varying kernel approach Spatial Dirichlet process: a mixture of Gaussian processes The inappropriate stationarity or the Gaussian assumption Generalized spatial Dirichlet process: Allows different surface selection at different sites Marginal distribution of the effect still comes from a DP

SDP Denote the stochastic process: We have replicate observations at each location: A random distribution on drawn from is almost surely discrete : A spatial Dirichlet process: replace with a realization of a random field so that is the n-variate distribution for SDP: the continuity of implies that is continuous

GSDP Drawbacks of SDP: The joint distribution of n locations uses the same set of stick-breaking probabilities; It cannot capture more flexible spatial effects. We define a random probability measure on the space of surfaces over D, for any set of locations : determine the site-specific joint selection probabilities

GSDP The weights need to satisfy a consistency condition in order to define properly a random process for ; For any set of and for all In addition, the weights satisfy a continuity property: random effects associated with and near to each other to be similar. e.g. for and , as , tends to the marginal probability when and to otherwise.

The spatially varying probabilities model GSDP Random effect model: where and is a Gaussian pure random error The spatially varying probabilities model A constructive approach is provided and can be viewed as multivariate stick-breaking: Gaussian thresholding. Assume is a countable collection of independent stationary Gaussian random fields on D, having variance 1 and correlation function . Assume the mean of the th process, , is unknown.

GSDP Consider the stochastic process : If and if in which . For example, for For any s, If are independent , the marginal distribution of is a Dirichlet process.

Model Specification

Model Specification For model fitting, the joint random distribution is approximated with a finite sum: For and , we sample the latent variables in stead of computing the weights

Simulation A set of locations in a given region: and replicates; For , let and 50 design locations and 40 independent replicates;

Simulation

Simulation

Thanks!