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

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

3. 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

4. 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

5. 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

6. 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.

7. 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.

8. 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.

9. Model Specification

10. 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

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

12. Simulation

14. Simulation

15. Thanks!