Generalized Spatial Dirichlet Process Models

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