Generalized Spatial Dirichlet Process Models Jason A. Duan Michele Guindani Alan E. Gelfand March, 2006

Generalized Dirichlet Process Models – 2 Agenda Overview of Bayesian Spatial Statistics Models. Spatial Dirichlet Processes (SDP) Generalized Spatial Dirichlet Processes (GSDP) –Multivariate Stick-Breaking Random Probability Measure –Spatially Varying Probability Model –Simulation Based Model Fitting –Illustrative Example –Dynamic Model with GSDP Conclusions

Generalized Dirichlet Process Models – 3 Bayesian Spatial Model Suppose we have point-referenced spatial data such that

Generalized Dirichlet Process Models – 4 Bayesian Spatial Model We can build a hierarchical model for Customary choice for priors.

Generalized Dirichlet Process Models – 5 Criticism of Gaussian Model Two problems with this model: Gaussian assumption: –Multi-modality. –Heavy tail behavior. Stationary assumption: –Inappropriate for most environmental issues: dispersion of pollutant, meteorology, etc. Nonparametric models add flexibility.

Generalized Dirichlet Process Models – 6 Spatial Dirichlet Process (SDP) Sethuraman’s construction for the SDP: This is a stick-breaking construction for the random weights.

Generalized Dirichlet Process Models – 7 Motivations for GSDP SDP implies the same surface selection for for all locations. GSDP has a flexible surface selection. –Fewer random surfaces needed. –Better for small data sets. –Applications: brain imaging, species distribution, house prices, etc.

Generalized Dirichlet Process Models – 8 Definition of GSDP How to construct a consistent distribution? Multivariate stick-breaking.

Generalized Dirichlet Process Models – 9 Association Structure of GSDP Moments conditional on G Y(s) has heterogeneous variance and is nonstationary.

Generalized Dirichlet Process Models – 10 Association Structure of GSDP

Generalized Dirichlet Process Models – 11 Consistency and Continuity Requirements  The first requirement is met by the consistency of the multivariate stick-breaking.  The second requirement is established by the continuity property of the GSDP.

Generalized Dirichlet Process Models – 12 Multivariate Stick-Breaking

Generalized Dirichlet Process Models – 13 Multivariate Stick-Breaking Multivariate extension: We exemplify with a bivariate case Similarly, we define two events at one location s

Generalized Dirichlet Process Models – 14 Multivariate Stick-Breaking

Generalized Dirichlet Process Models – 15 Multivariate Stick-Breaking

Generalized Dirichlet Process Models – 16 Spatially Varying Weights We createthrough latent Gaussian processes: Define How to construct?

Generalized Dirichlet Process Models – 17 Spatially Varying Weights Define joint weight This definition of joint weights satisfies the consistency and continuity requirements.

Generalized Dirichlet Process Models – 18 Hierarchical Spatial Model Our model has the following hierarchical structure:

Generalized Dirichlet Process Models – 19 Hierarchical Spatial Model

Generalized Dirichlet Process Models – 20 GSDP Mixture Model Note the first two hierarchies of our model define a GSDP mixture of multivariate normal distributions: where This construction alleviates the undesirable almost discreteness property of the GSDP. The GSDP mixture model resonates the typical “random effect + nugget” spatial model.

Generalized Dirichlet Process Models – 21 Moments of GSDP Mixture where

Generalized Dirichlet Process Models – 22 Model Fitting The marginal model at each location is a DP model. The joint model has NO Polya Urn Scheme. In practice, we have to use a finite approximation. We only need

Generalized Dirichlet Process Models – 23 Model Fitting Evaluation of weightsis difficult. We directly sample the auxiliary variable

Generalized Dirichlet Process Models – 24 Model Fitting The likelihood function is The posterior conditional distribution of is truncated normal.

Generalized Dirichlet Process Models – 25 Full Conditionals

Generalized Dirichlet Process Models – 26 Full Conditionals

Generalized Dirichlet Process Models – 27 A Simulation Example 50 locations, 40 replicates

Generalized Dirichlet Process Models – 28 A Simulation Example has a bimodal normal mixture distribution.

Generalized Dirichlet Process Models – 29 Simulation Results: Marginal Density Thick Curve: GSDP Dashed Curve: SDP Dotted Curve: True Distribution

Generalized Dirichlet Process Models – 30 Simulation Results: Joint Density Very close locations: 26 and 50. Predictive density: GSDPTrue DensityPredictive density: SDP

Generalized Dirichlet Process Models – 31 Simulation Results: Joint Density Slightly Distant locations: 33 and 50. Predictive density: GSDPTrue DensityPredictive density: SDP

Generalized Dirichlet Process Models – 32 Simulation Results: Joint Density Very Distant locations: 49 and 50. Predictive density: GSDPTrue DensityPredictive density: SDP

Generalized Dirichlet Process Models – 33 Results: Common Surface Selection

Generalized Dirichlet Process Models – 34 Dynamic GSDP Model We can model it by embeding GSDP in a dynamic linear model: In practice repeated measurements are made in consecutive time periods: has temporal correlation.

Generalized Dirichlet Process Models – 35 Dynamic GSDP: Simulation 54 locations, 41 periods

Generalized Dirichlet Process Models – 36 Dynamic GSDP: Simulation = 0.7  We fit the Dynamic GSDP to 50 locations and 40 periods, leaving the last period and 4 locations out to validate prediction.

Generalized Dirichlet Process Models – 37 Dynamic GSDP: Prediction

Generalized Dirichlet Process Models – 38 Dynamic GSDP: Prediction Predictive density: T+1True Density: T+1 Very close locations: 23 and 50.

Generalized Dirichlet Process Models – 39 Dynamic GSDP: Prediction Predictive density: T+1True Density: T+1 Very distant locations: 49 and 50.

Generalized Dirichlet Process Models – 40 Conclusion GSDP is a more flexible generalization of SDP Methodology Development –Number of surfaces for finite approximation –Discrete and multivariate data GSDP –Other constructions of random weights Future Applications of GSDP –Species distribution –Brain Imaging –Spatial demand function –House prices