Transcript Slide 1

Generalized Spatial Dirichlet Process Models
Jason A. Duan, Michele Guindani and Alan E. Gelfand
Presenter: Lu Ren
ECE@Duke
Oct 23, 2008
Outline
1. Introduction
2. Spatial Dirichlet process (SDP)
3. Generalized spatial Dirichlet process (GSDP)
4. The spatially varying probabilities model
5. Simulation-based model fitting
6. 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:
1. Allows different surface selection at different sites
2. Marginal distribution of the effect still comes from a DP
SDP
Denote the stochastic process: {Y (s) : s  D} D  R d s ( n)  (s1 ,, sn )
We have replicate observations at each location: Yt  {Yt (s1 ),, Yt (sn )}T
A random distribution on (, ) drawn from DP(G0 ) is almost surely
discrete :  pl *
l 1
l
A spatial Dirichlet process: replace l with a realization of a

random field l*,D  {l* (s) : s  D} so that G   pl *
*
l 1
l ,D
( n)
G( n) ~ DP(G0( n) ) G0 is the n-variate distribution for {Y (s1 ),, Y (sn )}
SDP: the continuity of  l*,D implies that G{Y ( s)} is continuous
GSDP
Drawbacks of SDP:
The joint distribution of n locations uses the same set of stickbreaking probabilities;
It cannot capture more flexible spatial effects.
We define a random probability measure G on the space of
surfaces over D, for any set of locations (s1 ,, sn )  D :
{ pi1 ,,in } 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 Y ;
For any set of (s1 ,, sn ) and for all k {1,, n}
In addition, the weights satisfy a continuity property: random
effects associated with s1 and s 2 near to each other to be similar.
e.g. for s and s 0 , as s  s0 ,
tends to the marginal probability
i1  i2 and to
0 otherwise.
when
GSDP
Random effect model: Y (s)  u(s)   (s)   (s)
where u(s)  X (s)T  and  (s) 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  Z () .
Assume the mean of the l th process, l (s) , is unknown.
GSDP
Consider the stochastic process
If
and
:
if
Zl (s)  Al (s)
in which Al (s)  {Zl (s)  0. }
For example, for n  2,
For any s,
If
are independent
distribution of  (s) is a Dirichlet process.
, the marginal
Model Specification
Model Specification
For model fitting, the joint random distribution G (n) is
approximated with a finite sum:
For t  1,, T and l  1,, K  1 , we sample the latent
variables Zl in stead of computing the weights Pi1,,in
Zt ,l (s)  {s  D, Zt ,1 (s)  0,, Zt ,l 1 (s)  0, Zt ,l (s)  0}
Zt ,K (s)  {s  D, Zt ,1 (s)  0,, Zt ,K 1 (s)  0}
Simulation
A set of locations in a given region: (s1 ,, sn ) and T replicates;
For t  1,, T , let
and
yt (si ) ~ 12 N (1) (,12 )  12 N (2) (, 22 )
50 design locations and 40
independent replicates;
1  2  3 1  2 2  2
1  2  0.3
K  20
  0.3
Simulation
Simulation
Thanks!