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Spatial Dynamic Factor Analysis
Hedibert Freitas Lopes, Esther Salazar, Dani Gamerman
Presented by Zhengming Xing
Jan 29 ,2010
* tables and figures are directly copied from the original paper.
Outline
• Introduction
• Spatial Dynamic Factor Analysis Model
• Inference and Application
• Experiment
• Future direction
Basic factor analysis
),0(~ Nfuy ttttt
Spatial dynamic FA model),...,( 1 Nttt yyy
Nss ,...,1 Tt ,...,1Locations: Times:
key idea:
Temporal dependence is modeled by latent factor score and spatial dependence is modeled by the factor loadings
Factor loading matrix mN
tf 1mFactor score
ty observations 1N
introduction
Spatial Dynamic Factor Analysis
),((.)),(~
),0(~
),0(~
22)(
1
**
*
jjRNGRF
Nff
Nfy
jjjjj
tttt
tttytt
)( ,....,1 mdiag
),...,( 221 Ndiag
),....,( 1 mdiag
|)(|:),( kllk ssrRofelementkljj
Model:
)/exp()( dd
Covariate effects*yt
),(~ 1 WN yt
yt
1.Constant mean
2.Regression model
3.Dynamic coefficient model
yyt
*
yyt
yt X
*
yyt
yt tX
*
0*
j
Njj 1*
jjj X*
1.
2.
3.
Mean level of the spatio-time process
Mean level of Gaussian process
* j
Prior information
)2/,2/(~2 snnIGi
)2/,2/(~ snnIGi
)()1(),(~
),(~
1)1,1(
)1,1(
jtrj
trj
smN
smN
Recall:
),((.)),(~
),0(~
),0(~
22)(
1
**
*
jjRNGRF
Nff
Nfy
jjjjj
tttt
tttytt
),...,( 221 Ndiag
),....,( 1 mdiag
)( ,....,1 mdiag
|)(|:),( kllk ssrRofelementkljj
Priors:
),(~ SmNj
),2(~ bIGj
),(~2 snnIGj
))05.0ln(2/(0 b
||max ,...,1,0 jiNji ss
Seasonal dynamic factorsGoal: capture the periodic or cyclical behavior
Example :
p=52 for weekly data and annual cycle
Spatio-temporal separability
),( tsZ
))|(cov)|(cov()|(cov)|(cov)),(),,(cov( 2211 horhtsZtsZ tsts
)),(()1)((),cov( 212,
jih
htjit yy
Random process indexed by space and time
)),(()1)((),cov( 212
1,
jkikkkkhkk
m
khtjit yy
Assume
if
then separable
Choose for convenience rather than for the ability to fit the data
SDFA model
m=1
m>1
MCMC InferenceAssume:
Posterior distribution:
Full conditional distribution of all parameters can be found in appendix
Model in matrix notation
Number of factorsReverse jump MCMC
accept With probability
Collect samples
Proposal distribution:
ApplicationsPrediction
Interpolation
Experiment
Sulfur dioxide concentration in eastern US
24 stations
342 observations (from the first week of 1998 to the 30th week of 2004)
2 station left out for interpolation and the last 30 weeks left out for prediction
Dataset available online:
http://www.epa.gov/castnet/data.html
Data description
ExperimentSpatial dynamic factor models
Benchmark model
Experiment
Experiment
Experiment
Experiment
Experiment
Future direction
• Time varying factor loadings
• Allow binomial and Poisson response
• Non-diagonal covariance matrix and more general dynamic structure.
