spatial dynamic factor analysis
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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.
![Page 2: Spatial Dynamic Factor Analysis](https://reader036.vdocument.in/reader036/viewer/2022062501/56816792550346895ddcca03/html5/thumbnails/2.jpg)
Outline
• Introduction• Spatial Dynamic Factor Analysis Model• Inference and Application• Experiment• Future direction
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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
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Spatial Dynamic Factor Analysis
),((.)),(~
),0(~),0(~
22)(
1**
*
jjRNGRF
NffNfy
jjjjj
tttt
tttytt
)( ,....,1 mdiag
),...,( 221 Ndiag
),....,( 1 mdiag
|)(|:),( kllk ssrRofelementkljj
Model:
)/exp()( dd
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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
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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
NffNfy
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
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Seasonal dynamic factorsGoal: capture the periodic or cyclical behavior
Example :
p=52 for weekly data and annual cycle
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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
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MCMC InferenceAssume:
Posterior distribution:
Full conditional distribution of all parameters can be found in appendix
Model in matrix notation
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Number of factorsReverse jump MCMC
accept With probability
Collect samples
Proposal distribution:
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ApplicationsPrediction
Interpolation
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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
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ExperimentSpatial dynamic factor models
Benchmark model
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Experiment
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Experiment
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Experiment
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Experiment
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Experiment
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Future direction
• Time varying factor loadings• Allow binomial and Poisson response • Non-diagonal covariance matrix and more
general dynamic structure.