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
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Introduction
Spatial Dynamic Factor Analysis Model
Inference and Application
Experiment
Future direction
introduction
Basic factor analysis
yt  ut  f t   t
yt

ft
 t ~ N (0, )
observations
Factor loading matrix
Factor score
N 1
N m
m 1
Spatial dynamic FA model
yt  ( y1t ,..., y Nt )
Locations:
s1 ,..., sN
Times:
t  1,..., T
key idea:
Temporal dependence is modeled by latent factor score and
spatial dependence is modeled by the factor loadings
Spatial Dynamic Factor Analysis
Model:
yt    f t   t
 t ~ N (0, )
 t ~ N (0, )
y*
t
f t  f t 1   t
*
*
 ( j ) ~ GRF (  j ,  (.))  N (  j , 2j R )
2
j
j
j
(l , k ) element of R j : rlk   j (| sl  sk |)
 (d )  exp( d /  )
  diag (1 ,...., m )
  diag ( 1, .... , m )
  diag ( 12 , ...,  N2 )
Covariate effects

y*
t
Mean level of the
spatio-time process
1.Constant mean
 
y*
t
j
1.
y
2.Regression model
2.
ty  X ty  y
*
3.Dynamic coefficient model
 X 
y*
t
y
t
y
t
ty ~ N ( ty1 ,W )
Mean level
of Gaussian
process
*
*
j  0
*
 j   j 1N
3.
*


j  X j j
Prior information
Recall:
yt    f t   t
 t ~ N (0, )
 t ~ N (0, )
y*
t
f t  f t 1   t
  diag ( 1, .... , m )
 ( j ) ~ GRF (  j , 2j  (.))  N (  j , 2j R )
*
  diag (1 ,...., m )
*
j
j
(l , k ) element of R j : rlk   j (| sl  sk |)
  diag ( 12 , ...,  N2 )
Priors:
 i2 ~ IG (n / 2, n s / 2)
i ~ IG (n / 2, n s / 2)
 j ~ Ntr ( 1,1) (m , s )
 j ~ Ntr ( 1,1) (m , s )  (1   )1 ( j )
 j ~ N (m , S )  2j ~ IG(n , n s )
 j ~ IG (2, b)
b  0 /( 2 ln( 0.05))
 0  max i , j 1,..., N | si  s j |
Seasonal dynamic factors
Goal: capture the periodic or cyclical behavior
Example :
p=52 for weekly data and annual cycle
Spatio-temporal separability
Assume
Z ( s, t )
Random process indexed by space and time
if
cov( Z ( s1 , t1 ), Z ( s2 , t2 ))  cov s ( |  ) covt (h |  ) or ( cov s ( |  )  covt (h |  ) )
then
separable
Choose for convenience rather than for the ability to fit the data
SDFA model
m=1
cov( yit , y j ,t h )  ( h )(1   2 )1 ( 2  (,  )  i  j )
m>1
m
cov( yit , y j ,t  h )   (k  k )(1   k ) 1 ( k  (  , k )  ik  jk )
k 1
h
2
2
MCMC Inference
Assume:
Model in matrix notation
Posterior distribution:
Full conditional distribution of all parameters can be found in appendix
Number of factors
Reverse jump MCMC
Collect samples
Proposal distribution:
accept
With probability
Applications
Prediction
Interpolation
Experiment
Data description
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
Experiment
Spatial 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.