Geoscience Reference
In-Depth Information
5.4.3
Spatial Effects
Assume now that
z
represents the location a particular observation pertains to.
If exact locations are available,
z
D
z
.1/
;
z
.2/
0
is two-dimensional, and the
components
z
.1/
and
z
.2/
correspond to the coordinates of the location. In this case,
the spatial effect f
z
.1/
;
z
.2/
could be modeled by two-dimensional extensions
of P-splines as described in Lang and Brezger (
2004
). An alternative approach
widely used in the geostatistics literature is to model the spatial effect by stationary
Gaussian random fields; see Kamman and Wand (
2003
).
If exact locations are not available as in our application, the correlated district-
specific heterogeneity effect f
mrf
5;6;2
.d i st / in Eq. (
5.5
) can be modeled by Markov
random fields (MRF). Suppose that
z
2f
1;:::;K
g
is the indicator for the
district in which a house is located. MRFs define one parameter for every discrete
geographical unit (districts in our case), that is, f.
z
/
D
ˇ
z
, and are defined via
the conditional distributions of ˇ
z
given the parameters ˇ
s
of neighboring sites s.
We denote the set of neighbors of site
z
by N.
z
/. Typically sites are assumed to
be neighbors if they share a common boundary. MRFs assume that the conditional
distribution of ˇ
z
given neighboring sites s
2
N.
z
/ is Gaussian with
0
1
X
2
j
N.
z
/
j
1
j
N.
z
/
j
@
A
;
ˇ
z
j
ˇ
s
;s
¤
z
N
ˇ
s
;
s2N.
z
/
where
j
N.
z
/
j
denotes the number of neighbors of site
z
.
The joint (prior) distribution of
ˇ
is of the form (
5.8
) with penalty matrix
K
given
by
<
1
z
¤
s; s
2
N.
z
/;
0
z
¤
s; s
62
N.
z
/
j
N.
z
/
j
z
D
s:
K
Œ
z
;s
D
(5.10)
:
If a Markov random field is used in the level-1 equation, the design matrix
Z
is
a 0/1 incidence matrix whose entry in the i -th row, and k-th column is 1 if the i -th
observed house is located in district k and 0 else. In our application, the MRF is
specified in the level-3 equation to model smooth district-specific heterogeneity. In
this case, the design matrix is the identity matrix, that is,
Z
5;6;2
D
I
.
5.4.4
Bayesian Inference Based on Markov Chain Monte
Carlo Simulations
It is beyond the scope of this chapter to present detailed algorithms for Bayesian
inference. Instead, we refer to the recent literature. An overview about Bayesian
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