Geoscience Reference
In-Depth Information
inference in additive models based on MCMC simulations is given in Fahrmeir et al.
(
2013
). Algorithms for hierarchical structured additive regression are given in Lang
et al. (
2014
). Bayesian distributional regression in the spirit of GAMLSS is treated in
Klein et al. (
2013
). Bayesian quantile regression is discussed in detail in Waldmann
et al. (
2013
) and Fahrmeir et al. (
2013
).
5.5
Generalized Random Slope Modeling
A common phenomenon observed for real estate data is spatial heterogeneity in the
sense that (possibly nonlinear) effects of covariates vary in size (and possibly also
shape) from one spatial unit to another. This is primarily the case if the spatial units,
for example, districts, can be regarded as submarkets of one larger market.
In order to model spatially heterogeneous effects, we discuss in this section
nonlinear generalizations of random slopes. Suppose that for a continuous covariate
z
a nonlinear effect f.
z
/ is assumed. Moreover, suppose that there might be
heterogeneity with respect to a cluster variable c
2f
1;:::;C
g
in the sense
that the nonlinear function is not homogeneous from cluster to cluster. However,
completely different functional forms in each cluster are not likely a priori. Instead,
one might think that only a particular feature of the function is subject to cluster-
specific heterogeneity. Here, we assume homogeneity for the functional form but
heterogeneity for the scaling of the function. This leads to a term of the form
.1
C
˛
c
i
/f .
z
i
/
D
f.
z
i
/
C
˛
c
i
f.
z
i
/;
where the possibly nonlinear function f of
z
is scaled by the factor .1
C
˛
c
/.In
matrix notation, we obtain
diag.1
C
˛
c
1
;:::;1
C
˛
c
n
/
Z
ˇ
;
(5.11)
where
Z
is the design matrix corresponding to the nonlinear function f .An
equivalent formulation in terms of the cluster-specific scaling parameter vector
˛ D
.˛
1
;:::;˛
C
/
0
and a 0/1 incidence (design) matrix
C
for the cluster-specific
scaling effect is given by
f
C
diag.f .
z
1
/;:::;f.
z
n
//
C
˛
:
(5.12)
Similar to pure additive models, the prior for the scaling parameter vector
˛ D
.˛
1
;:::;˛
C
/
0
may obey another structured additive model, that is,
˛ D
Z
˛;1
ˇ
˛;1
C
:::
C
Z
˛;q
˛
ˇ
˛;q
˛
C
X
˛
˛
C "
˛
:
Some care has to be taken regarding identifiability of the parameters. In
particular, there is an arbitrary multiplicative constant for the nonlinear function f .
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