342
Handbook of Discrete-Valued Time Series
60
50
60°N
40
30
45°N
165°W
90°W
20
150°W
135°W
120°W
105°W
10
0
(b)
40
35
30
60°N
25
20
45°N
15
165°W
90°W
150°W
105°W
10
135°W
120°W
5
(c)
FIGURE 15.2 (Continued)
Observed and (one-step-ahead forecasts) predicted counts for 2010 (using the median of the posterior predictive
distribution) and their median absolute deviation (MAD) for the Poisson model (Section 15.5.3) using 8 EOFs.
(b) Plot of
Y
ˆ
2010
, (c) MAD of in-sample prediction.
15.6 Conclusion
Modeling discrete-valued spatio-temporal data is often a challenging endeavor due to
model complexity and high dimensionality. That is, specifying realistic dependence struc-
ture for a real-world observed spatio-temporal process can be extremely difcult due to
inherent nonlinearities in time coupled with potential nonstationary behaviors in time
Hierarchical Dynamic Generalized Linear Mixed Models 343
1
1
1
1.5
1.0
2
2
0.5
2
3
3
3
1.0
0.5
0.0
4
4
4
5
5
5
−0.5
0.5
0.0
6
6
6
−1.0
7
7
7
0.0
−0.5
8
8
8
1 2 3
4 5 6 7 8
1 2 3
4 5 6 7 8
1 2 3
4 5 6 7 8
(a)
(b) (c)
1
1
1
0.5
1.0
2.0
2
2
2
0.0
3
0.5
3
3
1.5
−0.5
4
4
4
−1.0
0.0
5
5
1.0
5
−1.5
6
6
6
−0.5
0.5
7
−1.0
8
7
−2.0
7
8
8
1 2 3
4 5 6 7 8
1 2 3
4 5 6 7 8
1 2 3 4 5 6 7 8
(d) (e) (f)
1
1
1
1.0
0.5
2.5
2
2
2
0.0
3
2.0
3
3
0.5
−0.5
4
4
4
1.5
−1.0
5
0.0
5
5
−1.5
6
7
6
6
1.0
−0.5
7
−2.0
7
0.5
8
8
8
1 2 3 4 5 6 7 8
1 2 3
4 5 6 7 8
1 2 3 4 5 6 7 8
(g) (h) (i)
FIGURE 15.3
Image plots for element-wise posterior mean and 95% credible intervals of redistribution matrices M
i
, i = 1, 2, 3
for the Poisson model (Section 15.5.3) using 8 EOFs. Note that lower CI and upper CI denote the 2.5% and 97.5%
quantiles of the posterior distributions, respectively. (a) M
1
: Lower CI, (b) M
1
: Posterior mean, (c) M
1
: Upper CI,
(d) M
2
: Lower CI, (e) M
2
: Posterior mean, (f) M
2
: Upper CI, (g) M
3
: Lower CI, (h) M
3
: Posterior mean, and (i) M
3
:
Upper CI.
and/or space. Instead, utilizing a process-based approach and taking advantage of the
Bayesian hierarchical paradigm often prove remarkably advantageous.
Although, in principle, there are many approaches to modeling discrete-valued spatio-
temporal data, we presented a process-driven Bayesian hierarchical perspective. The key
to this paradigm is that the joint probability model for the data, process, and parameters
can be conditionally specied through linked model components. In particular, this for-
mulation facilitates the specication of complicated marginal dependence through a more
344 Handbook of Discrete-Valued Time Series
scientically motivated representation of the conditional mean, which can be modeled at
the process (or parameter) stage in the model hierarchy.
We have described BHMs for modeling discrete-valued spatio-temporal data. For
those less familiar with the hierarchical paradigm, we have provided a broad overview
(Section 15.2). To facilitate model development, we have outlined several discrete-valued
data model distributions, placing an emphasis on count-valued data model distribu-
tions (Section 15.3). Lastly, we presented an expansive framework for modeling dynamics
(Section 15.4). Combined, Sections 15.2 through 15.4 provide an extremely rich and exible
framework for modeling discrete-valued spatio-temporal data.
To illustrate the exibility and utility of this approach, we provided an application
to forecasting one-year-ahead migratory bird settling patterns across the north-central
United States and Canada. In this example, we showcased the ability to specify compli-
cated nonlinear dependence structures using a three-regime TVAR model based on the
PDSI, a climate-related covariate. In this context, efcient dimension reduction is facilitated
through EOFs. Further, the effectiveness of our model was evaluated through in-sample
and out-of-sample prediction. Finally, as a by-product of the scientically motivated model
specication, we considered the notion of site philopatry through the Markov transition
matrices.
There are several open areas of research for discrete-valued spatio-temporal modeling.
One natural direction is to extend the models described here to the multivariate context
(Fahrmeir et al., 1994). Doing so requires several modeling choices that would need to be
investigated for each given application. For example, for a given application, would speci-
cation of a multivariate data model be superior to a conditional specication, as presented
in Section 15.2? In our illustration, we specied the evolution of the latent process through
a TVAR model, which is nonlinear. Nevertheless, this only constitutes one specication.
Other specications to accommodate nonlinearity and/or nonstationarity are also open
areas of research.
Acknowledgments
This research was partially supported by the U.S. National Science Foundation (NSF) and
the U.S. Census Bureau under NSF grant SES-1132031, funded through the NSF-Census
Research Network (NCRN) program and through NSF grant DMS-1049093. The authors
would like to thank Guohui Wu for his assistance with various aspects concerning the data
analysis, including preparation of the gures.
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