Hierarchical Agent-Based Spatio-Temporal Dynamic Models for Discrete-Valued Data 359
X
1
Simulation time: 1
30
5
7
30
6
25
25
5
20
20
4
0
15
15
3
10
10
2
5
1
5
−5
10 20 30
10 20 30
(a) (d)
X
2
Simulation time: 51
30
5
8
30
25
25
6
20
20
0
4
15
15
10
10
2
5
5
−5
0
10 20 30 10 20 30
(b) (e)
p
Simulation time: 61
30
8
30
0.8
25
25
6
20
0.6
20
4
15
15
0.4
10
10
2
0.2
5
5
0
10 20 30 10 20 30
(c)
(f)
FIGURE 16.3
(a) X
1
covariate related to the probability of infection; (b) X
2
covariate related to the probability of infection;
(c) prior probability of infection associated with the X
1
and X
2
spatial covariates; (d) initial state (time 1)
for the SIR cellular ABM; (e) agent states at time t = 51 for the SIR cellular ABM; (f) agent state at time t = 61 for
the SIR cellular ABM. In plots (d–f), state 0 (white) corresponds to susceptible agents, states (1–4) correspond to
infected agents (black), and states (5–8) correspond to recovered agents (gray).
360 Handbook of Discrete-Valued Time Series
0.2
0
5
10
15
20
25
30
35
40
0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 −0.35
0
5
10
15
20
25
30
35
40
45
−0.3 −0.25 −0.2 −0.15
−0.1 −0.05 0 0.05
β
1
β
2
0
10
20
30
40
50
0
10
20
30
40
50
0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
β
1
β
2
FIGURE 16.4
Histograms of posterior means for β parameters in the SIR ABM for each of 200 simulations with emulator-
and ABC-based estimation approaches. Upper left panel: Emulator, β
1
; Lower left panel: Emulator, β
2
; Upper
right panel: ABC, β
1
; Lower right panel: ABC, β
2
(right panel). The true parameter values are β
1
= 0.3 and
β
2
=−0.1.
of infection areas is apparent in the simulated spatial patterns in Figure 16.3e and f. The
nal 100 iterations of this simulation serve as the “data” (i.e., y
D
in (16.7)), assumed to be
observed without error in this simple example.
To build the emulator, we generate 200 samples of β ≡ (β
1
, β
2
)
in the range of −0.8 to
0.8, using a Latin-hypercube design. We then run the ABM simulator, with the same ini-
tial condition as the original data-generating simulation, and build the rst-order emulator
assuming a very simple linear function for g(·),with �
η
= σ
2
η
I. With uniform prior distri-
butions on the β parameters (−0.8 to 0.8) and a diffuse prior on σ
2
η
, we sample from the
posterior distribution for β and σ
2
η
via MCMC. The posterior simulation is run for 10,000
iterations beyond a burn-in of 1,000 iterations; convergence is evaluated by visual inspec-
tion of the two chains. It is critical to note that the emulator runs in a fraction of the time
as the ABM simulator in this case. We replicate this simulation procedure 200 times. The
left panels of Figure 16.4 show histograms of the posterior means from these simulations,
and the left two panels of Figure 16.5 show the 95% credible regions for each simulation
relative to the truth. Even with this very simple linear emulator, the estimation procedure
does a reasonable job of estimating the true parameters, although the posterior means for
β
2
show a small negative bias.
16.5 Approximate Bayesian Computation (ABC) and ABMs
There is an increasing interest in the statistical sciences for performing inference and
calibration in situations where likelihoods are difcult to calculate or not available, or
if simulation models can easily be used to generate “data.” One of the most popular
Hierarchical Agent-Based Spatio-Temporal Dynamic Models for Discrete-Valued Data 361
Simulation
200
180
160
140
120
100
80
60
40
20
0
200
180
160
140
120
100
80
60
40
20
0
0.2 0.3 0.4 −0.4 −0.2 0 0.2
(a)
β
1
(b)
β
2
FIGURE 16.5
Posterior 95% credible intervals for β
1
(a) and β
2
(b) for each of 200 simulations for the SIR ABM. The true
parameter values are shown by dashed lines.
362 Handbook of Discrete-Valued Time Series
methods for such situations is ABC, originally introduced in genetics by Tavaré et al. (1997)
and popularized by Beaumont et al. (2002). This methodology has since been used across
a wide variety of applications (e.g., Beaumont, 2010; Csilléry et al., 2010; Lagarrigues et al.,
2014; Marin et al., 2012, and references therein). Given the simulation-intensive nature of
ABMs, it is natural to apply the ABC methodology for parameter estimation in such set-
tings. We present a brief and simple illustration of ABC for estimation of ABMs in the
following.
Let θ be an ABM parameter vector of interest with prior distribution [θ]. We also have
data, y
D
, and the data-generating ABM distribution [y
D
|θ]. In its simplest form, ABC
seeks to approximate the posterior distribution [θ|y
D
]∝[y
D
|θ][θ] in the following way
(Toni et al., 2009):
1. Sample a candidate vector θ
˜
from a proposal distribution f (θ).
2. Simulate a dataset y˜
D
from the data model given by [y
D
|θ
˜
].
3. Compare the simulated dataset (or its summary, h(y˜
D
))withthedata(orits
summary, h(y
D
)) using a distance metric, d(·), and tolerance, : procedurally, if
d(h(y
D
), h(y˜
D
)) ≤ , then accept θ
˜
. The tolerance ≥ 0 is an indication of the level
of “agreement” between y
D
and y˜
D
.
The output of such an ABC algorithm is a sample of θ from [θ| d(h(y
D
), h(y˜
D
)) ≤ ].Thus,
in general, if the summary measure is appropriate and is sufciently small, this distri-
bution will be a reasonable approximation of the posterior distribution of interest, [θ|y
D
].
Of course there are many issues with such a procedure, such as choosing the tolerance level
(), the summary measure (h(·)), and the distance metric (d(·)). In addition, this basic algo-
rithm often leads to too many rejections. Many algorithms have been proposed in recent
years to mitigate some of these subjective calibration issues and improve the procedure
(e.g., Marin et al., 2012).
16.5.1 ABC/ABM Example
We illustrate the basic ABC algorithm from Section 16.5 on the SIR ABM example presented
in Section 16.4.1. In this case, to facilitate dimension reduction on the model output, we
project the data and each simulation on the rst 40 spatial principal components derived
from the data, y
D
(e.g., called empirical orthogonal functions, EOFs, in the spatial statistics
literature as summarized in Cressie and Wikle, 2011). In this case, the EOFs correspond to
spatial basis functions and the projection coefcients are principal component time series.
Our summary measure was then the Euclidean distance in terms of these projection coef-
cients, and we chose an acceptance tolerance, , corresponding to the 10th percentile of d().
As with the emulator example, we replicate this simulation procedure 200 times. The right
panels of Figure 16.4 show histograms of the posterior means from these simulations, with
coverage similar to the emulator-based estimates. Note that we did not try to optimize our
selection of distance metric nor tolerance region for this example.
Although the simulation ABM in this simple example is quite fast to compute, in many
real-world situations this would not be the case. In such situations, the simulation model
could also be approximated. For example, one could use an emulator approach as described
in Section 16.4.
Hierarchical Agent-Based Spatio-Temporal Dynamic Models for Discrete-Valued Data 363
16.6 Conclusion
Rule-based ABMs can be an effective modeling paradigm to account for complex spatio-
temporal dynamics for discrete state spaces. With the addition of stochastic terms, and the
presence of data, this becomes a statistical modeling challenge. The Bayesian hierarchical
modeling approach can facilitate such statistical modeling due to its facility with dealing
with uncertainty in data, processes, and parameters. Estimation can be facilitated by ABC
and statistical emulators of the ABM. In addition, these models can be considered from a
network perspective, which also allows one to borrow many existing network modeling
methodologies. Although the use of complicated network models in the ABM context is
relatively new, we expect to see additional developments in this area. In addition, there
are potentially important issues associated with model selection in ABMs (e.g., Piou et al.,
2009), and this is likely to be an important area of future research.
Acknowledgments
Wikle acknowledges support from U.S. National Science Foundation (NSF) grant DMS-
1049093 and the NSF and U.S. Census Bureau under NSF grant SES-1132031, funded
through the NSF-Census Research Network (NCRN) program. We thank Scott Holan and
an anonymous reviewer for helpful comments on an early version of this chapter.
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