16
Hierarchical Agent-Based Spatio-Temporal Dynamic
Models for Discrete-Valued Data
Christopher K. Wikle and Mevin B. Hooten
CONTENTS
16.1 Introduction...................................................................................349
16.2 HierarchicalDynamic Spatio-TemporalABM Methodology...........................351
16.3 StatisticalABM for the Spread of Disease.................................................352
16.4 Hierarchical First-Order Emulators and ABMs..........................................355
16.4.1 Simple Simulated Epidemic ABM Emulator Example..........................357
16.5 Approximate Bayesian Computation (ABC) and ABMs................................360
16.5.1 ABC/ABM Example.................................................................362
16.6 Conclusion.....................................................................................363
References............................................................................................363
16.1 Introduction
Inthischapter, weareconcernedwithdiscrete-valuedspatio-temporalprocesses. Tofacilitate
presentation, we will restrict our attention to such processes in discrete time, yet allow
space to be continuous or discrete in principle. In spatio-temporal statistics, it is common to
consider such models from a generalized linear mixed-model perspective (e.g., see Cressie
and Wikle, 2011 and Holan and Wikle [2015; Chapter 15 in this volume]). This is a “top-
down” approachwhereby the spatio-temporal propertiesofthe system aremodeled in terms
of a latent Gaussian spatio-temporal dynamical process (e.g., Wikle, 2002). Alternatively,
one may consider such processes as Markov random elds (MRFs) using one of the classes of
spatio-temporal “auto” models (e.g., spatio-temporal auto-logistic) as described in Zhu and
Zheng (2015; Chapter 17 in this volume). The MRF approach is a local specication where
relationships between neighbors are specied conditionally in a way to guarantee a valid
joint distribution (see Section 16.2). In this chapter, we discuss an alternative “bottom-up”
modeling strategy for discrete-valued spatio-temporal dynamical processes, which is agent
based. Such agent-based models (ABMs) are prevalent in epidemiology and social sciences
(e.g., Filatova et al., 2013; Gilbert, 2008; Keeling and Rohani, 2008; Sattenspiel, 2009) and
also are called individual-based models in the ecological sciences (e.g., Grimm and Railsback,
2005), multi-agent models in engineering (e.g., Olfati-Saber, 2006), and cellular automata in
the physical and mathematical sciences (e.g., Wolfram, 1984). All of these paradigms are
characterized by autonomous agents (or individuals) that take on one of a discrete number
349
350 Handbook of Discrete-Valued Time Series
of states (hence, discrete valued) that vary with time and space depending on a set of deter-
ministic or probabilistic “rules.” As discussed later, from a stochastic perspective, ABMs
can sometimes be linked to MRF-based models through Markov network properties.
As a bottom-up modeling approach, ABMs have the property that the characteristics and
actions of the agents ultimately dene the properties and behavior of the system in which
they exist. The most important part of such a modeling approach is that the autonomous
agents interact with each other and with their environment. Although much of the liter-
ature associated with these models assumes that the agent interactions and evolution are
governed by deterministic rules, from a statistical perspective, a crucial component of the
interactions is stochasticity, and we focus on such probabilistic-based specications in this
chapter. In probabilistic ABMs, the evolution of the agent’s state is dened through a para-
metric probability distribution, and thus, each spatio-temporal realization of the system is
different (although, typically the properties of the system are the same in each realization).
Agents can be dened to represent various scales, for example, a virus, an individual
animal, a node in a network, or a spatial unit (e.g., census tract, county). From this per-
spective, much of the distinction between the various ABMs considered across different
disciplines (individual-based models, multi-agent models, and cellular automata) is lost,
which makes it an ideal framework to consider from a statistical modeling perspective. As
mentioned, ABMs are typically considered in discrete time. As such, one must decide if all
agent states are updated at once during each time step (i.e., synchronous updating) or in a
way so that as an agent is updated, its new state is available to the other agents (i.e., asyn-
chronous updating). As discussed in Caron-Lormier et al. (2008), this can make a difference
in the global system properties, but it is more common to consider synchronous updating,
thus we only consider such updates here.
The primary statistical challenges associated with probabilistic ABMs are related to spec-
ication of the parametric probability distributions that govern agent behavior, as well
as associated parameter (or rule) learning, and accounting realistically for uncertainty in
observations and parameters. In addition, sensitivity of the local and global properties of
the system to parameter variability is of interest, along with model selection. Of course,
given the individual nature of such models and the fact that there are typically a very large
number of autonomous agents, computational complexity can be daunting in a statistical
framework. Note that computation can also be daunting in a deterministic ABM frame-
work, especially if one is seeking to learn the rules of agent behavior. We have found that the
principles of hierarchical statistical modeling can help mitigate many of the issues related
to probabilistic ABMs (e.g., Hooten et al., 2010; Hooten and Wikle, 2010). In particular,
the hierarchical framework is ideal for managing the uncertainty associated with observa-
tions, processes, and parameters. Perhaps the most critical advantage of the hierarchical
paradigm in ABMs is that parameters describing individual interactions with other agents
and the environment can themselves be stochastic processes with spatial and/or temporal
variability that may include exogenous information about the environment. These models
are still potentially computationally prohibitive, yet approximate inference methods such
as statistical emulation and approximate Bayesian computation (ABC) can mitigate these
computational issues in some cases.
In this chapter, we rst present a somewhat general methodological framework for hier-
archical ABMs in Section 16.2, followed by a specic hierarchical modeling example related
to the spread of an animal disease in Section 16.3. This is followed by illustrations of the
so-called “rst-order emulator” in Section 16.4 and ABC in Section 16.5 to facilitate
parameter estimation. We conclude in Section 16.6 with a brief discussion.
Hierarchical Agent-Based Spatio-Temporal Dynamic Models for Discrete-Valued Data 351
16.2 Hierarchical Dynamic Spatio-Temporal ABM Methodology
Let y
i,t
(s
i,t
) correspond to a discrete-valued observation for the ith agent (i = 1, ..., n)at
time t = 1, 2, ..., T and spatial location s
i,t
∈ D, where D is some spatial domain that is a
subset of R
d
, typically with d = 2, where we assume that the agent locations in the spatial
domain can, in principle, vary with time (i.e., s
i,t
). Here, we allow s
i,t
to represent either a
point in continuous space or a spatial region (e.g., a Census tract or grid cell) with positive
support, depending on the application. We assume that y
i,t
(s
i,t
) can take one of K possible
values (i.e., states) at time t. As mentioned previously, there are many different types of
ABMs, with a wide variety of notation and mathematical representations. However, one
can consider most of these models to be some sort of network with each autonomous agent
as a node in the network graph. As an example to introduce notation, consider rst a static
network in which only the value of the agent’s state changes with time, not the location.
We then dene the graph G = (V, E) with a set of n vertices {v
i
∈ V, i = 1, ..., n} and
associated edges, E ⊂{(i, j) : i, j ∈ V, j = i}. We also dene an adjacency matrix A =[a
ij
]
with nonzero elements, a
ij
, specied if there exists an edge between the vertices v
i
and v
j
.
Finally, we denote the set of neighbors of the ith node by N
i
={j ∈ V : a
i,j
= 0} or, equiv-
alently, by N
i
={j ∈ V : (i, j) ∈ E}. These neighbors may be specied apriori or based
on some known proximity structure (e.g., county centroids or a regular grid). Dynamic
graphs further allow the locations of these vertices to potentially move with time (i.e.,
so-called dynamic agents). Necessarily, the edges and adjacency matrices would then also be
time dependent. For brevity of presentation in the following examples, we assume that only
the discrete-valued states and adjacency matrix change with time, not the spatial locations
of the network vertices.
We note that the graph structures associated with our ABM social network can have
directed or undirected edges, depending on the application, or it can be a combination of
both types of edges. An example of an undirected graphical structure is a Markov network
whose probability distribution for a set of variables, say X ={x
1
, ..., x
n
},isgivenasthe
product
1
C
p(x
1
, ..., x
n
) = φ
c
(X
c
),
Z
c=1
where X
c
⊂ X and φ
c
(X
c
) is known as a potential, which is a nonnegative function of its
argument. In this case, Z is a normalizing constant, called a partition function,that is a
function of the potentials (e.g., Barber, 2012, Chapter 4). A special case of this Markov net-
work is the MRF. The MRF is then dened by a set of conditional probability distributions,
p(x
i
|N
i
), i = 1, ..., n. That is, the model is dened such that each node is independent
of all of the other nodes, given its neighbors on the undirected graph, G. Note that one
can then use the Hammersly–Clifford theorem to specify the functional conditions neces-
sary for these conditional distributions to yield a valid joint distribution (e.g., Besag, 1974;
Cressie, 1993; Guyon, 1995). Such models are useful in a variety of network analysis appli-
cations. In spatial statistics and image analysis, MRF models are typically constructed on a
lattice. These models lead to the family of “auto” models (e.g., Besag, 1974; Cressie, 1993;
Cressie and Wikle, 2011) and can be used to model discrete-valued spatio-temporal data as
352 Handbook of Discrete-Valued Time Series
described in, for example, Besag (1972), Zhu et al. (2005), Besag and Tantrum (2003), Zheng
and Zhu (2008; Chapter 17 in this volume).
We now consider the MRF perspective to denote a stochastic ABM. To simplify the expo-
sition, assume that the agents exist on a xed network, so that s
i,t
= s
i
is xed for each
agent through time. We then denote the state of the random variable associated with the
ith agent at time t by Y
i,t
. We further consider our “neighborhood” in discrete time to be
the current time t and one previous time lag (t − 1), although higher lagged neighbor-
hoods could be considered, as could neighborhoods that consider the future (e.g., Zheng
and Zhu, 2008). An alternative perspective that considers continuous time can be found in
Rasmussen et al. (2007). In the discrete-time case, we then must specify the conditional dis-
tribution p(Y
i,t
|{Y
j,t
, Y
j,t−1
: j ∈ N
i
}),fori = 1, ..., n. For example, if the agents can take only
binary states, we might specify this distribution to be a Bernoulli distribution, in which case
the model becomes a spatio-temporal auto-binomial model (e.g., Besag, 1972), depending
on coefcients describing the weights of the current and previous neighborhood states, as
well as potential interactions. In this case, the normalizing constant of the joint distribu-
tion implied by the Hammersly–Clifford theorem is intractable in general, complicating
inference (e.g., Besag and Tantrum, 2003; Zhu et al., 2008). However, as discussed in Besag
and Tantrum (2003), this distribution is substantially simplied if the time conditioning
does not depend on the state of the neighbors at the current time t, but only the state of
the neighbors at the past time, t − 1. This is the typical situation in synchronous updating
of ABMs and is one of the distinctions between MRF-based models and ABMs. Thus, we
consider conditional distributions of this form, p(Y
i,t
|{Y
j,t−1
: j ∈ N
i
}).
Another major distinction between the spatio-temporal MRF approach and the hier-
archical stochastic ABM approach considered here is that the parameters that control
the conditional distributions, say given by the q-dimensional vectors θ
i
, are modeled
explicitly as random variables at the next stage of a model hierarchy. That is, if we
had p(Y
i,t
|{Y
j,t−1
: j ∈ N
i
}; θ
i
) for i = 1, ..., n, these would be given prior distributions
n
i=1
p(θ
i
|ν
i
) that might depend on the vector of hyperparameters, ν
i
(which could, in
turn, depend on other parameters). These parameters {θ
i
} are typically modeled them-
selves as processes (in space, and possibly time) and can depend on exogenous variables
that can allow the model to adapt to more complicated processes. That is, we are effec-
tively modeling conditional ABMs, given parameters. Inference can then be obtained using
Bayesian methods such as Markov chain Monte Carlo (MCMC) or ABC. These will be illus-
trated in the animal disease example in Section 16.3 and the epidemic model example in
Section 16.5. We note that in nonhierarchical forms of these models, one can perform esti-
mation through approximate likelihood methods (e.g., Besag and Tantrum, 2003; Zhu et al.,
2005, 2008).
16.3 Statistical ABM for the Spread of Disease
During the past few decades, a raccoon (Procyon lotor) rabies epidemic has been occurring
throughout the eastern United States This rabies epidemic has been the subject of many
epidemiological research and was discussed in detail by Smith et al. (2002) and Wheeler
and Waller (2008). Hooten and Wikle (2010) considered the data collected in Connecticut,
USA, from years 1991–1995. These data were binary valued and arranged on a regular
grid spanning the state of Connecticut (and a small portion of eastern New York, USA)
Hierarchical Agent-Based Spatio-Temporal Dynamic Models for Discrete-Valued Data 353
FIGURE 16.1
Presence (dark cells) or absence (light cells) of data for raccoon rabies in Connecticut, USA, during 1991–1995.
Time for each image increases from top to bottom and left to right, in that order.
consisting of 109 approximately township-sized cells. The binary data represented presence
or absence of rabies in each grid cell over a sequence of regularly spaced time periods
spanning 1991–1995 (Figure 16.1). These data clearly indicate a spreading dynamic process
from west to east throughout this time period. A unique aspect of the epidemic is that the
spread is not as smooth as that which might arise from a partial differential equation (PDE)
(e.g., Wikle, 2003) and it consists of several apparent long-distance dispersal events. In this
case, long-distance dispersal is dened as a noncontiguous “jump” in disease status beyond
that of the rst- and second-order neighboring grid cells where the disease is present.
In what follows, we describe a hierarchical Bayesian ABM for those data that allow
for heterogeneous spread of the disease similar to that presented by Hooten and Wikle
(2010). We use the term “ABM” to describe this model because one could envision the
areal units (i.e., grid cells or townships) themselves acting as “agents” which interact, effec-
tively spreading the disease from one spatial region to another over time. In the network
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