17
Autologistic Regression Models for Spatio-Temporal
Binary Data
Jun Zhu and Yanbing Zheng
CONTENTS
17.1 Introduction...................................................................................367
17.1.1 Markov RandomFieldand Autologistic Model.................................368
17.1.2 Spatio-Temporal Autologistic Model..............................................370
17.2 Spatio-Temporal Autologistic Regression Model........................................371
17.2.1 Model...................................................................................371
17.2.2 StatisticalInference...................................................................372
17.2.2.1 MaximumPseudo-Likelihood Estimation..............................373
17.2.2.2 Monte Carlo Maximum Likelihood Estimation....... .. .. .. .. .. .. .. .. .373
17.2.3 BayesianInference....................................................................374
17.2.4 Prediction..............................................................................375
17.3 Centered Autologistic RegressionModel.................................................376
17.3.1 Model with Centered Parameterization...........................................377
17.3.2 StatisticalInference...................................................................378
17.3.2.1 Expectation–Maximization Pseudo-Likelihood Estimator. . . ... . . ...378
17.3.2.2 Monte Carlo Expectation–Maximization Likelihood Estimator. . . . .378
17.3.2.3 Bayesian Inference.........................................................379
17.4 DataExample..................................................................................379
17.5 Discussion......................................................................................383
References............................................................................................384
17.1 Introduction
Binary data on a spatial lattice are often encountered in environmental and ecologi-
cal studies. Spatial statistical methods have been developed for modeling spatial binary
responses and their relations to covariates while properly accounting for spatial correla-
tion. In this chapter, we review autologistic models in the class of Markov random elds
that model spatial dependence via autoregression and consider extensions to autologis-
tic regression models for spatio-temporal binary data. In particular, the introduction of
autoregression in space and time results in an unknown normalizing constant in the likeli-
hood function, which makes estimation and statistical inference challenging. We describe
367
368 Handbook of Discrete-Valued Time Series
several approaches to the inference for spatio-temporal autologistic regression models and
illustrate them by an ecological data example.
17.1.1 Markov Random Field and Autologistic Model
For site i =1, ..., n,let Y
i
denote the response variable at site i. Besag (1974) developed
a Markov random eld model that conditionally species the distribution of Y
i
.Let Y =
(Y
1
, ..., Y
n
)
and Y
−i
= (Y
j
: j = i)
denote the response variables with y = (y
1
, ..., y
n
)
and y
−i
= (y
j
: j = i)
denoting a corresponding realization at all n sites and all except
site i, respectively. In a Markov random eld model for Y, the full conditional probability
density of Y
i
(conditional on all other sites) is assumed to depend only on the responses
at neighboring sites; that is, p(y
i
|y
−i
) = p(y
i
|y
j
: j ∈ N
i
), where N
i
denotes a prespeci-
ed neighborhood of site i. The conditional probability density is generally specied in an
exponential form
p(y
i
|y
−i
) = p(y
i
|y
j
: j ∈ N
i
) = exp
A
i
(y
−i
)y
i
− B
i
(y
−i
) + C
i
(y
i
)
(17.1)
where A
i
is a natural parameter function, B
i
is a function of the model parameters and y
−i
but free of y
i
,and C
i
is a function of y
i
but free of the model parameters.
To ensure that the resulting joint distribution of Y is valid, Besag (1974) dened a
negpotential function
Q(y) = ln
p(y)
, (17.2)
p(0)
which is essentially the logarithm of the joint probability density function p(y) up to a
normalizing constant since
p(y) =
exp{Q(y)}
, (17.3)
exp{Q(z)}
z∈
where denotes a suitable space of responses. It has been shown that Q(y) in (17.2)
can be uniquely expanded on and the expansion is made up of the conditional prob-
abilities p(y
i
|y
−i
) in (17.1) under a positivity condition (Besag, 1974; Cressie, 1993). The
Hammersley–Clifford Theorem and its corollary establish the sparsity of the expansion
and most importantly, the validity of the joint probability p(y) through the negpotential
function Q(y).
For binary data on a spatial lattice, Besag (1972) developed an autologistic model in the
framework of Markov random elds. In particular, the binary response variable Y
i
∈{0, 1}
has a conditional Bernoulli distribution. The pairwise-only dependence is among neigh-
boring sites according to the neighborhood N
i
. Thus, the natural parameter function is of
the form
A
i
(y
−i
) = α
i
+ θ
ij
y
j
, (17.4)
j∈N
i
369 Autologistic Regression Models for Spatio-Temporal Binary Data
B
i
(y
−i
) = ln[1 + exp{A
i
(y
−i
)}],and C
i
(y
i
) =0, where α
i
is a constant, θ
ij
s are spatial
dependence parameters such that θ
ij
=θ
ji
for j =i and θ
ii
=0forsite i =1, ..., n.The
corresponding negpotential function is
n
Q(y) =
y
i
α
i
+ (1/2) θ
ij
y
j
. (17.5)
i=1 j∈N
i
By the Hammersley–Clifford Theorem, the joint probability density is
n
exp
i=1
y
i
α
i
+ (1/2)
j∈N
i
θ
ij
y
j
p(y) =
. (17.6)
z∈
exp
i
n
=1
z
i
α
i
+ (1/2)
j∈N
i
θ
ij
z
j
In (17.6), the normalizing constant in the denominator involves the model parameters and
generally does not have an analytical form, which makes it a challenge to directly maximize
the likelihood function.
The traditional parameterization of autologistic models may not be intuitive when
incorporating regression. Similar to the parametrization used for auto-Gaussian models,
a centered parameterization of autologistic models was proposed recently (Caragea and
Kaiser, 2009; Kaiser et al., 2012), which is perhaps more suitable for regression purposes.
In the centered parameterization,
A
i
(y
−i
) = ln{κ
i
/(1 − κ
i
)}+ θ
ij
(y
j
− κ
j
),
j∈N
i
where κ
i
∈ (0, 1), i = 1, ..., n. A detailed description of the centered parameterization of
autologistic regression models is given in Section 17.3.
A special case of the autologistic model is the Ising model. The Ising model was rst
developed by Ernst Ising in his doctoral thesis as an attempt to describe phase transitions
in ferromagnets (Ising, 1924, 1925). The basic idea is that microscopic magnets are arranged
on a square lattice such that there is one magnet at each lattice site. Each magnet is assumed
to have two possible spin directions, generally labeled as up (y
i
=+1) or down (y
i
=−1),
and is assumed to only interact with its four nearest neighbors. In the Ising model, the total
energy, also known as the Hamiltonian, of the conguration is given by
n n
H(y) =−
θy
i
y
j
− αy
i
, (17.7)
i=1 j∈N
i
,j<i i=1
where N
i
denotes the neighborhood of site i comprising the four nearest neighbors, the
coefcient θ represents the strength of interactions among the nearest neighbors, and
the coefcient α represents an external magnetic eld. The cases θ > 0andθ < 0 correspond
to ferromagnetism and antiferromagnetism, respectively. The joint probability density of a
conguration is given by the so-called Boltzmann factor
Z
−1
exp{−βH(y)}, (17.8)
β
370 Handbook of Discrete-Valued Time Series
where β =(k
B
T)
−1
≥0with T denoting the Kelvin temperature and k
B
denoting the
Boltzmann constant, while Z
β
= exp{−βH(y)} is a partition function (or, normalizing
y
constant). When the parameter β in (17.8) surpasses a threshold value, a phase transition
from short-range to long-range interactions would occur, resulting in an ordered phase
with nonzero limiting correlation (see, e.g., Pickard, 1976, 1977).
17.1.2 Spatio-Temporal Autologistic Model
For spatio-temporal binary data, let Y
i,t
∈{±1} denote the binary response variable with
y
i,t
denoting a realization at site i = 1, ..., n and time t.Let Y
t
= (Y
1,t
, ..., Y
n,t
)
denote the
vector of binary responses with realizations y
t
= (y
1,t
, ..., y
n,t
)
at all sites and a given time
point t. Bartlett (1971, 1972) developed a Markov process with
P(Y
i,t+t
= y
i,t
|y
t
) = 1 − λ(t){1 − F
i
(y
t
)}, (17.9)
where λ ≥ 0, t ≥ 0, and F
i
(y
t
) is a function of y
t
. The joint probability density of Y
t
, when
the Markov process is at equilibrium, is
n n
p(y
t
) = c(α, θ) exp
−α
y
i,t
− θ y
i,t
z
i,t
, (17.10)
i=1 i=1
where α and θ are two coefcients, under the condition that
n n
∗
{1 − F
i
(y
t
)}= {1 − F
i
(y˜
i,t
)}exp(2αy
i,t
+ 4θy
i,t
z
i,t
), (17.11)
i=1 i=1
where z
i,t
denotes a linear combination of y
j,t
for j = i, z
i
∗
,t
is a symmetrized form of z
i,t
(e.g., in the one-dimensional space, if z
i,t
= y
i−1,t
, then 2z
i
∗
,t
= y
i−1,t
+ y
i+1,t
), and y˜
i,t
=
(y
1,t
, ..., y
i−1,t
, −y
i,t
, y
i+1,t
, ..., y
n,t
)
.
A direct and symmetric solution to Equation (17.11) is
1 − F
i
(y
t
) = exp(αy
i,t
+ 2θy
i,t
z
∗
i,t
)f (αy
i,t
, θy
i,t
z
∗
i,t
), (17.12)
where f is a suitable, positive function, and even in both y
i,t
and y
i,t
z
i
∗
,t
.
Now, on a square lattice, let Y
i,i
,t
∈{0, 1} denote the binary response with a realization
y
i,i
,t
at row i, column i
,and time t. Besag (1972) proposed a Markov process of binary
responses developing through time on the square lattice, which can be viewed as a special
case of Bartlett (1971, 1972). In particular, for xed α
y
, θ
y,1
,and θ
y,2
,
P
Y
i,i
,t+t
= y|Y
i,i
,t
= y, y
·,·,t
:t
≤t+t
, excluding y
i,i
,t+t
=
1 + t exp
α
y
+ θ
y,1
(y
i−1,i
,t
+ y
i+1,i
,t
) + θ
y,2
(y
i,i
−1,t
+ y
i,i
+1,t
)
−1
+ o(t)
= 1 − t exp
α
y
+ θ
y,1
(y
i−1,i
,t
+ y
i+1,i
,t
) + θ
y,2
(y
i,i
−1,t
+ y
i,i
+1,t
)
+ o(t), (17.13)
371 Autologistic Regression Models for Spatio-Temporal Binary Data
gives the probability that Y
i,i
,·
remains unchanged in the time interval (t, t + t],given all
other values at or before time t+t. Further, it can be shown that its stationary distribution
is an autologistic model with the full conditional distribution
p
y
i,i
,t
|y
i−1,i
,t
, y
i+1,i
,t
, y
i,i
−1,t
, y
i,i
+1,t
exp
y
i,i
,t
α + θ
1
(y
i−1,i
,t
+ y
i+1,i
,t
) + θ
2
(y
i,i
−1,t
+ y
i,i
+1,t
)
=
, (17.14)
1 + exp
α + θ
1
(y
i−1,i
,t
+ y
i+1,i
,t
) + θ
2
(y
i,i
−1,t
+ y
i,i
+1,t
)
where α = α
0
− α
1
, θ
1
= θ
0,1
− θ
1,1
,and θ
2
= θ
0,2
− θ
1,2
.
17.2 Spatio-Temporal Autologistic Regression Model
17.2.1 Model
For the analysis of spatio-temporal binary data in practice, it is often of interest to account
for possible effects of covariates. For example, Gumpertz et al. (1997) and Huffer and Wu
(1998) incorporated covariates in an autologistic model by replacing the constant α
i
in (17.6)
with a linear regression term and the spatial lattice can be either regular or irregular. The
resulting model is referred to as an autologistic regression model. Zhu et al. (2005) and
Zheng and Zhu (2008) extended the autologistic regression model to a spatio-temporal
autologistic regression model that accounts for covariates and spatio-temporal dependence
simultaneously for binary responses measured repeatedly over discrete time points on a
spatial lattice.
As earlier, let i = 1, ..., n denote sites on a spatial lattice. Further, let t ∈ Z index discrete
time points and Y
i,t
∈{0, 1} denote the binary response variable at site i and time t.Let
x
0,i,t
≡ 1andlet x
k,i,t
denote the kth covariate at site i and time t,for k = 1, ..., p and a total
of p covariates. Zhu et al. (2005) developed a spatio-temporal autologistic regression model
via the full conditional distributions:
p(y
i,t
|y
i
,t
: (i
, t
) = (i, t)) = p(y
i,t
|y
i
,t
: (i
, t
) ∈ N
i,t
)
exp
k
p
=0
θ
k
x
k,i,t
y
i,t
+
j∈N
i
θ
p+1
y
i,t
y
j,t
+ θ
p+2
y
i,t
(y
i,t−1
+ y
i,t+1
)
=
p
, (17.15)
1 + exp
k=0
θ
k
x
k,i,t
+
j∈N
i
θ
p+1
y
j,t
+ θ
p+2
(y
i,t−1
+ y
i,t+1
)
where N
i,t
={(j, t) : j ∈ N
i
}∪{(i, t−1), (i, t+1)} denotes a spatio-temporal neighborhood for
site i and time t and recall that N
i
={j :site j is a neighbor of site i}. The model parameters
are the intercept θ
0
,slope θ
k
for covariate x
k
with k =1, ..., p, a spatial autoregressive coef-
cient θ
p+1
, and a temporal autoregressive coefcient θ
p+2
.Let θ =(θ
0
, ..., θ
p+2
)
denote
the vector of parameters in the model (17.15).
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