=
,
377 Autologistic Regression Models for Spatio-Temporal Binary Data
17.3.1 Model with Centered Parameterization
For site i = 1, ..., n and time t,letπ
i,t
denote the probability of Y
i,t
= 1 under spatio-temporal
independence. That is,
exp
k
p
=0
θ
k
x
k,i,t
π
i,t
=
.
1 + exp
p
k=0
θ
k
x
k,i,t
Let y
∗
i,t
= y
i,t
− π
i,t
denote a centered response at site i and time t, centering around π
i,t
. For
pairwise-only dependence, Wang and Zheng (2013) dened a centered spatio-temporal
autologistic regression model via the following 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
)
. (17.20)
1 + exp
p
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
)
By the Hammersley–Clifford theorem and its corollary, the joint likelihood function of
Y
2
, ..., Y
T−1
conditioned on Y
1
and Y
T
is
L(Y; θ) = p(y
2
, ..., y
T−1
|y
1
, y
T
; θ)
T−1 n
p
n T
n
1
∗ −1 ∗ ∗ ∗ ∗ ∗
= c
(θ) exp
θ
k
x
k,i,t
y
i,t
+
2
θ
p+1
y
i,t
y
j,t
+
θ
p+2
y
i,t
y
i,t−1
,
t=2
i=1 k=0 i=1 j∈N
i
t=2
i=1
(17.21)
where c
∗
(θ) is the normalizing constant. When the temporal autocorrelation coefcient
is zero (i.e., θ
p+2
= 0), the model reduces to a spatio-only autologistic regression model
(Caragea and Kaiser, 2009; Hughes et al., 2011).
Thus, the conditional expectation of Y
i,t
given its neighbors is
E(Y
i,t
|Y
i
,t
= y
i
,t
: (i
, t
) ∈ N
i,t
)
∗ ∗ ∗
exp
k
p
=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
)
1 + exp
k
p
=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
)
which we denote as π
∗
i,t
. Suppose that the spatial autoregressive coefcient θ
p+1
and the
temporal autoregressive coefcient θ
p+2
are positive. Then, π
∗
i,t
> π
i,t
when
θ
p+1
y
j,t
+ θ
p+2
(y
i,t−1
+ y
i,t+1
)>θ
p+1
π
j,t
+ θ
p+2
(π
i,t−1
+ π
i,t+1
),
j∈N
i
j∈N
i
where
j∈N
i
π
j,t
and π
i,t−1
+ π
i,t+1
are the expected numbers of nonzero spatial and tem-
poral neighbors under the independence model, respectively. Specically, if θ
p+2
= 0, then
378 Handbook of Discrete-Valued Time Series
π
∗
i,t
> π
i,t
only when the observed number of nonzero spatial neighbors is greater than the
expected number of nonzero spatial neighbors under independence. That is,
y
j,t
>
j∈N
i
j∈N
i
π
j,t
.If θ
p+1
= 0, then π
i
∗
,t
> π
i,t
only when the observed number of nonzero tempo-
ral neighbors is greater than the expected number of nonzero temporal neighbors under
independence. That is, y
i,t−1
+ y
i,t+1
> π
i,t−1
+ π
i,t+1
. Thus, the interpretation of θ
p+1
and
θ
p+2
as local dependence parameters is more sensible. Further, the simulation study in
Wang (2013) showed that the marginal expectation of Y
i,t
under the centered parameter-
ization remains constant over moderate levels of spatial and temporal dependence (i.e.,
E(Y
i,t
|x
k,i,t
, k = 1, ..., p) ≈ π
i,t
). The interpretation of regression coefcients as effects of
covariates is more sensible as well.
17.3.2 Statistical Inference
For the model with centered parameterization, its statistical inference has been devel-
oped based on expectation–maximization pseudo-likelihood, Monte Carlo expectation–
maximization likelihood, and Bayesian inference (Wang and Zheng, 2013).
17.3.2.1 Expectation–Maximization Pseudo-Likelihood Estimator
To obtain the maximum pseudo-likelihood estimates of the model parameters, the combi-
nation of an expectation–maximization (EM) algorithm and a Newton–Raphson algorithm,
called the expectation–maximization pseudo-likelihood estimator (EMPLE), is considered.
Specically, update π
i,t
, the expectation of Y
i,t
under the independent model, at the E step
and then at the M step, update
θ
ˆ
l
by maximizing
exp
p
k=0
θ
k
x
k,i,t
y
i,t
+
j∈N
i
θ
p+1
y
i,t
y
∗
j,
(
t
l−1)
+ θ
p+2
y
i,t
(y
∗
i,
(
t−
l−
1
1)
+ y
∗
i,
(
t+
l−
1
1)
)
i,t
1 + exp
k
p
=0
θ
k
x
k,i,t
+
j∈N
i
θ
p+1
y
j
∗
,
(
t
l−1)
+ θ
p+2
(y
i
∗
,
(
t−
l−
1
1)
+ y
i
∗
,t
(
+
l−
1
1)
)
,
where y
i
∗
,t
(l)
is the centered response at the lth iteration. The M step can be carried out
by a Newton–Raphson algorithm using the standard logistic regression and the E and
M steps are repeated until convergence. A parametric bootstrap can be used to compute
the standard error of the EMPLE. For the starting value θ
0
at the start of the algorithm, dif-
ferent starting points can impact how long it takes to convergence. The maximum MPLE
from the uncentered autologistic regression model is a natural choice.
17.3.2.2 Monte Carlo Expectation–Maximization Likelihood Estimator
Let z
∗
= (
∗
...,
∗
1
y
∗
y
∗
∗
y
∗
. We consider a
θ
i,t
x
0,i,t
y
i,t
,
i,t
x
p,i,t
y
i,t
,
2
i,t j∈N
i
i,t j,t
,
i,t
y
i,t i,t−1
)
rescaled version of the likelihood function
c
∗
(ψ)
exp(θ
z
∗
)
−1
c
∗
(ψ)L(Y; θ) =
c
∗
(θ)
exp(θ
z
θ
∗
) =
E
ψ
θ
∗
exp(θ
z
θ
∗
),
exp(ψ
z
ψ
)
379 Autologistic Regression Models for Spatio-Temporal Binary Data
where ψ is a reference parameter and z
∗
is z
∗
with centers evaluated at ψ.MonteCarlo
ψ
expectation–maximization likelihood (MCEML) estimator can be used by combining an
EM algorithm and a Newton–Raphson algorithm. Specically, rst choose a reference
parameter vector ψ and generate M Monte Carlo samples of Y from the likelihood func-
tion evaluated at ψ. Then for the lth iteration, at the E step, we update π
(
i,
l
t
−1)
and set
y
∗
i,
(
t
l−1)
= y
i,t
− π
(
i,
l
t
−1)
. At the M step, we maximize the rescaled version of the likelihood
function
M
−1
∗(m) ∗(m)
exp(θ
z
∗
ˆ
l−1
) M
−1
exp θ
z
ˆ
l−1
− ψ
z
ψ
,
θ
θ
m=1
∗
∗
∗(l−1) ∗(m) ∗(m)
∗
where z
ˆ
l−1
is z with centered responses y
i,t
and z
ˆ
l−1
and z
ψ
are z evaluated
θ
θ
at the mth Monte Carlo sample of Y generated at the beginning of the algorithm with
centers computed at θ
ˆ
(l−1)
and ψ, respectively. The M step can be carried out using a
Newton–Raphson algorithm. We compute the observed Fisher information matrix and
obtain the standard errors of the MCEMLE as a by-product of the MCEML estimation.
17.3.2.3 Bayesian Inference
We consider an MH algorithm to generate Monte Carlo samples of θ from the posterior
distribution p(θ|y) (Zheng and Zhu, 2008), where the likelihood ratio in α(θ
∗
|θ) in the
acceptance probability is approximated as
∗
c
p(y
2
, ..., y
T−1
|y
1
, y
T
, θ
∗
)
exp(θ
∗
z
∗
)
∗
(θ)
θ
c
∗
(ψ)
p(y
2
, ..., y
T−1
|y
1
, y
T
, θ)
=
exp(θ
z
∗
)
×
c
∗
(θ
∗
)
θ
c
∗
(ψ)
∗
M
∗(m)
− ψ
∗(m)
exp(θ
∗
z
θ
∗
)
m=1
exp(θ
z z )
≈
∗
×
M
θ
∗(m)
ψ
∗(m)
,
exp(θ
z
θ
)
m=1
exp(θ
∗
z
θ
∗
− ψ
z
ψ
)
where z
∗(m)
, z
∗(m)
,and z
∗(m)
are z
∗
evaluated at the mth Monte Carlo sample of Y with
θ
θ
∗
ψ
centers computed based on θ, θ
∗
,and ψ, respectively.
17.4 Data Example
For illustration, we consider the outbreak of southern pine beetle (SPB) in North Carolina.
The data consist of indicators of outbreak or not (0 = no outbreak; 1 = outbreak) in the
100 counties of North Carolina from 1960 to 1996. Figure 17.1 gives a time series of the
county-level outbreak maps. The average precipitation in the fall (in cm) will be the covari-
ate and is mapped in Figure 17.2. We use the data from 1960 to 1991 for model tting and
set aside the data from 1992 to 1996 for model validation. Two counties are considered to
be neighbors if the corresponding county seats are within 30 miles of each other.
380
1960 1968 1976 1984 1992
1961 1969 1977 1985 1993
1962
1963
1964
1965
1966
1967
1970
1971
1972
1973
1974
1975
1978
1979
1980
1981
1982
1983
1986
1987
1988
1989
1990
1991
1994
1995
1996
FIGURE 17.1
Maps of southern pine beetle outbreak from 1960 to 1996 in the counties of North Carolina. A county is lled black if there was an outbreak and is unlled otherwise.
Handbook of Discrete-Valued Time Series
381 Autologistic Regression Models for Spatio-Temporal Binary Data
2.5 − 3
3 − 3.5
3.5 − 4
4 − 4.5
4.5 − 5
5 − 5.5
5.5 − 6
6 − 6.5
6.5 − 7
FIGURE 17.2
Map of mean fall precipitation in the counties of North Carolina.
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