382 Handbook of Discrete-Valued Time Series
TABLE 17.1
Comparison of Model Parameter Estimation for the Spatio-Temporal Autologistic Model with
Uncentered Parameterization Using Maximum Pseudo-likelihood (MPL), Monte Carlo Maximum
Likelihood (MCML), and Bayesian Inference
MPL MCML Bayesian
Parameters Estimate SE Estimate SE Mean SD
Intercept θ
0
Slope θ
1
Spatial θ
2
Temporal θ
3
−5.16
0.25
1.45
1.71
0.66
0.18
0.14
0.24
−2.71
−0.14
0.91
1.02
0.20
0.05
0.05
0.11
−2.71
−0.14
0.91
1.03
0.20
0.05
0.06
0.13
TABLE 17.2
Comparison of Model Parameter Estimation for the Spatio-Temporal Autologistic Model with
Centered Parameterization Using Expectation–Maximization Pseudo-likelihood (EMPL), Monte
Carlo Expectation–Maximization Likelihood (MCEML), and Bayesian Inference
EMPL MCEML Bayesian
Parameters Estimate SE Estimate SE Mean SD
Intercept θ
0
Slope θ
1
Spatial θ
2
Temporal θ
2
−4.96
0.21
1.47
1.75
0.32
0.08
0.13
0.18
−2.40
−0.13
0.95
0.89
0.16
0.05
0.05
0.07
−2.86
−0.13
0.95
0.89
0.29
0.08
0.06
0.10
Tables 17.1 and 17.2 give the model parameter estimates and standard errors from tting
the spatio-temporal autologistic regression models with uncentered and centered parame-
terization, respectively (see Wang and Zheng, 2013; Zheng and Zhu, 2008). The parameter
estimates and the corresponding standard errors for both models using all of the three
inference approaches, MPLE, MCMLE, and Bayesian inference, are quite close. One pos-
sible reason for this is that for this data set, the inuence of the center is small relative to
the strength of spatio-temporal dependence. The average of the centers π
i,t
evaluated at
the MCEMLE is only 0.05, and thus, the spatio-temporal autoregressive terms dominate
the outbreak probabilities.
For comparison among various statistical inference approaches, the results suggest that
the inference for the model parameters using the posterior distribution matches well with
MCML, but the inference from pseudo-likelihood is different from both Bayesian inference
and MCML for both uncentered and centered parameterization models. In addition, esti-
mation based on pseudo-likelihood results in higher variance than the other approaches.
In terms of computing time, Bayesian inference is more time consuming compared with
the other two approaches. Further, we predict the SPB outbreak from 1992 to 2001 in
North Carolina (Table 17.3). The responses at the end time point (here, y
i,2002
) are gen-
1991
erated from independent Bernoulli trials with probability of outbreak
t=1960
y
i,t
/31 for
i = 1, ..., 100. The prediction performances based on models with uncentered and cen-
tered parameterization are comparable. Overall, our recommendation is to use a model
with uncentered parameterization if prediction is of primary interest, since the two
383 Autologistic Regression Models for Spatio-Temporal Binary Data
TABLE 17.3
Comparison of the Prediction Performance between Models with Centered and Uncentered
Parameterization
Centered Uncentered
Year EMPL MCEML Bayesian MPL MCML Bayesian
1992 0.65 0.18 0.14 0.66 0.09 0.09
1993 0.72 0.19 0.12 0.65 0.13 0.13
1994 0.70 0.20 0.14 0.74 0.06 0.08
1995 0.63 0.23 0.13 0.68 0.13 0.14
1996 0.62 0.24 0.09 0.61 0.17 0.16
Note: For centered parameterization, the prediction is based on statistical inference obtained using expectation–
maximization pseudo-likelihood (EMPL), Monte Carlo expectation–maximization likelihood (MCEML),
and Bayesian inference. For uncentered parameterization, the prediction is based on statistical infer-
ence obtained using maximum pseudo-likelihood (MPL), Monte Carlo maximum likelihood (MCML), and
Bayesian inference. Reported are the prediction error rates for each year in 1992–1996.
parameterizations provide comparable performance in prediction but the centered param-
eterization is computationally more intensive. If the focus is on the interpretation of the
regression coefcients, however, the centered parameterization is recommended.
17.5 Discussion
In this chapter, we have reviewed spatio-temporal autologistic regression models for
spatio-temporal binary data. Alternatively, a generalized linear mixed model (GLMM)
framework can be adopted for modeling such spatial data (Diggle and Ribeiro, 2007;
Holan and Wikle [2015; Chapter 15 in this volume]). The response variable is modeled by a
distribution in the exponential family and is related to covariates and spatial random effects
in a link function. Thus, GLMM is exible, as it is suitable for both Gaussian responses
and non-Gaussian responses such as binomial and Poisson random variables. Statistical
inference can be carried out using Bayesian hierarchical modeling, which is exible as more
complex structures can be readily placed on the model parameters. With suitable reduction
of dimensionality for the spatio-temporal random effects, computation is generally feasible.
In particular, faster computational algorithms are emerging such as integrated nested
Laplace approximations (INLA) (Rue et al., 2009). Although likelihood-based approaches
aresuitable, it is sometimes a challenge to attain a full specication of the likelihood function,
due to a lack of sufcient information and complex interactions among responses. In this
case, an estimating equation approach may be attractive. For spatial binary data, Lin et al.
(2008) developed a central limit theorem for a random eld under various L
p
metrics and
derived the consistency and asymptotic normality of quasi-likelihood estimators. Lin (2010)
further developed a generalized estimating equation (GEE) method for spatio-temporal
binary data, but only a single binary covariate was considered and the spatio-temporal
dependence is limited to be separable. Moreover, variable selection methods for identifying
the suitable set of covariates are of interest. For example, Fu et al. (2013) developed adaptive
Lasso for the selection of covariates in an autologistic regression model and extension to
384 Handbook of Discrete-Valued Time Series
spatio-temporal autologistic regression model is discussed. Further research on this and
other related topics will be worthwhile.
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