Bayesian dynamic models for space–time point processes
Introduction
Spatial point processes is the area of Statistics concerned with the study of observations of events in a given geographic location. Studies in this area have been performed from both theoretical point of view, where the probabilistic properties of these processes are analyzed (Cox and Isham, 1980), and through the study of statistical properties, where emphasis is given in the estimation the intensity of events in the region of interest (Diggle et al., 2003).
An example of problem of this type is the study of residence locations of people who suffer from a particular contagious disease. This analysis helps to determine possible patterns in the geographic distribution of the contamination risk. Various studies were developed in this area, from both Bayesian and frequentist points of view, in many fields of application, such as epidemiology (Diggle, 2000), criminology (Liu and Brown, 2003) and geology (Ogata, 1998), amongst others.
A relevant extension to this problem is the analysis of the variation of the observations in the temporal dimension as well as in the spatial dimension. In this case, not only the place, but also the time of occurrence is registered. For the example of disease mapping described above, these processes are of great utility as they allow for the analysis of the spread of the risk in space over time. That way, it is possible to create an alarm system to detect new focuses of the disease and establish a control strategy for the dispersion of the disease in the region. It is also possible to obtain a prediction of the spatial pattern of the disease for future times.
Brix and Diggle (2001) describe a flexible class of space–time point processes based on log-Gaussian Cox models. In the disease mapping context, the intensity of the process in space and time is defined by the product of a deterministic process describing the spatial variation of the population, and a risk function. The risk function is defined by a Ornstein–Uhlenbeck space–time process, given by stochastic differential equations. Inference is a difficult task to perform in this context, and Brix and Diggle (2001) made it via the method of moments. In their model specification, both time and space are defined as continuous, but they are discretized for inference purposes.
Also in the disease mapping context, Paez and Diggle (2009) work with the intensity of the point process aggregated in space, with the objective of analyzing time variation only. They proposed a time continuous Cox process, where for each fixed period of time the intensity of the process represents the mean intensity in space (for that particular time). One way of defining this intensity is through the product of the populational intensity, assumed to be known, and a risk function. The authors proposed a log-Gaussian model for the risk function incorporating an autoregressive process in time, and covariates that help treating the temporal variation not explained by the risk function.
In the context of continuous spatial processes, Gelfand et al. (2005) model space–time data through Gaussian processes, where the space is seen as continuous and time as discrete. Their idea is to view the data as a time series of spatial processes, by adapting the structure of dynamic models to space–time models with space-varying coefficients.
In this paper the approaches of Brix and Diggle (2001) and Paez and Diggle (2009) for point processes and the approach of Gelfand et al. (2005) for continuous processes are combined. Our aim is to analyze space–time point processes where the sequence of intensity surfaces (varying in space) are linked through time, as in Gelfand et al. (2005). We define the evolution of this sequence to be dynamic in time and call the resulted models dynamic spatial point processes.
Inference for these processes is performed under the Bayesian approach. The resulting posterior distributions are not analytically tractable and different approximating methods are described and used: Markov Chain Monte Carlo (MCMC) sampling schemes for simulation-based approximations and Laplace method for analytic approximations. That way, it is possible to obtain posterior distributions for the sequence of intensities and unknown model parameters as, for example, means and variances of the temporal evolution and measurements of correlation of the spatial dispersion. Based on the posterior distributions, not only point estimations but also credibility intervals can be obtained for these quantities. Also, the predictive distribution for the intensity of the process for future times can be obtained and used to predict future observations.
The paper is organized as follows: in the next section, the proposed space–time model is specified in its general form and a discretized version of the model is presented. Computational aspects of the Bayesian inference for these models, performed via MCMC and Laplace approximations are outlined in Section 3. Section 4 presents the application to the incidence of impulses in the small intestine of cats. Finally, in Section 5, some concluding remarks are drawn and some possible research extensions for this work are discussed.
Section snippets
Model formulation
Consider a space–time point process observed over a region in space and in time. Typically represents an observation process over the plane, but observations over spaces in other dimensions can also be considered. The time span need not start at 0 but any time interval can be represented as without loss of generality. Assume that observations of this process occur according to an inhomogeneous space–time Poisson process with intensity function
Computations
The resulting posterior distributions are too complex to allow for exact calculation of summaries of interest. Approximating techniques are required and two approaches can be entertained: MCMC and integrated Laplace (Rue et al., 2009). Calculations are outlined with model (17), (18), (19). Relatively straightforward modifications lead to the calculations for the other models in the general class presented in this paper.
In the MCMC sampling scheme, samples are drawn through the construction of a
Application
In this section, we applied the proposed models to a real neuro-gastroenterology data-set. The problem is presented in Section 4.1, the methods are presented in Section 4.2, and the results are presented in Section 4.3.
Discussion
In this work, a new methodology is presented to model the incidence of impulses in the small intestine. Space–time models were proposed for the intensity of point processes, specified by a sequence of surfaces of spatial intensity linked in time through a dynamic evolution. The temporal trend can be freely modeled. It can be assumed constant, for example, or assumed to be a deterministic function of covariates measured for each period of time, or even a dynamic model, among other possibilities.
Acknowledgments
The research of the authors was supported by grants from CNPq-Brazil and FAPERJ-Brazil. The research of the first author was also supported by grants from CAPES-Brazil.
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