Spatio-temporal Cox Process Research Articles

Corrigendum: Exact Bayesian inference in spatiotemporal Cox processes driven by multivariate Gaussian processes

Corrigendum: Exact Bayesian inference in spatiotemporal Cox processes driven by multivariate Gaussian processes

Exact Bayesian Inference in Spatiotemporal Cox Processes Driven by Multivariate Gaussian Processes

SummaryWe present a novel inference methodology to perform Bayesian inference for spatiotemporal Cox processes where the intensity function depends on a multivariate Gaussian process. Dynamic Gaussian processes are introduced to enable evolution of the intensity function over discrete time. The novelty of the method lies on the fact that no discretization error is involved despite the non-tractability of the likelihood function and infinite dimensionality of the problem. The method is based on a Markov chain Monte Carlo algorithm that samples from the joint posterior distribution of the parameters and latent variables of the model. A particular choice of the dominating measure to obtain the likelihood function is shown to be crucial to devise a valid Markov chain Monte Carlo algorithm. The models are defined in a general and flexible way but they are amenable to direct sampling from the relevant distributions because of careful characterization of its components. The models also enable the inclusion of regression covariates and/or temporal components to explain the variability of the intensity function. These components may be subject to relevant interaction with space and/or time. Real and simulated examples illustrate the methodology, followed by concluding remarks.

Cox process representation and inference for stochastic reaction-diffusion processes.

Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction–diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate to observational data. Here we use ideas from statistical physics and machine learning to provide a solution to the inverse problem of learning a stochastic reaction–diffusion process from data. Our solution relies on a non-trivial connection between stochastic reaction–diffusion processes and spatio-temporal Cox processes, a well-studied class of models from computational statistics. This connection leads to an efficient and flexible algorithm for parameter inference and model selection. Our approach shows excellent accuracy on numeric and real data examples from systems biology and epidemiology. Our work provides both insights into spatio-temporal stochastic systems, and a practical solution to a long-standing problem in computational modelling.

Modeling space-time data using stochastic differential equations

This paper demonstrates the use and value of stochastic difierential equations for modeling space-time data in two common settings. The flrst consists of point-referenced or geostatistical data where observations are collected at flxed locations and times. The second considers random point pattern data where the emergence of locations and times is random. For both cases, we employ stochas- tic difierential equations to describe a latent process within a hierarchical model for the data. The intent is to view this latent process mechanistically and endow it with appropriate simple features and interpretable parameters. A motivating problem for the second setting is to model urban development through observed locations and times of new home construction; this gives rise to a space-time point pattern. We show that a spatio-temporal Cox process whose intensity is driven by a stochastic logistic equation is a viable mechanistic model that afiords mean- ingful interpretation for the results of statistical inference. Other applications of stochastic logistic difierential equations with space-time varying parameters in- clude modeling population growth and product difiusion, which motivate our flrst, point-referenced data application. We propose a method to discretize both time and space in order to flt the model. We demonstrate the inference for the geosta- tistical model through a simulated dataset. Then, we flt the Cox process model to a real dataset taken from the greater Dallas metropolitan area.