Bayesian Analysis

Bayesian Inference and Model Assessment for Spatial Point Patterns Using Posterior Predictive Samples

Thomas J. Leininger and Alan E. Gelfand

Full-text: Open access

Abstract

Spatial point pattern data describes locations of events observed over a given domain, with the number of and locations of these events being random. Historically, data analysis for spatial point patterns has focused on rejecting complete spatial randomness and then on fitting a richer model specification. From a Bayesian standpoint, the literature is growing but primarily considers versions of Poisson processes, focusing on specifications for the intensity. However, the Bayesian literature on, e.g., clustering or inhibition processes is limited, primarily attending to model fitting. There is little attention given to full inference and scant with regard to model adequacy or model comparison.

The contribution here is full Bayesian analysis, implemented through generation of posterior point patterns using composition. Model features, hence broad inference, can be explored through functions of these samples. The approach is general, applicable to any generative model for spatial point patterns.

The approach is also useful in considering model criticism and model selection both in-sample and, when possible, out-of-sample. Here, we adapt or extend familiar tools. In particular, for model criticism, we consider Bayesian residuals, realized and predictive, along with empirical coverage and prior predictive checks through Monte Carlo tests. For model choice, we propose strategies using predictive mean square error, empirical coverage, and ranked probability scores. For simplicity, we illustrate these methods with standard models such as Poisson processes, log-Gaussian Cox processes, and Gibbs processes. The utility of our approach is demonstrated using a simulation study and two real datasets.

Article information

Source
Bayesian Anal., Volume 12, Number 1 (2017), 1-30.

Dates
First available in Project Euclid: 30 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.ba/1448899901

Digital Object Identifier
doi:10.1214/15-BA985

Mathematical Reviews number (MathSciNet)
MR3597565

Zentralblatt MATH identifier
1384.62091

Keywords
Cox process cross-validation Gibbs process Markov chain Monte Carlo nonhomogeneous Poisson process predictive residuals ranked probability scores realized residuals Strauss process

Citation

Leininger, Thomas J.; Gelfand, Alan E. Bayesian Inference and Model Assessment for Spatial Point Patterns Using Posterior Predictive Samples. Bayesian Anal. 12 (2017), no. 1, 1--30. doi:10.1214/15-BA985. https://projecteuclid.org/euclid.ba/1448899901


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Supplemental materials

  • Online Supplementary Material for Bayesian Inference and Model Assessment for Spatial Point Patterns Using Posterior Predictive Samples. Further analysis of the Duke Forest example is given in the online supplementary material, showing posterior distributions for first- and second-order marginal intensities, the pairwise correlation function, and other features of interest. Then a simulation study is presented, showing the performance of predictive residual coverage and model choice using RPS for various models under several data-generating scenarios. Data is generated under an HPP, an NHPP, and different specifications of LGCPs and then each model is fit to each data scenario. With many observations, the simpler models show signs of lack of fit when the data-generating process is more complex.