Annals of Applied Statistics

Space and circular time log Gaussian Cox processes with application to crime event data

Shinichiro Shirota and Alan E. Gelfand

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We view the locations and times of a collection of crime events as a space–time point pattern. Then, with either a nonhomogeneous Poisson process or with a more general Cox process, we need to specify a space–time intensity. For the latter, we need a random intensity which we model as a realization of a spatio-temporal log Gaussian process. Importantly, we view time as circular not linear, necessitating valid separable and nonseparable covariance functions over a bounded spatial region crossed with circular time. In addition, crimes are classified by crime type. Furthermore, each crime event is recorded by day of the year, which we convert to day of the week marks.

The contribution here is to develop models to accommodate such data. Our specifications take the form of hierarchical models which we fit within a Bayesian framework. In this regard, we consider model comparison between the nonhomogeneous Poisson process and the log Gaussian Cox process. We also compare separable vs. nonseparable covariance specifications.

Our motivating dataset is a collection of crime events for the city of San Francisco during the year 2012. We have location, hour, day of the year, and crime type for each event. We investigate models to enhance our understanding of the set of incidences.

Article information

Ann. Appl. Stat., Volume 11, Number 2 (2017), 481-503.

Received: September 2015
Revised: July 2016
First available in Project Euclid: 20 July 2017

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Zentralblatt MATH identifier

Derived covariates hierarchical model marked point pattern Markov chain Monte Carlo separable and nonseparable covariance functions wrapped circular variables


Shirota, Shinichiro; Gelfand, Alan E. Space and circular time log Gaussian Cox processes with application to crime event data. Ann. Appl. Stat. 11 (2017), no. 2, 481--503. doi:10.1214/16-AOAS960.

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Supplemental materials

  • Supplement to “Space and circular time log Gaussian Cox processes with application to crime event data”. In this online supplement article, we provide (1) proof of the validity of our proposed nonseparable covariance function on $\mathbb{R}^{2}\times S^{1}$ and (2) additional figures and tables to see posterior mean intensity estimates under different models.