## Annals of Applied Statistics

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

#### Abstract

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

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

Dates
Revised: July 2016
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.aoas/1500537712

Digital Object Identifier
doi:10.1214/16-AOAS960

Mathematical Reviews number (MathSciNet)
MR3693535

Zentralblatt MATH identifier
06775881

#### Citation

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. https://projecteuclid.org/euclid.aoas/1500537712

#### References

• Adams, R. P., Murray, I. and MacKay, D. J. C. (2009). Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities. In Proceedings of the 26th International Conference on Machine Learning MIT Press, Cambridge, MA.
• Andrieu, C. and Thoms, J. (2008). A tutorial on adaptive MCMC. Stat. Comput. 18 343–373.
• Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2015). Hierarchical Modeling and Analysis for Spatial Data, 2nd ed. Monographs on Statistics and Applied Probability 135. CRC Press, Boca Raton, FL.
• Brantingham, P. and Brantingham, P. (1995). Criminality of place: Crime generators and crime attractors. Eur. J. Crim. Policy Res. 3 5–26.
• Brix, A. and Diggle, P. J. (2001). Spatiotemporal prediction for log-Gaussian Cox processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 823–841.
• Chainey, S., Tompson, L. and Uhlig, S. (2008). The utility of hotspot mapping for predicting spatial patterns of crime. Secur. J. 21 4–28.
• Cressie, N. and Huang, H. C. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions. J. Amer. Statist. Assoc. 94 1330–1340.
• Czado, C., Gneiting, T. and Held, L. (2009). Predictive model assessment for count data. Biometrics 65 1254–1261.
• Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd ed. Springer, New York.
• Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd ed. Springer, Berlin.
• Doornik, J. (2007). Ox: Object Oriented Matrix Programming. Timberlake Consultants Press.
• Fisher, N. I. (1993). Statistical Analysis of Circular Data. Cambridge Univ. Press, Cambridge.
• Girolami, M. and Calderhead, B. (2011). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 123–214.
• Gneiting, T. (2002). Nonseparable, stationary covariance functions for space–time data. J. Amer. Statist. Assoc. 97 590–600.
• Gneiting, T. (2013). Strictly and non-strictly positive definite functions on spheres. Bernoulli 19 1327–1349.
• Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
• Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Statistics in Practice. Wiley, Chichester.
• Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics. World Scientific, River Edge, NJ.
• Jona-Lasinio, G., Gelfand, A. E. and Jona-Lasinio, M. (2012). Spatial analysis of wave direction data using wrapped Gaussian processes. Ann. Appl. Stat. 6 1478–1498.
• Leininger, T. J. (2014). Bayesian analysis of spatial point patterns. PhD dissertation.
• Leininger, T. J. and Gelfand, A. E. (2016). Bayesian inference and model assessment for spatial point patterns using posterior predictive samples. Bayesian Anal. 12 1–30.
• Liang, W. W. J., Colvin, J. B., Sansó, B. and Lee, H. K. H. (2014). Modeling and anomalous cluster detection for point processes using process convolutions. J. Comput. Graph. Statist. 23 129–150.
• Mardia, K. V. (1972). Statistics of Directional Data. Academic Press, London.
• Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Wiley, Chichester.
• Mohler, G. O. (2013). Modeling and estimation of multi-source clustering in crime and security data. Ann. Appl. Stat. 7 1825–1839.
• Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P. and Tita, G. E. (2011). Self-exciting point process modeling of crime. J. Amer. Statist. Assoc. 106 100–108.
• Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Stat. 25 451–482.
• Murray, I. and Adams, R. P. (2010). Slice sampling covariance hyperparameters of latent Gaussian models. In Advances in Neural Information Processing Systems 23. MIT Press, Cambridge, MA.
• Murray, I., Adams, R. P. and Graham, M. M. (2010). Elliptical slice sampling. In Proceedings of the 13th International Conference on Artifical Intelligence and Statistics (AISTAT). AISTAT Press.
• Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Ann. Inst. Statist. Math. 50 379–402.
• Porcu, E., Bevilacqua, M. and Genton, M. G. (2016). Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. J. Amer. Statist. Assoc. 111 888–898.
• Rodrigues, A. and Diggle, P. J. (2012). Bayesian estimation and prediction for inhomogeneous spatiotemporal log-Gaussian Cox processes using low-rank models, with application to criminal surveillance. J. Amer. Statist. Assoc. 107 93–101.
• Shirota, S. and Gelfand, A. E. (2017). Supplement to “Space and circular time log Gaussian Cox processes with application to crime event data.” DOI:10.1214/16-AOAS960SUPP.
• Taddy, M. A. (2010). Autoregressive mixture models for dynamic spatial Poisson processes: Application to tracking intensity of violent crime. J. Amer. Statist. Assoc. 105 1403–1417.
• Wang, F. and Gelfand, A. E. (2014). Modeling space and space–time directional data using projected Gaussian processes. J. Amer. Statist. Assoc. 109 1565–1580.
• Zhang, H. (2004). Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Amer. Statist. Assoc. 99 250–261.

#### 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.