Skip to main content
SpringerLink
Log in

Multivariate Hawkes processes with spatial covariates for spatiotemporal event data analysis

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Spatiotemporal events occur in many disciplines, including economics, sociology, criminology, and seismology, with different patterns in space and time related to environmental characteristics, policing, and human behavior. In this paper, we propose a class of multivariate Hawkes processes with spatial covariates to consider the influence structure of spatial features in spatiotemporal events and the spatiotemporal patterns such as clustering. Baseline intensities are assumed to be a spatial Poisson regression model to explain spatial feature influence. The transfer functions are considered unknown but smooth and decreasing to explain the clustering phenomena. A semiparametric estimation method based on time discretization and local constant approximation is introduced. Transfer function estimators are shown to be consistent, and baseline intensity estimators are consistent and asymptotically normal. We examine the numerical performance of the proposed estimators with extensive simulation and illustrate the application of the proposed model to crime data obtained from Pittsburgh, Pennsylvania.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  • Adelfio, G., Chiodi, M. (2015). Alternated estimation in semi-parametric space-time branching-type point processes with application to seismic catalogs. Stochastic Environmental Research and Risk Assessment, 29(2), 443–450.

    Article  Google Scholar 

  • Bacry, E., Delattre, S., Hoffmann, M., Muzy, J. F. (2012). Scaling limits for Hawkes processes and application to financial statistics. Stochastic Processes and Their Applications, 123(7), 251–265.

    Google Scholar 

  • Bernasco, W., Nieuwbeerta, P. (2004). How do residential burglars select target areas? A new approach to the analysis of criminal location choice. British Journal of Criminology, 45(3), 296–315.

    Article  Google Scholar 

  • Clauset, A., Woodard, R. (2013). Estimating the historical and future probabilities of large terrorist events. The Annals of Applied Statistics, 7(4), 1838–1865.

    MathSciNet  Google Scholar 

  • Clements, R. A., Schoenberg, F. P., Veen, A. (2012). Evaluation of space-time point process models using super-thinning. Environmetrics, 23(7), 606–616.

    Article  MathSciNet  Google Scholar 

  • Daley, D. J., Vere-Jones, D. (2003). An introduction to the theory of point processes: Volume I: Elementary theory and methods. New York: Springer-Verlag.

    Google Scholar 

  • Egesdal, M., Fathauer, C., Louie, K., Neuman, J. (2010). Statistical modeling of gang violence in Los Angeles. Society for Industrial and Applied Mathematics Undergraduate Research Online.

  • Eichler, M., Dahlhaus, R., Dueck, J. (2016). Graphical modeling for multivariate Hawkes processes with nonparametric link functions. Journal of Time Series Analysis, 38, 225–242.

    Article  MathSciNet  Google Scholar 

  • Embrechts, P., Liniger, T., Lin, L. (2011). Multivariate Hawkes processes: an application to financial data. Journal of Applied Probability, 48A, 367–378.

    Article  MathSciNet  Google Scholar 

  • Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1), 83–90.

    Article  MathSciNet  Google Scholar 

  • Kirchner, M. (2017). An estimation procedure for the Hawkes process. Quantitative Finance, 17(4), 571–595.

    Article  MathSciNet  Google Scholar 

  • Lewis, E., Mohler, G. (2011). A nonparametric EM algorithm for multiscale Hawkes processes. Journal of Nonparametric Statistics, 1(1), 1–20.

    Google Scholar 

  • Lewis, E., Mohler, G., Brantingham, P. J., Bertozzi, A. (2010). Self-exciting point process models of insurgency in Iraq. Security Journal, 25(3), 244–264.

    Article  Google Scholar 

  • Lewis, E. A. (2012). Estimation techniques for self-exciting point processes with applications to criminal behavior. PhD thesis, University of California, Los Angeles.

  • Liniger, T. J. (2009). Multivariate Hawkes processes. PhD thesis, ETH Zurich.

  • Marsan, D., Lengline, O. (2008). Extending earthquakes’ reach through cascading. Science, 319(5866), 1076–1079.

    Article  Google Scholar 

  • Mohler, G. O. (2013). Modeling and estimation of multi-source clustering in crime and security data. The Annals of Applied Statistics, 7(3), 1525–1539.

    Article  MathSciNet  Google Scholar 

  • Mohler, G. O., Short, M. B. (2012). Geographic profiling from kinetic models of criminal behavior. SIAM Journal on Applied Mathematics, 72(1), 163–180.

    Article  MathSciNet  Google Scholar 

  • Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P., Tita, G. E. (2011). Self-exciting point process modeling of crime. Journal of the American Statistical Association, 106(493), 100–108.

    Article  MathSciNet  Google Scholar 

  • Moller, J., Waagepetersen, R. P. (2003). Statistical inference and simulation for spatial point processes. London: CRC Press.

    Book  Google Scholar 

  • Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE Transactions on Information Theory, 27(1), 23–31.

    Article  Google Scholar 

  • Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401), 9–27.

    Article  Google Scholar 

  • Park, J., Schoenberg, F. P., Bertozzi, A. L., Brantingham, P. J. (2021). Investigating clustering and violence interruption in gang-related violent crime data using spatial-temporal point processes with covariates. Journal of the American Statistical Association, 116(536), 1674–1687.

    Article  MathSciNet  Google Scholar 

  • Reinhart, A. (2018). A review of self-exciting Spatio-temporal point processes and their applications. Statistical Science, 33(3), 299–318.

    MathSciNet  Google Scholar 

  • Reinhart, A., Greenhouse, J. (2017). Self-exciting point processes with spatial covariates: modeling the dynamics of crime. Journal of the Royal Statistical Society, 67(5), 1305–1329.

    Google Scholar 

  • Schoenberg, F. P. (2003). Multidimensional residual analysis of point process models for earthquake occurrences. Journal of the American Statistical Association, 98(464), 789–795.

    Article  MathSciNet  Google Scholar 

  • Short, M. B., Mohler, G. O., Brantingham, P. J., Tita, G. E. (2017). Gang rivalry dynamics via coupled point process networks. Discrete and Continuous Dynamical Systems-Series B, 19(5), 1459–1477.

    Article  MathSciNet  Google Scholar 

  • Tita, G., Ridgeway, G. (2007). The impact of gang formation on local patterns of crime. Journal of Research in Crime and Delinquency, 44(2), 208–237.

    Article  Google Scholar 

  • Veen, A., Schoenberg, F. P. (2008). Estimation of space-time branching process models in seismology using an EM-type algorithm. Journal of the American Statistical Association, 103(482), 614–624.

    Article  MathSciNet  Google Scholar 

  • Vere-Jones, D. (1992). Statistical methods for the description and display of earthquake catalogues, Statistics in the Environmental and Earth Sciences (eds. A. T. Walden and P. Guttorp), 220–246, Edward Arnold, London.

    Google Scholar 

  • Vere-Jones, D., Davies, R. B. (1966). A statistical survey of earthquakes in the main seismic region of New Zealand: Part 2-Time series analyses. New Zealand Journal of Geology and Geophysics, 9(3), 251–284.

    Article  Google Scholar 

  • Yuan, B., Schoenberg, F. P., Bertozzi, A. L. (2021). Fast estimation of multivariate spatiotemporal Hawkes processes and network reconstruction. Annals of the Institute of Statistical Mathematics, 73(6), 1127–1152.

    Article  MathSciNet  Google Scholar 

  • Zhuang, J. C., Mateu, J. (2019). A semiparametric spatiotemporal Hawkes-type point process model with periodic background for crime data. Journal of the Royal Statistical Society Series A, 182(3), 919–942.

    Article  MathSciNet  Google Scholar 

  • Zhuang, J. C., Ogata, Y., Vere-Jones, D. (2004). Analyzing earthquake clustering features by using stochastic reconstruction. Journal of Geophysical Research: Solid Earth, 109(B5), B05301.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, China

    Chenlong Li

  2. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China

    Kaiyan Cui

Authors
  1. Chenlong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kaiyan Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kaiyan Cui.

Ethics declarations

Funding

The first author was supported by the Applied Basic Research Programs of Shanxi Province (Grant No. 201901D211105). The second author was supported by the Fundamental Research Program of Shanxi Province (Grant No. 202103021223023).

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, C., Cui, K. Multivariate Hawkes processes with spatial covariates for spatiotemporal event data analysis. Ann Inst Stat Math (2024). https://doi.org/10.1007/s10463-023-00894-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10463-023-00894-2

Keywords

Search

Navigation