Statistical Science

A Review of Self-Exciting Spatio-Temporal Point Processes and Their Applications

Alex Reinhart

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Self-exciting spatio-temporal point process models predict the rate of events as a function of space, time, and the previous history of events. These models naturally capture triggering and clustering behavior, and have been widely used in fields where spatio-temporal clustering of events is observed, such as earthquake modeling, infectious disease, and crime. In the past several decades, advances have been made in estimation, inference, simulation, and diagnostic tools for self-exciting point process models. In this review, I describe the basic theory, survey related estimation and inference techniques from each field, highlight several key applications, and suggest directions for future research.

Article information

Source
Statist. Sci., Volume 33, Number 3 (2018), 299-318.

Dates
First available in Project Euclid: 13 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ss/1534147221

Digital Object Identifier
doi:10.1214/17-STS629

Mathematical Reviews number (MathSciNet)
MR3843374

Zentralblatt MATH identifier
06991118

Keywords
Epidemic-Type Aftershock Sequence conditional intensity Hawkes process stochastic declustering

Citation

Reinhart, Alex. A Review of Self-Exciting Spatio-Temporal Point Processes and Their Applications. Statist. Sci. 33 (2018), no. 3, 299--318. doi:10.1214/17-STS629. https://projecteuclid.org/euclid.ss/1534147221


Export citation

References

  • Adelfio, G. and Chiodi, M. (2015a). Alternated estimation in semi-parametric space–time branching-type point processes with application to seismic catalogs. Stoch. Environ. Res. Risk Assess. 29 443–450.
  • Adelfio, G. and Chiodi, M. (2015b). FLP estimation of semi-parametric models for space–time point processes and diagnostic tools. Spat. Stat. 14 119–132.
  • Bacry, E., Mastromatteo, I. and Muzy, J.-F. (2015). Hawkes processes in finance. Mark. Microstruct. Liq. 1 1550005.
  • Baddeley, A., Møller, J. and Pakes, A. G. (2007). Properties of residuals for spatial point processes. Ann. Inst. Statist. Math. 60 627–649.
  • Baddeley, A., Rubak, E. and Møller, J. (2011). Score, pseudo-score and residual diagnostics for spatial point process models. Statist. Sci. 26 613–646.
  • Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005). Residual analysis for spatial point processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 67 617–666.
  • Bauwens, L. and Hautsch, N. (2009). Modelling financial high frequency data using point processes. In Handbook of Financial Time Series (T. Mikosch, J.-P. Kreiß, R. A. Davis and T. G. Andersen, eds.) 953–979. Springer, Berlin.
  • Bernasco, W., Johnson, S. D. and Ruiter, S. (2015). Learning where to offend: Effects of past on future burglary locations. Appl. Geogr. 60 120–129.
  • Braga, A. A., Papachristos, A. V. and Hureau, D. M. (2014). The effects of hot spots policing on crime: An updated systematic review and meta-analysis. Justice Q. 31 633–663.
  • Bray, A. and Schoenberg, F. P. (2013). Assessment of point process models for earthquake forecasting. Statist. Sci. 28 510–520.
  • Bray, A., Wong, K., Barr, C. D. and Schoenberg, F. P. (2014). Voronoi residual analysis of spatial point process models with applications to California earthquake forecasts. Ann. Appl. Stat. 8 2247–2267.
  • Brockmann, D., Hufnagel, L. and Geisel, T. (2006). The scaling laws of human travel. Nature 439 462–465.
  • Chen, J. M., Hawkes, A. G., Scalas, E. and Trinh, M. (2017). Performance of information criteria used for model selection of Hawkes process models of financial data. Available at arXiv:1702.06055.
  • Chiodi, M. and Adelfio, G. (2011). Forward likelihood-based predictive approach for space–time point processes. Environmetrics 22 749–757.
  • Clarke, R. V. and Cornish, D. B. (1985). Modeling offenders’ decisions: A framework for research and policy. Crime Justice 6 147–185.
  • Clements, R. A., Schoenberg, F. P. and Veen, A. (2012). Evaluation of space–time point process models using super-thinning. Environmetrics 23 606–616.
  • Cohen, L. E. and Felson, M. (1979). Social change and crime rate trends: A routine activity approach. Am. Sociol. Rev. 44 588–608.
  • Cohen, J., Gorr, W. L. and Olligschlaeger, A. M. (2007). Leading indicators and spatial interactions: A crime-forecasting model for proactive police deployment. Geogr. Anal. 39 105–127.
  • Cowling, A. and Hall, P. (1996). On pseudodata methods for removing boundary effects in kernel density estimation. J. Roy. Statist. Soc. Ser. B 58 551–563.
  • Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data. Wiley, New York.
  • Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Volume I: Elementary Theory and Methods, 2nd ed. Springer, New York.
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximimum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1–38.
  • Diggle, P. J. (2014). Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, 3rd ed. Monographs on Statistics and Applied Probability 128. CRC Press, Boca Raton, FL.
  • Diggle, P. J., Moraga, P., Rowlingson, B. and Taylor, B. M. (2013). Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm. Statist. Sci. 28 542–563.
  • Fotheringham, A. S. and Wong, D. W. S. (1991). The modifiable areal unit problem in multivariate statistical analysis. Environ. Plann. A. 23 1025–1044.
  • Fox, E. W., Schoenberg, F. P. and Gordon, J. S. (2016). Spatially inhomogeneous background rate estimators and uncertainty quantification for nonparametric Hawkes point process models of earthquake occurrences. Ann. Appl. Stat. 10 1725–1756.
  • Fox, E. W., Short, M. B., Schoenberg, F. P., Coronges, K. D. and Bertozzi, A. L. (2016). Modeling e-mail networks and inferring leadership using self-exciting point processes. J. Amer. Statist. Assoc. 111 564–584.
  • Freed, A. M. (2005). Earthquake triggering by static, dynamic, and postseismic stress transfer. Annu. Rev. Earth Planet. Sci. 33 335–367.
  • González, J. A., Rodríguez-Cortés, F. J., Cronie, O. and Mateu, J. (2016). Spatio-temporal point process statistics: A review. Spat. Stat. 18 505–544.
  • Gorr, W. L. and Lee, Y. (2015). Early warning system for temporary crime hot spots. J. Quant. Criminol. 31 25–47.
  • Gray, A. G. and Moore, A. W. (2003). Nonparametric density estimation: Toward computational tractability. In SIAM International Conference on Data Mining 203–211. SIAM, Philadelphia, PA.
  • Green, B., Horel, T. and Papachristos, A. V. (2017). Modeling contagion through social networks to explain and predict gunshot violence in Chicago, 2006 to 2014. JAMA Intern. Med. 177 326–333.
  • Harte, D. S. (2012). Bias in fitting the ETAS model: A case study based on New Zealand seismicity. Geophys. J. Int. 192 390–412.
  • Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 51 83–90.
  • Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Probab. 11 493–503.
  • Johnson, D. H. (1996). Point process models of single-neuron discharges. J. Comput. Neurosci. 3 275–299.
  • Kumazawa, T. and Ogata, Y. (2014). Nonstationary ETAS models for nonstandard earthquakes. Ann. Appl. Stat. 8 1825–1852.
  • Lewis, P. A. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Nav. Res. Logist. Q. 26 403–413.
  • Lewis, E., Mohler, G., Brantingham, P. J. and Bertozzi, A. L. (2011). Self-exciting point process models of civilian deaths in Iraq. Secur. J. 25 244–264.
  • Lippiello, E., Giacco, F., de Arcangelis, L., Marzocchi, W. and Godano, C. (2014). Parameter estimation in the ETAS model: Approximations and novel methods. Bull. Seismol. Soc. Am. 104 985–994.
  • Loeffler, C. and Flaxman, S. (2016). Is gun violence contagious? Available at arXiv:1611.06713.
  • Marsan, D. and Lengliné, O. (2008). Extending earthquakes’ reach through cascading. Science 319 1076–1079.
  • Marsan, D. and Lengliné, O. (2010). A new estimation of the decay of aftershock density with distance to the mainshock. J. Geophys. Res. 115 B09302.
  • McLachlan, G. J. and Krishnan, T. (2008). The EM Algorithm and Extensions, 2nd ed. Wiley, New York.
  • Meyer, S. (2010). Spatio-temporal infectious disease epidemiology based on point processes. Master’s thesis, Ludwig-Maximilians-Univ. München.
  • Meyer, S., Elias, J. and Höhle, M. (2012). A space–time conditional intensity model for invasive meningococcal disease occurrence. Biometrics 68 607–616.
  • Meyer, S. and Held, L. (2014). Power-law models for infectious disease spread. Ann. Appl. Stat. 8 1612–1639.
  • Meyer, S., Warnke, I., Rössler, W. and Held, L. (2016). Model-based testing for space–time interaction using point processes: An application to psychiatric hospital admissions in an urban area. Spat. Spatio-temporal Epidemiol. 17 15–25.
  • Mohler, G. O. (2014). Marked point process hotspot maps for homicide and gun crime prediction in Chicago. Int. J. Forecast. 30 491–497.
  • 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. and Rasmussen, J. G. (2005). Perfect simulation of Hawkes processes. Adv. in Appl. Probab. 37 629–646.
  • Musmeci, F. and Vere-Jones, D. (1992). A space–time clustering model for historical earthquakes. Ann. Inst. Statist. Math. 44 1–11.
  • Nandan, S., Ouillon, G., Wiemer, S. and Sornette, D. (2017). Objective estimation of spatially variable parameters of epidemic type aftershock sequence model: Application to California. J. Geophys. Res. Solid Earth 122 5118–5143.
  • Nsoesie, E. O., Brownstein, J. S., Ramakrishnan, N. and Marathe, M. V. (2013). A systematic review of studies on forecasting the dynamics of influenza outbreaks. Influenza Other Respir. Viruses 8 309–316.
  • Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Ann. Inst. Statist. Math. 30 243–261.
  • Ogata, Y. (1998). Space–time point-process models for earthquake occurrences. Ann. Inst. Statist. Math. 50 379–402.
  • Ogata, Y. (1999). Seismicity analysis through point-process modeling: A review. Pure Appl. Geophys. 155 471–507.
  • Ogata, Y. and Katsura, K. (1988). Likelihood analysis of spatial inhomogeneity for marked point patterns. Ann. Inst. Statist. Math. 40 29–39.
  • Ogata, Y., Katsura, K. and Tanemura, M. (2003). Modelling heterogeneous space–time occurrences of earthquakes and its residual analysis. J. R. Stat. Soc. Ser. C. Appl. Stat. 52 499–509.
  • Ogata, Y. and Zhuang, J. (2006). Space–time ETAS models and an improved extension. Tectonophysics 413 13–23.
  • Peng, R. D., Schoenberg, F. P. and Woods, J. A. (2005). A space–time conditional intensity model for evaluating a wildfire hazard index. J. Amer. Statist. Assoc. 100 26–35.
  • Porter, M. D. and White, G. (2012). Self-exciting hurdle models for terrorist activity. Ann. Appl. Stat. 6 106–124.
  • Rasmussen, J. G. (2013). Bayesian inference for Hawkes processes. Methodol. Comput. Appl. Probab. 15 623–642.
  • Ratcliffe, J. H. and Rengert, G. F. (2008). Near-repeat patterns in Philadelphia shootings. Secur. J. 21 58–76.
  • Rathbun, S. L. (1996). Asymptotic properties of the maximum likelihood estimator for spatio-temporal point processes. J. Statist. Plann. Inference 51 55–74.
  • Ripley, B. D. (1977). Modelling spatial patterns. J. Roy. Statist. Soc. Ser. B 39 172–212.
  • Ross, G. J. (2016). Bayesian estimation of the ETAS model for earthquake occurrences. Preprint.
  • Sarma, S. V., Nguyen, D. P., Czanner, G., Wirth, S., Wilson, M. A., Suzuki, W. and Brown, E. N. (2011). Computing confidence intervals for point process models. Neural Comput. 23 2731–2745.
  • Schoenberg, F. P. (2003). Multidimensional residual analysis of point process models for earthquake occurrences. J. Amer. Statist. Assoc. 98 789–795.
  • Schoenberg, F. P. (2013). Facilitated estimation of ETAS. Bull. Seismol. Soc. Am. 103 601–605.
  • Schoenberg, F. P. (2016). A note on the consistent estimation of spatial-temporal point process parameters. Statist. Sinica 26 861–879.
  • Schoenberg, F. P., Hoffman, M. and Harrigan, R. (2017). A recursive point process model for infectious diseases. Available at arXiv:1703.08202.
  • Short, M. B., D’Orsogna, M. R., Brantingham, P. J. and Tita, G. E. (2009). Measuring and modeling repeat and near-repeat burglary effects. J. Quant. Criminol. 25 325–339.
  • Silverman, B. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall, Boca Raton, FL.
  • Stan Development Team (2016). Stan modeling language users guide and reference manual. Available at http://mc-stan.org.
  • Townsley, M., Homel, R. and Chaseling, J. (2003). Infectious burglaries: A test of the near repeat hypothesis. Br. J. Criminol. 43 615–633.
  • Veen, A. and Schoenberg, F. P. (2008). Estimation of space–time branching process models in seismology using an EM-type algorithm. J. Amer. Statist. Assoc. 103 614–624.
  • Vere-Jones, D. (2009). Some models and procedures for space–time point processes. Environ. Ecol. Stat. 16 173–195.
  • Wang, Q., Schoenberg, F. P. and Jackson, D. D. (2010). Standard errors of parameter estimates in the ETAS model. Bull. Seismol. Soc. Am. 100 1989–2001.
  • Weisburd, D. (2015). The law of crime concentration and the criminology of place. Criminology 53 133–157.
  • Zhuang, J. (2006). Second-order residual analysis of spatiotemporal point processes and applications in model evaluation. J. Roy. Statist. Soc. Ser. B 68 635–653.
  • Zhuang, J. (2011). Next-day earthquake forecasts for the Japan region generated by the ETAS model. Earth Planets Space 63 207–216.
  • Zhuang, J., Ogata, Y. and Vere-Jones, D. (2002). Stochastic declustering of space–time earthquake occurrences. J. Amer. Statist. Assoc. 97 369–380.
  • Zhuang, J., Ogata, Y. and Vere-Jones, D. (2004). Analyzing earthquake clustering features by using stochastic reconstruction. J. Geophys. Res. 109 B05301.
  • Zipkin, J. R., Schoenberg, F. P., Coronges, K. and Bertozzi, A. L. (2016). Point-process models of social network interactions: Parameter estimation and missing data recovery. European J. Appl. Math. 27 502–529.