A semiparametric spatiotemporal Hawkes‐type point process model with periodic background for crime data

Past studies have shown that crime events are often clustered. This study proposes a spatiotemporal Hawkes‐type point process model, which includes a background component with daily and weekly periodization, and a clustering component that is triggered by previous events. We generalize the non‐parametric stochastic reconstruction method so that we can estimate each component in the background rate and the triggering response that appears in the model conditional intensity: the background rate includes a daily and a weekly periodicity, a separable spatial component and a long‐term background trend. Two relaxation coefficients are introduced to stabilize and secure the estimation process. This model is used to describe the occurrences of violence or robbery cases in Castellon, Spain, during 2 years. The results show that robbery crime is highly influenced by daily life rhythms, revealed by its daily and weekly periodicity, and that about 3% of such crimes can be explained by clustering. Further diagnostic analysis shows that the model could be improved by considering the following ingredients: the daily occurrence patterns are different between weekends and working days; in the city centre, robbery activity shows different temporal patterns, in both weekly periodicity and long‐term trend, from other suburb areas.

[1]  Yosihiko Ogata,et al.  Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes , 1988 .

[2]  G. Ost,et al.  Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels , 2017, Stochastic Processes and their Applications.

[3]  George Mohler,et al.  Marked point process hotspot maps for homicide and gun crime prediction in Chicago , 2014 .

[4]  P. Brémaud,et al.  STABILITY OF NONLINEAR HAWKES PROCESSES , 1996 .

[5]  Lingjiong Zhu,et al.  Large deviations for Markovian nonlinear Hawkes processes , 2011, 1108.2432.

[6]  M. Felson,et al.  Crime and Everyday Life , 1998 .

[7]  M. Bartlett.  The Spectral Analysis of Point Processes , 1963 .

[8]  D. Phan,et al.  Self-exciting point process models for political conflict forecasting , 2017, European Journal of Applied Mathematics.

[9]  A. Hawkes,et al.  A cluster process representation of a self-exciting process , 1974, Journal of Applied Probability.

[10]  Jerry H. Ratcliffe,et al.  The Hotspot Matrix: A Framework for the Spatio‐Temporal Targeting of Crime Reduction , 2004 .

[11]  J. Neyman,et al.  Statistical Approach to Problems of Cosmology , 1958 .

[12]  Alan E. Gelfand,et al.  Space and circular time log Gaussian Cox processes with application to crime event data , 2016, 1611.08719.

[13]  J Neyman,et al.  Frequency of Separation and of Interlocking of Clusters of Galaxies. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Lingjiong Zhu.  Nonlinear Hawkes Processes , 2013, 1304.7531.

[15]  D. Vere-Jones,et al.  Stochastic Declustering of Space-Time Earthquake Occurrences , 2002 .

[16]  Wilpen L. Gorr,et al.  Leading Indicators and Spatial Interactions: A Crime‐Forecasting Model for Proactive Police Deployment , 2007 .

[17]  George E. Tita,et al.  Self-Exciting Point Process Modeling of Crime , 2011 .

[18]  M. Townsley,et al.  Infectious Burglaries. A Test of the Near Repeat Hypothesis , 2003 .

[19]  Jiancang Zhuang,et al.  Stability of earthquake clustering models: criticality and branching ratios. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Alex Reinhart,et al.  A Review of Self-Exciting Spatio-Temporal Point Processes and Their Applications , 2017, Statistical Science.

[21]  Lingjiong Zhu.  Process-level large deviations for nonlinear Hawkes point processes , 2011, 1108.2431.

[22]  Jiancang Zhuang,et al.  Second‐order residual analysis of spatiotemporal point processes and applications in model evaluation , 2006 .

[23]  D. Marsan,et al.  Extending Earthquakes' Reach Through Cascading , 2008, Science.

[24]  Frederic Paik Schoenberg,et al.  On Rescaled Poisson Processes and the Brownian Bridge , 2002 .

[25]  S. Delattre,et al.  HAWKES PROCESSES ON LARGE NETWORKS , 2016 .

[26]  Yosihiko Ogata,et al.  Statistical model for standard seismicity and detection of anomalies by residual analysis , 1989 .

[27]  G. Torrisi.  Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes , 2017 .

[28]  Tao Cheng,et al.  Improving the Robustness and Accuracy of Crime Prediction with the Self-Exciting Point Process Through Isotropic Triggering , 2016, Applied Spatial Analysis and Policy.

[29]  Peter J. Diggle,et al.  Bayesian Estimation and Prediction for Inhomogeneous Spatiotemporal Log-Gaussian Cox Processes Using Low-Rank Models, With Application to Criminal Surveillance , 2012 .

[30]  D. Vere-Jones,et al.  Analyzing earthquake clustering features by using stochastic reconstruction , 2004 .

[31]  P. Lewis.  A Branching Poisson Process Model for the Analysis of Computer Failure Patterns , 1964 .

[32]  Ned Levine,et al.  CrimeStat: A Spatial Statistical Program for the Analysis of Crime Incidents , 2008, Encyclopedia of GIS.

[33]  Ulrike Goldschmidt,et al.  An Introduction To The Theory Of Point Processes , 2016 .

[34]  Ken Pease,et al.  Prospective hot-spotting - The future of crime mapping? , 2004 .

[35]  A. Hawkes.  Point Spectra of Some Mutually Exciting Point Processes , 1971 .

[36]  A. Hawkes.  Spectra of some self-exciting and mutually exciting point processes , 1971 .

[37]  Andrea L. Bertozzi,et al.  Randomized Controlled Field Trials of Predictive Policing , 2015 .

[38]  Joel Greenhouse,et al.  Self‐exciting point processes with spatial covariates: modelling the dynamics of crime , 2017, 1708.03579.

[39]  Y. Ogata.  Space-Time Point-Process Models for Earthquake Occurrences , 1998 .

[40]  G. Torrisi.  Gaussian approximation of nonlinear Hawkes processes , 2016, 1609.08899.