Skip to main content
SpringerLink
Log in

Space–time inhomogeneous background intensity estimators for semi-parametric space–time self-exciting point process models

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

Abstract

Histogram maximum likelihood estimators of semi-parametric space–time self-exciting point process models via expectation–maximization algorithm can be biased when the background process is inhomogeneous. We explore an alternative estimation method based on the variable bandwidth kernel density estimation (KDE) and EM algorithm. The proposed estimation method involves expanding the semi-parametric models by incorporating an inhomogeneous background process in space and time and applying the variable bandwidth KDE to estimate the background intensity function. Using an example, we show how the variable bandwidth KDE can be estimated this way. Two simulation examples based on residual analysis are designed to evaluate and validate the ability of our methods to recover the background intensity function and parametric triggering intensity function.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

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., Dayri, K., Muzy, J. F. (2012). Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data. The European Physical Journal B, 85(5), 1–12.

    Article  Google Scholar 

  • Bacry, E., Gaïffas, S., Muzy, J. F. (2015). A generalization error bound for sparse and low-rank multivariate Hawkes processes. arXiv:1501.00725.

  • Bray, A., Schoenberg, F. P. (2013). Assessment of point process models for earthquake forecasting. Statistical Science, 28(4), 510–520.

    Article  MathSciNet  Google Scholar 

  • Bray, A., Wong, K., Barr, C. D., Schoenberg, F. P. (2014). Voronoi residual analysis of spatial point process models with applications to California earthquake forecasts. The Annals of Applied Statistics, 8(4), 2247–2267.

    Article  MathSciNet  Google Scholar 

  • Brix, A., Diggle, P. J. (2011). Spatiotemporal prediction for log-Gaussian Cox processes. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(4), 823–841.

  • 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.

  • Diggle, P. J. (2006). Spatio-temporal point processes: Methods and applications. In Semstat2004 (pp. 1–45). London: CRC Press.

  • Diggle, P., Moraga, B., Rowlingson, B. M., Taylor, et al. (2013). Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm. Statistical Science, 28(4), 542–563.

    Article  MathSciNet  Google Scholar 

  • Fox, E. W., Schoenberg, F. P., Gordon, J. S. (2016a). Spatially inhomogeneous background rate estimators and uncertainty quantification for nonparametric Hawkes point process models of earthquake occurrences. The Annals of Applied Statistics, 10(3), 1725–1756.

    Article  MathSciNet  Google Scholar 

  • Fox, E. W., Short, M. B., Schoenberg, F. P., Coronges, K. D., Bertozzi, A. L. (2016b). Modeling e-mail networks and inferring leadership using self-exciting point processes. Journal of the American Statistical Association, 111(514), 564–584.

    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 

  • Kagan, Y. Y. (2003). Accuracy of modern global earthquake catalogs. Physics of the Earth and Planetary Interiors, 135(2–3), 173–209.

    Article  Google Scholar 

  • Liniger, T. J. (2009). Multivariate Hawkes processes. Ph.D. 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., 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 

  • Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50(2), 379–402.

    Article  MathSciNet  Google Scholar 

  • Ogata, Y., Katsura, K., Tanemura, M. (2003). Modelling heterogeneous space-time occurrences of earthquakes and its residual analysis. Journal of the Royal Statistical Society: Series C (Applied Statistics), 52(4), 499–509.

  • Ripley, B. D. (1977). Modelling spatial patterns. Journal of the Royal Statistical Society. Series B (Methodological), 39(2), 172–212.

    Article  MathSciNet  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 

  • 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. In A. T. Walden, P. Guttorp (Eds.), Statistics in the environmental and earth sciences, pp. 220–246. London: Edward Arnold.

  • Vere-Jones, D. (1995). Forecasting earthquakes and earthquake risk. International Journal of Forecasting, 11(4), 503–538.

    Article  Google Scholar 

  • Yang, Y. X., Etesami, J., He, N., Kiyavash, N. (2018). Nonparametric Hawkes processes: Online estimation and generalization bounds. arXiv:1801.08273.

  • Zhuang, J. C. (2011). Next-day earthquake forecasts for the Japan region generated by the ETAS model. Earth, Planets and Space, 63(3), 207–216.

    Article  Google Scholar 

  • Zhuang, J. C., Ogata, Y., Vere-Jones, D. (2002). Stochastic declustering of space-time earthquake occurrences. Journal of the American Statistical Association, 97(458), 369–380.

    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.

Download references

Acknowledgements

The authors would like to express their sincere gratitude to the reviewers and editor for very helpful suggestions and comments which greatly improved this paper. The authors would like to thank the National Natural Science Foundation of China (No. 91746107, 91746205); the State Scholarship Fund of China Scholarship Council (CSC); and the National Science and Engineering Research Council (NSERC) of Canada, for their funding and support.

Author information

Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300072, China

    Chenlong Li & Zhanjie Song

  2. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, N2L 3C5, Canada

    Chenlong Li

  3. Visual Pattern Analysis Research Lab, Tianjin University, Tianjin, 300072, China

    Zhanjie Song

  4. School of Computer Science and Technology, Tianjin University, Tianjin, 300072, China

    Wenjun Wang

Authors
  1. Chenlong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhanjie Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenjun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhanjie Song.

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., Song, Z. & Wang, W. Space–time inhomogeneous background intensity estimators for semi-parametric space–time self-exciting point process models. Ann Inst Stat Math 72, 945–967 (2020). https://doi.org/10.1007/s10463-019-00715-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-019-00715-5

Keywords

Search

Navigation