Fifty years later: new directions in Hawkes processes (invited article)

• John Worrall ^[1] ; Raiha Browning ^[1] ; Paul Wu ^[1] ; Kerrie Mengersen ^[1]
  1. [1] Queensland University of Technology

     Queensland University of Technology

     Australia

     
• Localización: Sort: Statistics and Operations Research Transactions, ISSN 1696-2281, Vol. 46, Nº. 1, 2022, págs. 3-38
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • The Hawkes process is a self-exciting Poisson point process, characterised by a conditional intensity function. Since its introduction fifty years ago, it has been the subject of numerous research directions and continues to inspire new methodological and theoretical developments as well as new applications. This paper marks half a century of interest in Hawkes processes by presenting a snapshot of four state-of-the-art research directions, categorised as frequentist and Bayesian methods, other modelling approaches and notable theoretical developments. A particular focus is on nonparametric approaches, with advances in kernel estimation and computational efficiencies. A survey of real world applications is provided to illustrate the breadth of application of this remarkable approach.

• Referencias bibliográficas
  • Achab, M., Bacry, E., Gaiffas, S., Mastromatteo, I. and Muzy, J. F. (2018). Uncovering causality from multivariate hawkes integrated cumulants....
  • Aldous, D. J. (1985). Exchangeability and related topics. Ecole d’ ́ e de Probabilit´Et´ es de Saint-Flour XIII—1983 1–198. Lecture Notes...
  • Bacry, E., Dayri, K. and Muzy, J. F. (2012). Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency...
  • Bacry, E., Mastromatteo, I. and Muzy, J. F. (2015). Hawkes Processes in Finance. Market Microstructure and Liquidity, 01 (01), 1550005.
  • Bacry, E. and Muzy, J. F. (2016). Firstand second-order statistics characterization of hawkes processes and non-parametric estimation. IEEE...
  • Bacry, E., Gaı̈ffas, S., Mastromatteo, I. and Muzy, J. F. (2016a). Mean-feld inference of Hawkes point processes. Journal of Physics A:...
  • Bacry, E., Jaisson, T. and Muzy, J. F. (2016b). Estimation of slowly decreasing Hawkes kernels: application to high-frequency order book dynamics....
  • Bacry, E., Bompaire, M., Gaı̈ffas, S. and Muzy, J. F. (2020). Sparse and low-rank multivariate Hawkes processes. Journal of Machine Learning...
  • Blundell, C., Heller, K. A. and Beck, J. M. (2012). Modelling reciprocating relationships with Hawkes processes. Advances in Neural Information...
  • Broderick, T., Boyd, N., Wibisono, A., Wilson, A. C. and Jordan, M. I. (2013). Streaming variational Bayes. Advances in Neural Information...
  • Browning, R., Sulem, D., Mengersen, K., Rivoirard, V. and Rousseau, J. (2021). Simple discrete-time self-exciting models can describe complex...
  • Chen, J., Hawkes, A. G. and Scalas, E. (2021). A Fractional Hawkes Process. SEMA SIMAI Springer Series, 26, 121–131.
  • Chérief-Abdellatif, B. E., Alquier, P. and Khan, M. E. (2019). A generalization bound for online variational inference. Asian Conference...
  • Chiang, W. H., Liu, X. and Mohler, G. (2021). Hawkes process modeling of COVID19 with mobility leading indicators and spatial covariates....
  • Cox, D. R. (1955). Some statistical methods connected with series of events. Journal of the Royal Statistical Society: Series B (Methodological),...
  • Cui, L., Hawkes, A. and Yi, H. (2020). An elementary derivation of moments of Hawkes processes. Advances in Applied Probability, 52 (1), 102–137.
  • Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Springer-Verlag.
  • Donnet, S., Rivoirard, V. and Rousseau, J. (2020). Nonparametric Bayesian estimation for multivariate Hawkes processes. The Annals of statistics,...
  • Du, H., Zhou, Y., Ma, Y. and Wang, S. (2021). Astrologer: Exploiting graph neural Hawkes process for event propagation prediction with spatio-temporal...
  • Du, N., Dai, H., Trivedi, R., Upadhyay, U., Gomez-Rodriguez, M. and Song, L. (2016). Recurrent marked temporal point processes: Embedding...
  • Du, N., Farajtabar, M., Ahmed, A., Smola, A. J. and Song, L. (2015). Dirichlet-Hawkes processes with applications to clustering continuous-time...
  • Embrechts, P. and Kirchner, M. (2018). Hawkes graphs. Theory of Probability and its Applications, 62 (1), 132–155.
  • Fox, E. W., Schoenberg, F. P. and Gordon, J. S. (2016). Spatially inhomogeneous background rate estimators and uncertainty quantifcation for...
  • Gao, F. and Zhu, L. (2018a). Some asymptotic results for nonlinear Hawkes processes. Stochastic Processes and their Applications, 128 (12),...
  • Gao, X. and Zhu, L. (2018b). Limit theorems for Markovian Hawkes processes with a large initial intensity. Stochastic Processes and their...
  • Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711–732.
  • Guo, F., Blundell, C., Wallach, H. and Heller, K. (2015). The Bayesian echo chamber: Modeling social infuence via linguistic accommodation....
  • Hall, E. C. and Willett, R. M. (2016). Tracking dynamic point processes on networks. IEEE Transactions on Information Theory, 62 (7), 4327–4346.
  • Halpin, P. F. (2013). An EM algorithm for Hawkes process. New Developments in Quantitative Psychology: Proceedings of the 77th International...
  • Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. Journal of the Royal Statistical Society: Series B (Methodological),...
  • Holbrook, A. J., Ji, X. and Suchard, M. A. (2022). From viral evolution to spatial contagion: a biologically modulated Hawkes model. Bioinformatics...
  • Holbrook, A. J., Loeffer, C. E., Flaxman, S. R. and Suchard, M. A. (2021). Scalable Bayesian inference for self-excitatory stochastic processes...
  • Jovanovic,´ S., Hertz, J. and Rotter, S. (2015). Cumulants of Hawkes point processes. Physical Review E Statistical, Nonlinear, and Soft Matter...
  • Kalair, K., Connaughton, C. and Di Loro, P. A. (2021). A non-parametric Hawkes process model of primary and secondary accidents on a UK smart...
  • Kanazawa, K. and Sornette, D. (2020). Field master equation theory of the self-excited Hawkes process. Physical Review Research, 2 (3), 33442.
  • Kelly, J. D., Park, J., Harrigan, R. J., Hoff, N. A., Lee, S. D., Wannier, R., Selo, B., Mossoko, M., Njoloko, B., Okitolonda-Wemakoy, E.,...
  • Kemp, C., Tenenbaum, J. B., Griffths, T. L., Yamada, T. and Ueda, N. (2006). Learning systems of concepts with an infnite relational model....
  • Kirchner, M. (2017). An estimation procedure for the Hawkes process. Quantitative Finance, 17 (4), 571–595.
  • Kobayashi, R. and Lambiotte, R. (2016). TiDeH: Time-dependent Hawkes process for predicting retweet dynamics. Tenth International AAAI Conference...
  • Laub, P. J., Lee, Y. and Taimre, T. (2021). The elements of Hawkes processes. Springer International Publishing. Cham.
  • Lee, Y., Lim, K. W. and Ong, C. S. (2016). Hawkes processes with stochastic excitations. 33rd International Conference on Machine Learning,...
  • Lewis, E. and Mohler, G. (2011). A nonparametric EM algorithm for multiscale Hawkes processes. Journal of Nonparametric Statistics, 1 (1),...
  • Li, Z., Cui, L. and Chen, J. (2018). Traffc accident modelling via self-exciting point processes. Reliability Engineering and System Safety,...
  • Linderman, S. W., and Adams, R. P. (2014). Discovering latent network structure in point process data. 31st International Conference on Machine...
  • Linderman, S. W., and Adams, R. P. (2015). Scalable Bayesian inference for excitatory point process networks. Computer Science. 1–16.
  • Linderman, S. W., Wang, Y., and Blei, D. M. (2017). Bayesian inference for latent Hawkes processes. Advances in Approximate Bayesian Inference...
  • Liu, Y., Yan, T. and Chen, H. (2018). Exploiting graph regularized multi-dimensional Hawkes processes for modeling events with spatio-temporal...
  • Malem-shinitski, N., Ojeda, C. and Opper, M. (2022). Variational Bayesian inference for non-linear Hawkes process with Gaussian process self-effects....
  • Markwick, D. (2020). Bayesian nonparametric Hawkes processes with applications. Doctoral thesis (Ph.D), UCL (University College London).
  • Marsan, D. and Lengliné, O. (2008). Extending earthquakes’ reach through cascading. Science, 319 (5866), 1076–1079.
  • Mavroforakis, C., Valera, I. and Rodriguez, M. G. (2017). Modeling the dynamics of online learning activity. In Proceedings of the 26th International...
  • Mei, H. and Eisner, J. (2017). The neural Hawkes process: A neurally self-modulating multivariate point process. Advances in Neural Information...
  • Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P. and Tita, G. E. (2011). Self-exciting point process modeling of crime....
  • Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics,...
  • Nichols, K., Trevino, E., Ikeda, N., Philo, D., Garcia, A. and Bowman, D. (2018). Interdependency amongst earthquake magnitudes in Southern...
  • Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE Transactions on Information Theory, 27 (1), 23–31.
  • Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of Computational and Graphical...
  • Ogata, Y. (1998). Space-Time Point-Process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50 (2),...
  • Okawa, M., Iwata, T., Tanaka, Y., Toda, H., Kurashima, T. and Kashima, H. (2021). Dynamic Hawkes processes for discovering time-evolving communities’...
  • Ozaki, T. (1979). Maximum likelihood estimation of hawkes’ self-exciting point processes. Annals of the Institute of Statistical Mathematics,...
  • Park, J., Chaffee, A. W., Harrigan, R. J. and Schoenberg, F. P. (2020). A non-parametric Hawkes model of the spread of Ebola in west Africa....
  • Polson, N. G., Scott, J. G. and Windle, J. (2013). Bayesian inference for logistic models using Pólya-Gamma latent variables. Journal of...
  • Porter, M. D. and White, G. (2010). Self-exciting hurdle models for terrorist activity. Annals of Applied Statistics, 4 (1), 106–124.
  • Rambaldi, M., Bacry, E. and Lillo, F. (2017). The role of volume in order book dynamics: a multivariate Hawkes process analysis. Quantitative...
  • Rasmussen, J. G. (2013). Bayesian inference for Hawkes processes. Methodology and Computing in Applied Probability, 15 (3), 623–642.
  • Rasmussen, J. G. (2018). Lecture Notes: Temporal point processes and the conditional intensity function. arXiv:1806.00221v1.
  • Reinhart, A. (2018). A review of self-exciting spatio-temporal point processes and their applications. Statistical Science, 33 (3), 299–318.
  • Schoenberg, F. P. (2016). A note on the consistent estimation of spatial-temporal point process parameters. Statistica Sinica, 26, 861-879.
  • Schoenberg, F. P., Gordon, J. S. and Harrigan, R. J. (2018). Analytic computation of nonparametric Marsan–Lengliné estimates for Hawkes...
  • Seonwoo, Y., Oh, A. and Park, S. (2018). Hierarchical Dirichlet Gaussian marked Hawkes process for narrative reconstruction in continuous...
  • Shang, J. and Sun, M. (2019). Geometric Hawkes processes with graph convolutional recurrent neural networks. 33rd AAAI Conference on Artifcial...
  • Sulem, D., Rivoirard, V. and Rousseau, J. (2021). Bayesian estimation of nonlinear Hawkes process. https://arxiv.org/abs/2103.17164v2
  • Titsias, M. K. (2009). Variational learning of inducing variables in sparse Gaussian processes. Artifcial intelligence and statistics, 5,...
  • Torrisi, G. L. (2016). Gaussian approximation of nonlinear Hawkes processes. Annals of Applied Probability, 26 (4), 2106–2140.
  • Torrisi, G. L. (2017). Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes....
  • Walder, C. J. and Bishop, A. N. (2017). Fast Bayesian intensity estimation for the permanental process. 34th International Conference on Machine...
  • White, G., Porter, M. D. and Mazerolle, L. (2012). Terrorism Risk, Resilience and Volatility: A Comparison of Terrorism Patterns in Three...
  • Xu, H. and Zha, H. (2017). A Dirichlet mixture model of Hawkes processes for event sequence clustering. Advances in Neural Information Processing...
  • Xu, Z., Tresp, V., Yu, K. and Kriegel, H.-P. (2006). Infnite hidden relational models. Proceedings of the Twenty-Second Conference on Uncertainty...
  • Yang, Y., Etesami, J., He, N. and Kiyavash, N. (2017). Online learning for multivariate Hawkes processes. Advances in Neural Information Processing...
  • Yang, Y., Etesami, J., He, N. and Kiyavash, N. (2018). Nonparametric Hawkes processes: online estimation and generalization bounds. (Nips...
  • Zhang, R., Walder, C., Rizoiu, M. A. and Xie, L. (2019). Effcient non-parametric Bayesian Hawkes processes. IJCAI International Joint Conference...
  • Zhang, Q., Lipani, A., Kirnap, O. and Yilmaz, E. (2020a). Self-attentive Hawkes process. 37th International Conference on Machine Learning,...
  • Zhang, R., Walder, C. and Rizoiu, M.-A. (2020b). Variational inference for sparse Gaussian process modulated Hawkes process. Proceedings of...
  • Zhou, F., Li, Z., Fan, X., Wang, Y., Sowmya, A. and Chen, F. (2020a). Effcient inference for nonparametric Hawkes processes using auxiliary...
  • Zhou, F., Li, Z., Fan, X., Wang, Y., Sowmya, A. and Chen, F. (2020b). Fast multiresolution segmentation for nonstationary Hawkes process using...
  • Zhou, F., Luo, S., Li, Z., Fan, X., Wang, Y., Sowmya, A. and Chen, F. (2021). Effcient EM-variational inference for nonparametric Hawkes process....
  • Zhou, K., Zha, H. and Song, L. (2013). Learning social infectivity in sparse low-rank networks using multi-dimensional Hawkes processes. Journal...
  • Zhu, L. (2015). Large deviations for Markovian nonlinear Hawkes processes. Annals of Applied Probability, 25 (2), 548–581.
  • Zhuang, J., Ogata, Y. and Vere-Jones, D. (2002). Stochastic declustering of space-time earthquake occurrences. Journal of the American Statistical...
  • Zhuang, J. and Mateu, J. (2019). A semiparametric spatiotemporal Hawkes-type point process model with periodic background for crime data....
  • Zuo, S., Jiang, H., Li, Z., Zhao, T. and Zha, H. (2020). Transformer Hawkes process. 37th International Conference on Machine Learning, ICML...