Hawkes-Knowledge: Exploring the Dynamics of Hawkes Processes

This repository provides an overview of papers on Hawkes processes, tailored primarily for researchers. Our aim is to categorize papers into different applications, provide links to official and unofficial code resources, and explore the general properties of Hawkes processes.

A Hawkes process falls under the umbrella term of temporal point processes, which generally model events occurring randomly over time. While the simplest form of these processes is arguably the homogeneous Poisson process, the Hawkes process distinguishes itself due to its main property of self-excitation. This unique feature enables the process to effectively incorporate the influence of past events, thereby giving it a memory of recent activities. This memory is captured in the conditional intensity function, which adjusts the expected rate of future events based on the occurrence of past events. Originally, this led to the process being referred to as a self-exciting point process.

If this sounds Greek to you, don't worry—we will provide you with some resources to learn more!

Currently, the repository is incomplete. For those with limited knowledge about stochastic processes and Hawkes processes, we highly recommend starting with Laub et al. [4] or Rizoiu et al. [5] to familiarize yourself with the basics before delving into the original papers [1, 2, 3].

Original Papers

• [1] Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1), 83-90.
• [2] Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. Journal of the Royal Statistical Society Series B: Statistical Methodology, 33(3), 438-443.
• [3] Hawkes, A. G., & Oakes, D. (1974). A cluster process representation of a self-exciting process. Journal of Applied Probability, 11(3), 493-503.

Fascinating insights into the origins of Hawkes processes:

Hawkes, A., & Chen, J. (2021). A personal history of Hawkes process. Proceedings of the Institute of Statistical Mathematics (統計数理), 69(2), 123-143.

Books & Book Chapters

• [4] Laub, P. J., Lee, Y., & Taimre, T. (2021). The elements of Hawkes processes. Springer Cham, Switzerland. [Code]
• [5] Rizoiu, M. A., Lee, Y., Mishra, S., & Xie, L. (2017). Hawkes processes for events in social media. In Frontiers of multimedia research (pp. 191-218).

For a general introduction into stochastic processes refer to:

• [6] Daley, D. J., & Vere-Jones, D. (2003). An introduction to the theory of point processes: Volume I: Elementary theory and methods. Springer New York, NY.
• [7] Daley, D. J., & Vere-Jones, D. (2008). An introduction to the theory of point processes: Volume II: General theory and structure. Springer New York, NY.

Literature Reviews

If you're searching for a reference and not interested in reading a book, here are a couple of literature reviews, with most focusing on a selected topic:

• Overview of Hawkes processes:
  • [8] Lima, R. (2023). Hawkes processes modeling, inference, and control: An overview. SIAM Review, 65(2), 331-374.
• Review with a focus on spatio-temporal processes:
  • [9] Reinhart, A. (2018). A review of self-exciting spatio-temporal point processes and their applications. Statistical Science, 33(3), 299-318.
• Reviews with a focus on financial applications:
  • [10] Bacry, E., Mastromatteo, I., & Muzy, J. F. (2015). Hawkes processes in finance. Market Microstructure and Liquidity, 1(01), 1550005.
  • [11] Hawkes, A. G. (2018). Hawkes processes and their applications to finance: A review. Quantitative Finance, 18(2), 193-198.
  • [12] Hawkes, A. G. (2022). Hawkes jump-diffusions and finance: A brief history and review. European Journal of Finance, 28(7), 627-641.
• Review on neural Hawkes processes:
  • [13] Shchur, O., T ̈urkmen, A. C., Januschowski, T., & G ̈unnemann, S. (2021). Neural temporal point processes: A review. In Proceedings of the 30th International Joint Conference on Artificial Intelligence (pp. 4585–4593).

Theoretical Properties

• [14] Oakes, D. (1975). The Markovian self-exciting process. Journal of Applied Probability, 12(1), 69-77.
• [15] Brémaud, P., & Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Annals of Probability, 24(3), 1563-1588.
• [16] Brémaud, P., Nappo, G., & Torrisi, G. L. (2002). Rate of convergence to equilibrium of marked Hawkes processes. Journal of Applied Probability, 39(1), 123-136.
• [17] Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. Journal of Applied Probability, 50(3), 760-771.
• [18] Jovanović, S., Hertz, J., & Rotter, S. (2015). Cumulants of Hawkes point processes. Physical Review E, 91(4), 042802.
• [19] Cui, L., Hawkes, A., & Yi, H. (2020). An elementary derivation of moments of Hawkes processes. Advances in Applied Probability, 52(1), 102-137.
• [20] Daw, A. (2024). Conditional uniformity and Hawkes processes. Mathematics of Operations Research, 49(1), 40-57.

Properties relevant for queueing theory:

• [21] Daw, A., & Pender, J. (2018). Queues driven by Hawkes processes. Stochastic Systems, 8(3), 192-229.
• [22] Koops, D. T., Saxena, M., Boxma, O. J., & Mandjes, M. (2018). Infinite-server queues with Hawkes input. Journal of Applied Probability, 55(3), 920-943.
• [23] Chen, X. (2021). Perfect sampling of Hawkes processes and queues with Hawkes arrivals. Stochastic Systems, 11(13), 264-283.

Libraries

As previously mentioned, the book by Laub et al. [4] contains code snippets, and there are code implementations available in the Python package hawkesbook, which serves as an accompanying library to the book. Additionally, notable libraries include tick, a Python library mainly focusing on the fast simulation and estimation of Hawkes processes, and the EasyTPP library, which serves as a benchmarking tool, particularly for "neural" temporal point processes.

• [24] Bacry, E., Bompaire, M., Deegan, P., Gaïffas, S., & Poulsen, S. V. (2018). tick: a Python library for statistical learning, with an emphasis on Hawkes processes and time-dependent models. Journal of Machine Learning Research, 18(214), 1-5. [Code]
• [25] Xue, S., Shi, X., Chu, Z., Wang, Y., Zhou, F., Hao, H., ... & Mei, H. (2024). EasyTPP: Towards open benchmarking temporal point processes. In International Conference on Learning Representations. [Code]

Simulation

• [26] Ogata, Y. (1981). On Lewis' simulation method for point processes. IEEE Transactions on Information Theory, 27(1), 23-31.
• [27] Dassios, A., & Zhao, H. (2013). Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. Probab., 18, 1-13

Statistical Inference

Fitting Hawkes processes is a notoriously difficult task. The selection of a statistical inference method should always depend on the amount of data at hand, the specific application domain, and the task to be accomplished. General statistical inference methods include the following:

• Maximimum-likelihood estimation (MLE)
  • [28] Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics, 30, 243-261
  • [29] Ozaki, T. (1979). Maximum likelihood estimation of Hawkes' self-exciting point processes. Annals of the Institute of Statistical Mathematics, 31, 145-155.
• Expectation maximization (EM)
  • [30] 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.
  • [31] Salehi, F., Trouleau, W., Grossglauser, M., & Thiran, P. (2019). Learning Hawkes processes from a handful of events. Advances in Neural Information Processing Systems, 32.
  • [32] Mark, M., & Weber, T. A. (2020). Robust identification of controlled Hawkes processes. Physical Review E, 101(4), 043305.
• Bayesian
  • [33] Rasmussen, J. G. (2013). Bayesian inference for Hawkes processes. Methodology and Computing in Applied Probability, 15, 623-642.
• Non-parametric:
  • [34] Lewis, E., & Mohler, G. (2011). A nonparametric EM algorithm for multiscale Hawkes processes. Technical Report.
  • [35] Bacry, E., & Muzy, J. F. (2016). First-and second-order statistics characterization of Hawkes processes and non-parametric estimation. IEEE Transactions on Information Theory, 62(4), 2184-2202.
  • [36] Kirchner, M. (2017). An estimation procedure for the Hawkes process. Quantitative Finance, 17(4), 571-595.
  • [37] Achab, M., Bacry, E., Gaïffas, S., Mastromatteo, I., & Muzy, J. F. (2017). Uncovering causality from multivariate Hawkes integrated cumulants. In Proceedings of the 34th International Conference on Machine Learning, PMLR 70:1-10. [Code]
  • [38] Donnet, S., Rivoirard, V., & Rousseau, J. (2020). Nonparametric Bayesian estimation for multivariate Hawkes processes. Annals of Statistics, 48(5), 2698-2727.
• Estimation from time-censored information
  • [39] Shlomovich, L., Cohen, E. A., Adams, N., & Patel, L. (2022). Parameter estimation of binned Hawkes processes. Journal of Computational and Graphical Statistics, 31(4), 990-1000. [Code]
  • [40] Cheysson, F., & Lang, G. (2022). Spectral estimation of Hawkes processes from count data. Annals of Statistics, 50(3), 1722-1746. [Code]
  • [41] Schneider, P. J., & Weber, T. A. (2023). Estimation of self-exciting point processes from time-censored data. Physical Review E, 108(1), 015303.

Neural Hawkes processes

There are several limitations to modeling temporal point processes for extremely high-dimensional data, such as large networks, where each dimension typically represents the activity of an individual, using parametric models following the general Hawkes process paradigm. Consequently, several scholars have demonstrated how to utilize various neural network architectures to overcome the computational challenges faced when adapting these processes.

• [42] Mei, H., & Eisner, J. M. (2017). The neural Hawkes process: A neurally self-modulating multivariate point process. Advances in Neural Information Processing Systems, 30.
• [43] Du, N., Dai, H., Trivedi, R., Upadhyay, U., Gomez-Rodriguez, M., & Song, L. (2016). Recurrent marked temporal point processes: Embedding event history to vector. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1555-1564). [Code]
• [44] Upadhyay, U., De, A., & Gomez Rodriguez, M. (2018). Deep reinforcement learning of marked temporal point processes. Advances in Neural Information Processing Systems, 31. [Code]
• [45] Zuo, S., Jiang, H., Li, Z., Zhao, T., & Zha, H. (2020). Transformer Hawkes process. In Proceedings of the 37th International Conference on Machine Learning, PMLR 119:11692-11702. [Code]
• [46] Zhang, Q., Lipani, A., Kirnap, O., & Yilmaz, E. (2020). Self-attentive Hawkes process. In Proceedings of the 37th International Conference on Machine Learning, PMLR 119:11183-11193. [Code]
• [47] Mei, H., Yang, C., & Eisner, J. (2022). Transformer embeddings of irregularly spaced events and their participants. In International Conference on Learning Representations. [Code]
• [48] Zhang, S., Zhou, C., Liu, Y. A., Zhang, P., Lin, X., & Ma, Z. M. (2024). Neural jump-diffusion temporal point processes. In Proceedings of the 41st International Conference on Machine Learning, PMLR 235:60541-60557.[Code]

Extensions of Hawkes processes

• Renewal Hawkes process
  • [49] Wheatley, S., Filimonov, V., & Sornette, D. (2016). The Hawkes process with renewal immigration & its estimation with an EM algorithm. Computational Statistics & Data Analysis, 94, 120-135.
  • [50] Chen, F., & Stindl, T. (2018). Direct likelihood evaluation for the renewal Hawkes process. Journal of Computational and Graphical Statistics, 27(1), 119-131.
• Discrete-time Hawkes process
  • [51] Browning, R., Sulem, D., Mengersen, K., Rivoirard, V., & Rousseau, J. (2021). Simple discrete-time self-exciting models can describe complex dynamic processes: A case study of COVID-19. PLOS One, 16(4), e0250015.
  • [52] Wang, H. (2022). Limit theorems for a discrete-time marked Hawkes process. Statistics & Probability Letters, 184, 109368.
  • [53] Wang, H. (2023). Large and moderate deviations for a discrete-time marked Hawkes process. Communications in Statistics-Theory and Methods, 52(17), 6037-6062.
• Recursive extension of Hawkes process
  • [54] Schoenberg, F. P., Hoffmann, M., & Harrigan, R. J. (2019). A recursive point process model for infectious diseases. Annals of the Institute of Statistical Mathematics, 71, 1271-1287.
• Quadratic Hawkes process
  • [55] Blanc, P., Donier, J., & Bouchaud, J. P. (2017). Quadratic Hawkes processes for financial prices. Quantitative Finance, 17(2), 171-188.
  • [56] Aubrun, C., Benzaquen, M., & Bouchaud, J. P. (2023). Multivariate quadratic Hawkes processes—part I: theoretical analysis. Quantitative Finance, 23(5), 741-758.
• Ephemerally Hawkes process
  • [57] Daw, A., & Pender, J. (2022). An ephemerally self-exciting point process. Advances in Applied Probability, 54(2), 340-403.

Applications

In the following, we aim to highlight the general applicability of Hawkes processes to a plethora of applications. The provided references represent just a hand-selected subset of publications. In the subdirectories, we strive to provide a more comprehensive overview of the significant contributions towards adopting Hawkes processes.

• Earthquake modeling:
  • [58] Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401), 9-27.
  • [59] Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50, 379-402.
• Social media:
  • [60] Zhao, Q., Erdogdu, M. A., He, H. Y., Rajaraman, A., & Leskovec, J. (2015). SEISMIC: A self-exciting point process model for predicting tweet popularity. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 1513-1522). [Code]
  • [61] Rizoiu, M. A., Xie, L., Sanner, S., Cebrian, M., Yu, H., & Van Hentenryck, P. (2017). Expecting to be hip: Hawkes intensity processes for social media popularity. In Proceedings of the 26th International Conference on World Wide Web (pp. 735-744).[Code]
  • [62] Schneider, P. J., & Rizoiu, M. A. (2023). The effectiveness of moderating harmful online content. Proceedings of the National Academy of Sciences, 120(34), e2307360120. [Code]
• Criminology:
  • [63] 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.
• Finance:
  • [64] Aït-Sahalia, Y., Cacho-Diaz, J., & Laeven, R. J. (2015). Modeling financial contagion using mutually exciting jump processes. Journal of Financial Economics, 117(3), 585-606.
  • [65] Rambaldi, M., Pennesi, P., & Lillo, F. (2015). Modeling foreign exchange market activity around macroeconomic news: Hawkes-process approach. Physical Review E, 91(1), 012819.
  • [66] Mark, M., Sila, J., & Weber, T. A. (2022). Quantifying endogeneity of cryptocurrency markets. European Journal of Finance, 28(7), 784-799.
• Epidemiology:
  • [67] Rizoiu, M. A., Mishra, S., Kong, Q., Carman, M., & Xie, L. (2018). SIR-Hawkes: Linking epidemic models and Hawkes processes to model diffusions in finite populations. In Proceedings of the 2018 World Wide Web Conference (pp. 419-428). [Code]
  • [68] Kim, M., Paini, D., & Jurdak, R. (2019). Modeling stochastic processes in disease spread across a heterogeneous social system. Proceedings of the National Academy of Sciences, 116(2), 401-406.
  • [69] Bertozzi, A. L., Franco, E., Mohler, G., Short, M. B., & Sledge, D. (2020). The challenges of modeling and forecasting the spread of COVID-19. Proceedings of the National Academy of Sciences, 117(29), 16732-16738. [Code]
• Cyber insurance:
  • [70] Bessy-Roland, Y., Boumezoued, A., & Hillairet, C. (2021). Multivariate Hawkes process for cyber insurance. Annals of Actuarial Science, 15(1), 14-39.
• Advertising:
  • [71] Xu, L., Duan, J. A., & Whinston, A. (2014). Path to purchase: A mutually exciting point process model for online advertising and conversion. Management Science, 60(6), 1392-1412.

PhD Theses on Hawkes processes

There have been a couple of entire PhD theses, or at least chapters, focused on Hawkes processes and their extensions.

• Liniger, T. (2009). Multivariate Hawkes processes [Doctoral dissertation, ETH Zurich (Switzerland)].
• Chehrazi, N. (2013). Identification and optimization of stochastic systems [Doctoral dissertation, Stanford University (USA)].
• Zhu, L. (2013). Nonlinear Hawkes processes [Doctoral dissertation, New York University (USA)].
• Kirchner, M. (2017). Perspectives on Hawkes processes [Doctoral dissertation, ETH Zurich (Switzerland)].
• Achab, M. (2017). Learning from sequences with point processes [Doctoral dissertation, Université Paris Saclay (France)].
• Stindl, T. (2019). Statistical inference for renewal Hawkes self-exciting point processes [Doctoral dissertation, UNSW Sydney (Australia)].
• Daw, A. M. (2020). Batches, bursts, and service systems [Doctoral dissertation, Cornell University (USA)].
• Trouleau, W. (2021). Learning self-exciting temporal point processes under noisy observations (Publication No. 7143). [Doctoral dissertation, EPFL (Switzerland)].
• Kong, Q. (2022). Linking epidemic models and self-exciting processes for online and offline event diffusions [Doctoral dissertation, Australian National University (Australia)].
• Mark, M. (2022). Self-exciting point processes: Identification and control (Publication No. 10991). [Doctoral dissertation, EPFL (Switzerland)].
• Browning, R. (2023). Bayesian approaches for modelling discrete-time self-exciting processes and their applications [Doctoral dissertation, Queensland University of Technology (Australia)].
• Kwan, J. (2023). Asymptotic analysis and ergodicity of the Hawkes process and its extensions [Doctoral dissertation, UNSW Sydney (Australia)].