Hawkes processes



An intersection of point processes and branching processes is the Hawkes process. The classic is the univariate linear Hawkes process. For now we’ll assume it to be indexed by time.

Recall the log likelihood of a generic point process, with occurrence times \(\{t_i\}.\)

\[ \begin{aligned} L_\theta(t_{1:N}) &:= -\int_0^T\lambda^*_\theta(t)dt + \int_0^T\log \lambda^*_\theta(t) dN_t\\ &= -\int_0^T\lambda^*_\theta(t)dt + \sum_{j} \log \lambda^*_\theta(t_j) \end{aligned} \]

\(\lambda^*(t)\) is shorthand for \(\lambda^*(t|\mathcal{F}_t)\), and we call this the intensity. This term is what distinguishes various point processes. For the Hawkes process is in particular we have

\[ \lambda^*(t) = \mu + \int_{-\infty}^t \eta\phi(t-s)dNs. \] where \(\phi_\kappa(t)\) is the influence kernel with parameter \(\kappa\), \(\eta\) the branching ratio, and

Time-inhomogeneous extension

Partial notes to an extension that I have looked at. Introduce an additional convolution kernel \(\psi\), and functions of the form

\[ \mu(t) = \mu + \sum_{1 \leq j \leq p}\omega_i\psi_{\nu_j}(t-t_j) \]

for some set of kernel bandwidths \(\{\nu_j\}_{1 \leq j \leq p}\), kernel weights \(\{\omega_{\nu_j}\}_{1 \leq j \leq p}\), kernel locations \(\{\tau_j\}_{1 \leq j \leq p}\).

There are many kernels available. We start with the top hat kernel, the piecewise-constant function.

\[ \psi_{\nu}(t):= \frac{\mathbb{I}_{0< t \leq \nu}}{\nu} \]

giving the following background intensity

\[ \mu(t) = \mu + \sum_{1\leq j\le p}\omega_j\frac{\mathbb{I}_{(0, \nu_j]}(t-\tau_j)}{\nu_j}. \]

I augment the parameter vector to include the kernel weights \(\theta':=( \mu,\eta,\kappa, \boldsymbol\omega).\) We could also try to infer kernel locations and bandwidths.

The hypothesized generating model now has conditional intensity process

\[ \lambda_{\theta'}(t|\mathcal{F}_t) = \mu + \sum_{j=2}^n \omega_j \mathbb{I}_{[\tau_{j-1},\tau_j)}(t) + \eta \sum_{t_i< t}\phi_\kappa(t-t_i). \]

References

Achab, Massil, Emmanuel Bacry, StΓ©phane GaΓ―ffas, Iacopo Mastromatteo, and Jean-Francois Muzy. 2017. β€œUncovering Causality from Multivariate Hawkes Integrated Cumulants.” In PMLR.
Adcock, Ben, Anders Hansen, Bogdan Roman, and Gerd Teschke. 2014. β€œGeneralized Sampling: Stable Reconstructions, Inverse Problems and Compressed Sensing over the Continuum.” In Advances in Imaging and Electron Physics, edited by Peter W. Hawkes, 182:187–279. Elsevier.
Bacry, E., K. Dayri, and J. F. Muzy. 2012. β€œNon-Parametric Kernel Estimation for Symmetric Hawkes Processes. Application to High Frequency Financial Data.” The European Physical Journal B 85 (5): 1–12.
Bacry, E., S. Delattre, M. Hoffmann, and J. F. Muzy. 2013a. β€œModelling Microstructure Noise with Mutually Exciting Point Processes.” Quantitative Finance 13 (1): 65–77.
β€”β€”β€”. 2013b. β€œSome Limit Theorems for Hawkes Processes and Application to Financial Statistics.” Stochastic Processes and Their Applications, A Special Issue on the Occasion of the 2013 International Year of Statistics, 123 (7): 2475–99.
Bacry, Emmanuel, Thibault Jaisson, and Jean-Francois Muzy. 2014. β€œEstimation of Slowly Decreasing Hawkes Kernels: Application to High Frequency Order Book Modelling.” arXiv:1412.7096 [q-Fin, Stat], December.
Bacry, Emmanuel, and Jean-FranΓ§ois Muzy. 2014. β€œHawkes Model for Price and Trades High-Frequency Dynamics.” Quantitative Finance 14 (7): 1147–66.
β€”β€”β€”. 2016. β€œFirst- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation.” IEEE Transactions on Information Theory 62 (4): 2184–2202.
Bordenave, Charles, and Giovanni Luca Torrisi. 2007. β€œLarge Deviations of Poisson Cluster Processes.” Stochastic Models 23 (4): 593–625.
Bowsher, Clive G. 2007. β€œModelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models.” Journal of Econometrics 141 (2): 876–912.
BrΓ©maud, Pierre, and Laurent MassouliΓ©. 2001. β€œHawkes Branching Point Processes Without Ancestors.” Journal of Applied Probability 38 (1): 122–35.
BrΓ©maud, Pierre, Laurent MassouliΓ©, and Andrea Ridolfi. 2005. β€œPower Spectra of Random Spike Fields and Related Processes.” Advances in Applied Probability 37 (4): 1116–46.
BrΓ©maud, P., and L. MassouliΓ©. 2002. β€œPower Spectra of General Shot Noises and Hawkes Point Processes with a Random Excitation.” Advances in Applied Probability 34 (1): 205–22.
Chen, Feng, and Peter Hall. 2013. β€œInference for a Nonstationary Self-Exciting Point Process with an Application in Ultra-High Frequency Financial Data Modeling.” Journal of Applied Probability 50 (4): 1006–24.
Chen, Feng, and Tom Stindl. 2017. β€œDirect Likelihood Evaluation for the Renewal Hawkes Process.” Journal of Computational and Graphical Statistics 27 (1): 1–13.
Dassios, Angelos, and Hongbiao Zhao. 2013. β€œExact Simulation of Hawkes Process with Exponentially Decaying Intensity.” Electronic Communications in Probability 18 (62).
Du, Nan, Mehrdad Farajtabar, Amr Ahmed, Alexander J. Smola, and Le Song. 2015. β€œDirichlet-Hawkes Processes with Applications to Clustering Continuous-Time Document Streams.” In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 219–28. KDD ’15. New York, NY, USA: ACM.
Eichler, Michael, Rainer Dahlhaus, and Johannes Dueck. 2016. β€œGraphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions.” Journal of Time Series Analysis, January, n/a–.
Embrechts, Paul, Thomas Liniger, and Lu Lin. 2011. β€œMultivariate Hawkes Processes: An Application to Financial Data.” Journal of Applied Probability 48A (August): 367–78.
Fan, Xuhui, Bin Li, Feng Zhou, and Scott A. Sisson. 2021. β€œContinuous-Time Edge Modelling Using Non-Parametric Point Processes.” In.
Filimonov, Vladimir, David Bicchetti, Nicolas Maystre, and Didier Sornette. 2014. β€œQuantification of the High Level of Endogeneity and of Structural Regime Shifts in Commodity Markets.” Journal of International Money and Finance, Understanding International Commodity Price Fluctuations, 42 (April): 174–92.
Filimonov, Vladimir, and Didier Sornette. 2012. β€œQuantifying Reflexivity in Financial Markets: Toward a Prediction of Flash Crashes.” Physical Review E 85 (5): 056108.
β€”β€”β€”. 2013. β€œApparent Criticality and Calibration Issues in the Hawkes Self-Excited Point Process Model: Application to High-Frequency Financial Data.” SSRN Scholarly Paper ID 2371284. Rochester, NY: Social Science Research Network.
Filimonov, Vladimir, Spencer Wheatley, and Didier Sornette. 2015. β€œEffective Measure of Endogeneity for the Autoregressive Conditional Duration Point Processes via Mapping to the Self-Excited Hawkes Process.” Communications in Nonlinear Science and Numerical Simulation 22 (1–3): 23–37.
GarcΓ­a, Antonio G. 2002. β€œA Brief Walk Through Sampling Theory.” In Advances in Imaging and Electron Physics, edited by Peter W. Hawkes, 124:63–137. Elsevier.
Godoy, Boris I., Victor Solo, Jason Min, and Syed Ahmed Pasha. 2016. β€œLocal Likelihood Estimation of Time-Variant Hawkes Models.” In, 4199–4203. IEEE.
Halpin, Peter F. 2012. β€œAn EM Algorithm for Hawkes Process.” Psychometrika 2.
Halpin, Peter F., and Paul De Boeck. 2013. β€œModelling Dyadic Interaction with Hawkes Processes.” Psychometrika 78 (4): 793–814.
Hansen, Niels Richard, Patricia Reynaud-Bouret, and Vincent Rivoirard. 2015. β€œLasso and Probabilistic Inequalities for Multivariate Point Processes.” Bernoulli 21 (1): 83–143.
Hardiman, Stephen J., Nicolas Bercot, and Jean-Philippe Bouchaud. 2013. β€œCritical Reflexivity in Financial Markets: A Hawkes Process Analysis.” The European Physical Journal B 86 (10): 1–9.
Hardiman, Stephen J., and Jean-Philippe Bouchaud. 2014. β€œBranching-Ratio Approximation for the Self-Exciting Hawkes Process.” Physical Review E 90 (6): 062807.
Hawkes, Alan G. 1971a. β€œPoint Spectra of Some Mutually Exciting Point Processes.” Journal of the Royal Statistical Society. Series B (Methodological) 33 (3): 438–43.
β€”β€”β€”. 1971b. β€œSpectra of Some Self-Exciting and Mutually Exciting Point Processes.” Biometrika 58 (1): 83–90.
Hawkes, Alan G., and David Oakes. 1974. β€œA Cluster Process Representation of a Self-Exciting Process.” Journal of Applied Probability 11 (3): 493.
Jaisson, Thibault, and Mathieu Rosenbaum. 2015. β€œLimit Theorems for Nearly Unstable Hawkes Processes.” The Annals of Applied Probability 25 (2): 600–631.
JovanoviΔ‡, Stojan, John Hertz, and Stefan Rotter. 2015. β€œCumulants of Hawkes Point Processes.” Physical Review E 91 (4): 042802.
Karabash, Dmytro, and Lingjiong Zhu. 2012. β€œLimit Theorems for Marked Hawkes Processes with Application to a Risk Model.” arXiv:1211.4039 [Math], November.
Kong, Quyu, Marian-Andrei Rizoiu, and Lexing Xie. 2020. β€œModeling Information Cascades with Self-Exciting Processes via Generalized Epidemic Models.” In Proceedings of the 13th International Conference on Web Search and Data Mining, 286–94. WSDM ’20. New York, NY, USA: Association for Computing Machinery.
KwieciΕ„ski, Andrzej, and Ryszard Szekli. 1996. β€œSome Monotonicity and Dependence Properties of Self-Exciting Point Processes.” The Annals of Applied Probability 6 (4): 1211–31.
Laub, Patrick J., Thomas Taimre, and Philip K. Pollett. 2015. β€œHawkes Processes.” arXiv:1507.02822 [Math, q-Fin, Stat], July.
Lewis, Erik, and George Mohler. 2011. β€œA Nonparametric EM Algorithm for Multiscale Hawkes Processes.” Preprint.
Li, Liangda, and Hongyuan Zha. 2014. β€œLearning Parametric Models for Social Infectivity in Multi-Dimensional Hawkes Processes.” In Twenty-Eighth AAAI Conference on Artificial Intelligence.
Liniger, Thomas Josef. 2009. β€œMultivariate Hawkes Processes.” Diss., EidgenΓΆssische Technische Hochschule ETH ZΓΌrich, Nr. 18403, 2009.
Mishra, Swapnil, Marian-Andrei Rizoiu, and Lexing Xie. 2016. β€œFeature Driven and Point Process Approaches for Popularity Prediction.” In Proceedings of the 25th ACM International Conference on Information and Knowledge Management, 1069–78. CIKM ’16. New York, NY, USA: ACM.
Mitchell, Lawrence, and Michael E. Cates. 2010. β€œHawkes Process as a Model of Social Interactions: A View on Video Dynamics.” Journal of Physics A: Mathematical and Theoretical 43 (4): 045101.
Mohler, George. 2013. β€œModeling and Estimation of Multi-Source Clustering in Crime and Security Data.” The Annals of Applied Statistics 7 (3): 1525–39.
MΓΈller, Jesper, and Jakob G. Rasmussen. 2005. β€œPerfect Simulation of Hawkes Processes.” Advances in Applied Probability, 629–46.
β€”β€”β€”. 2006. β€œApproximate Simulation of Hawkes Processes.” Methodology and Computing in Applied Probability 8 (1): 53–64.
MΓΈller, Jesper, and Giovanni Luca Torrisi. 2007. β€œThe Pair Correlation Function of Spatial Hawkes Processes.” Statistics & Probability Letters 77 (10): 995–1003.
Ozaki, T. 1979. β€œMaximum Likelihood Estimation of Hawkes’ Self-Exciting Point Processes.” Annals of the Institute of Statistical Mathematics 31 (1): 145–55.
Pinto, Julio Cesar Louzada, and Tijani Chahed. 2014. β€œModeling Multi-Topic Information Diffusion in Social Networks Using Latent Dirichlet Allocation and Hawkes Processes.” In Proceedings of the 2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems, 339–46. SITIS ’14. Washington, DC, USA: IEEE Computer Society.
Rambaldi, Marcello, Paris Pennesi, and Fabrizio Lillo. 2015. β€œModeling FX Market Activity Around Macroeconomic News: A Hawkes Process Approach.” Physical Review E 91 (1): 012819.
Rasmussen, Jakob Gulddahl. 2013. β€œBayesian Inference for Hawkes Processes.” Methodology and Computing in Applied Probability 15 (3): 623–42.
Reynaud-Bouret, Patricia, and Emmanuel Roy. 2007. β€œSome Non Asymptotic Tail Estimates for Hawkes Processes.” Bulletin of the Belgian Mathematical Society - Simon Stevin 13 (5): 883–96.
Reynaud-Bouret, Patricia, and Sophie Schbath. 2010. β€œAdaptive Estimation for Hawkes Processes; Application to Genome Analysis.” The Annals of Statistics 38 (5): 2781–2822.
Rizoiu, Marian-Andrei, Swapnil Mishra, Quyu Kong, Mark Carman, and Lexing Xie. 2018. β€œSIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations.” In Proceedings of the 2018 World Wide Web Conference, 419–28. Republic and Canton of Geneva, CHE: International World Wide Web Conferences Steering Committee.
Rizoiu, Marian-Andrei, Lexing Xie, Scott Sanner, Manuel Cebrian, Honglin Yu, and Pascal Van Hentenryck. 2017. β€œExpecting to Be HIP: Hawkes Intensity Processes for Social Media Popularity.” In World Wide Web 2017, International Conference on, 1–9. WWW ’17. Perth, Australia: International World Wide Web Conferences Steering Committee.
Saichev, A., and D. Sornette. 2011a. β€œHierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.1611 [Cond-Mat, Physics:physics], January.
β€”β€”β€”. 2011b. β€œGenerating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.5564 [Cond-Mat, Physics:physics], January.
Wang, Yichen, Bo Xie, Nan Du, and Le Song. 2016. β€œIsotonic Hawkes Processes.” In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, 2226–34. ICML’16. New York, NY, USA: JMLR.org.
Wheatley, Spencer. 2013. β€œQuantifying Endogeneity in Market Prices with Point Processes: Methods & Applications.” Masters Thesis. ETH ZΓΌrich.
Wheatley, Spencer, Vladimir Filimonov, and Didier Sornette. 2016. β€œThe Hawkes Process with Renewal Immigration & Its Estimation with an EM Algorithm.” Comput. Stat. Data Anal. 94 (C): 120–35.
Zadeh, Hassan, Amir, and Ramesh Sharda. 2013. β€œA Point Process Framework for Predicting Dynamics of Popularity of Content in Online Social Networks.” SSRN Scholarly Paper ID 2331565. Rochester, NY: Social Science Research Network.
Zhou, Ke, Hongyuan Zha, and Le Song. 2013. β€œLearning Triggering Kernels for Multi-Dimensional Hawkes Processes.” In Proceedings of the 30th International Conference on Machine Learning (ICML-13), 1301–9.
Zhu, Lingjiong. 2013. β€œModerate Deviations for Hawkes Processes.” Statistics & Probability Letters 83 (3): 885–90.

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