Hawkes processes

December 22, 2019 — December 22, 2019

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

0.1 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

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

1 Tools

2 References

Achab, Bacry, Gaïffas, et al. 2017. Uncovering Causality from Multivariate Hawkes Integrated Cumulants.” In PMLR.
Adcock, Hansen, Roman, et al. 2014. Generalized Sampling: Stable Reconstructions, Inverse Problems and Compressed Sensing over the Continuum.” In Advances in Imaging and Electron Physics.
Bacry, E., Dayri, and Muzy. 2012. Non-Parametric Kernel Estimation for Symmetric Hawkes Processes. Application to High Frequency Financial Data.” The European Physical Journal B.
Bacry, E., Delattre, Hoffmann, et al. 2013a. Modelling Microstructure Noise with Mutually Exciting Point Processes.” Quantitative Finance.
———, et al. 2013b. Some 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,.
Bacry, Emmanuel, Jaisson, and Muzy. 2014. Estimation of Slowly Decreasing Hawkes Kernels: Application to High Frequency Order Book Modelling.” arXiv:1412.7096 [q-Fin, Stat].
Bacry, Emmanuel, and Muzy. 2014. Hawkes Model for Price and Trades High-Frequency Dynamics.” Quantitative Finance.
———. 2016. First- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation.” IEEE Transactions on Information Theory.
Bordenave, and Torrisi. 2007. Large Deviations of Poisson Cluster Processes.” Stochastic Models.
Bowsher. 2007. Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models.” Journal of Econometrics.
Brémaud, Pierre, and Massoulié. 2001. Hawkes Branching Point Processes Without Ancestors.” Journal of Applied Probability.
Brémaud, P., and Massoulié. 2002. Power Spectra of General Shot Noises and Hawkes Point Processes with a Random Excitation.” Advances in Applied Probability.
Brémaud, Pierre, Massoulié, and Ridolfi. 2005. Power Spectra of Random Spike Fields and Related Processes.” Advances in Applied Probability.
Chen, and Hall. 2013. Inference for a Nonstationary Self-Exciting Point Process with an Application in Ultra-High Frequency Financial Data Modeling.” Journal of Applied Probability.
Chen, and Stindl. 2017. Direct Likelihood Evaluation for the Renewal Hawkes Process.” Journal of Computational and Graphical Statistics.
Dassios, and Zhao. 2013. Exact Simulation of Hawkes Process with Exponentially Decaying Intensity.” Electronic Communications in Probability.
Du, Farajtabar, Ahmed, et al. 2015. Dirichlet-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. KDD ’15.
Eichler, Dahlhaus, and Dueck. 2016. Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions.” Journal of Time Series Analysis.
Embrechts, Liniger, and Lin. 2011. Multivariate Hawkes Processes: An Application to Financial Data.” Journal of Applied Probability.
Fan, Li, Zhou, et al. 2021. Continuous-Time Edge Modelling Using Non-Parametric Point Processes.” In.
Filimonov, Bicchetti, Maystre, et al. 2014. Quantification 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,.
Filimonov, and Sornette. 2012. Quantifying Reflexivity in Financial Markets: Toward a Prediction of Flash Crashes.” Physical Review E.
———. 2013. Apparent Criticality and Calibration Issues in the Hawkes Self-Excited Point Process Model: Application to High-Frequency Financial Data.” SSRN Scholarly Paper ID 2371284.
Filimonov, Wheatley, and Sornette. 2015. Effective 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.
García. 2002. A Brief Walk Through Sampling Theory.” In Advances in Imaging and Electron Physics.
Godoy, Solo, Min, et al. 2016. Local Likelihood Estimation of Time-Variant Hawkes Models.” In.
Halpin. 2012. An EM Algorithm for Hawkes Process.” Psychometrika.
Halpin, and Boeck. 2013. Modelling Dyadic Interaction with Hawkes Processes.” Psychometrika.
Hansen, Reynaud-Bouret, and Rivoirard. 2015. Lasso and Probabilistic Inequalities for Multivariate Point Processes.” Bernoulli.
Hardiman, Bercot, and Bouchaud. 2013. Critical Reflexivity in Financial Markets: A Hawkes Process Analysis.” The European Physical Journal B.
Hardiman, and Bouchaud. 2014. Branching-Ratio Approximation for the Self-Exciting Hawkes Process.” Physical Review E.
Hawkes. 1971a. Point Spectra of Some Mutually Exciting Point Processes.” Journal of the Royal Statistical Society. Series B (Methodological).
———. 1971b. Spectra of Some Self-Exciting and Mutually Exciting Point Processes.” Biometrika.
Hawkes, and Oakes. 1974. A Cluster Process Representation of a Self-Exciting Process.” Journal of Applied Probability.
Jaisson, and Rosenbaum. 2015. Limit Theorems for Nearly Unstable Hawkes Processes.” The Annals of Applied Probability.
Jovanović, Hertz, and Rotter. 2015. Cumulants of Hawkes Point Processes.” Physical Review E.
Karabash, and Zhu. 2012. Limit Theorems for Marked Hawkes Processes with Application to a Risk Model.” arXiv:1211.4039 [Math].
Kong, Rizoiu, and Xie. 2020. Modeling Information Cascades with Self-Exciting Processes via Generalized Epidemic Models.” In Proceedings of the 13th International Conference on Web Search and Data Mining. WSDM ’20.
Kwieciński, and Szekli. 1996. Some Monotonicity and Dependence Properties of Self-Exciting Point Processes.” The Annals of Applied Probability.
Laub, Taimre, and Pollett. 2015. Hawkes Processes.” arXiv:1507.02822 [Math, q-Fin, Stat].
Lewis, and Mohler. 2011. A Nonparametric EM Algorithm for Multiscale Hawkes Processes.” Preprint.
Liniger. 2009. Multivariate Hawkes Processes.”
Li, and Zha. 2014. Learning Parametric Models for Social Infectivity in Multi-Dimensional Hawkes Processes.” In Twenty-Eighth AAAI Conference on Artificial Intelligence.
Mishra, Rizoiu, and Xie. 2016. Feature Driven and Point Process Approaches for Popularity Prediction.” In Proceedings of the 25th ACM International Conference on Information and Knowledge Management. CIKM ’16.
Mitchell, and Cates. 2010. Hawkes Process as a Model of Social Interactions: A View on Video Dynamics.” Journal of Physics A: Mathematical and Theoretical.
Mohler. 2013. Modeling and Estimation of Multi-Source Clustering in Crime and Security Data.” The Annals of Applied Statistics.
Møller, and Rasmussen. 2005. Perfect Simulation of Hawkes Processes.” Advances in Applied Probability.
———. 2006. Approximate Simulation of Hawkes Processes.” Methodology and Computing in Applied Probability.
Møller, and Torrisi. 2007. The Pair Correlation Function of Spatial Hawkes Processes.” Statistics & Probability Letters.
Ozaki. 1979. Maximum Likelihood Estimation of Hawkes’ Self-Exciting Point Processes.” Annals of the Institute of Statistical Mathematics.
Pinto, and Chahed. 2014. Modeling 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. SITIS ’14.
Rambaldi, Pennesi, and Lillo. 2015. Modeling FX Market Activity Around Macroeconomic News: A Hawkes Process Approach.” Physical Review E.
Rasmussen. 2013. Bayesian Inference for Hawkes Processes.” Methodology and Computing in Applied Probability.
Reynaud-Bouret, and Roy. 2007. “Some Non Asymptotic Tail Estimates for Hawkes Processes.” Bulletin of the Belgian Mathematical Society - Simon Stevin.
Reynaud-Bouret, and Schbath. 2010. Adaptive Estimation for Hawkes Processes; Application to Genome Analysis.” The Annals of Statistics.
Rizoiu, Mishra, Kong, et al. 2018. SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations.” In Proceedings of the 2018 World Wide Web Conference.
Rizoiu, Xie, Sanner, et al. 2017. Expecting to Be HIP: Hawkes Intensity Processes for Social Media Popularity.” In World Wide Web 2017, International Conference on. WWW ’17.
Saichev, and Sornette. 2011a. Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.1611 [Cond-Mat, Physics:physics].
———. 2011b. Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes.” arXiv:1101.5564 [Cond-Mat, Physics:physics].
Wang, Xie, Du, et al. 2016. Isotonic Hawkes Processes.” In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48. ICML’16.
Wheatley. 2013. “Quantifying Endogeneity in Market Prices with Point Processes: Methods & Applications.” Masters Thesis.
Wheatley, Filimonov, and Sornette. 2016. “The Hawkes Process with Renewal Immigration & Its Estimation with an EM Algorithm.” Comput. Stat. Data Anal.
Zadeh, Amir, and Sharda. 2013. A Point Process Framework for Predicting Dynamics of Popularity of Content in Online Social Networks.” SSRN Scholarly Paper ID 2331565.
Zhou, Zha, and Song. 2013. Learning Triggering Kernels for Multi-Dimensional Hawkes Processes.” In Proceedings of the 30th International Conference on Machine Learning (ICML-13).
Zhu. 2013. Moderate Deviations for Hawkes Processes.” Statistics & Probability Letters.