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In probability theory and statistics, a Hawkes process is an age-dependent branching process driven by immigration from an inhomogeneous Poisson process. The process, named after Alan G. Hawkes, is also called a self-exciting point process.^[1]. It has arrivals at times ${\textstyle 0<t_{1}<t_{2}<t_{3}<\cdots }$ where the infinitesimal probability of an arrival during the time interval ${\textstyle [t,t+dt)}$ is

\lambda _{t}\,dt=\left(\mu (t)+\sum _{t_{k}\,:\,t_{k}\,<\,t}\phi (t-t_{k})\right)\,dt.

The function ${\textstyle \mu }$ is the intensity of an underlying Poisson process. The first arrival occurs at time ${\textstyle t_{1}}$ and immediately after that, the intensity becomes ${\textstyle \mu (t)+\phi (t-t_{1})}$ , and at the time ${\textstyle t_{2}}$ of the second arrival the intensity jumps to ${\textstyle \mu (t)+\phi (t-t_{1})+\phi (t-t_{2})}$ and so on.^[2]

During the time interval ${\textstyle (t_{k},t_{k+1})}$ , the process is the sum of ${\textstyle k+1}$ independent processes with intensities ${\textstyle \mu (t),\phi (t-t_{1}),\ldots ,\phi (t-t_{k}).}$ The arrivals in the process whose intensity is ${\textstyle \phi (t-t_{k})}$ are the “daughters” of the arrival at time ${\textstyle t_{k}.}$ The integral $\int _{0}^{\infty }\phi (t)\,dt$ is the average number of daughters of each arrival and is called the branching ratio. Thus viewing some arrivals as descendants of earlier arrivals, we have a Galton–Watson branching process. The number of such descendants is finite with probability 1 if branching ratio is 1 or less. If the branching ratio is more than 1, then each arrival has positive probability of having infinitely many descendants.

Multivariate extension

In its seminal paper^[2], Hawkes also considered mutually exciting processes which are now called multivariate Hawkes processes. The ${\textstyle n}$ point processes have arrival times denoted by ${\textstyle 0<t_{1}^{i}<t_{2}^{i}<t_{3}^{i}<\cdots }$ for each type ${\textstyle i=1,\dots ,n}$ . The probability of shared points is null, i.e. ${\textstyle \mathbb {P} (\exists (i,k)\neq (j,\ell ):t_{k}^{i}=t_{\ell }^{j})=0}$ , and for each ${\textstyle i}$ , the infinitesimal probability of an arrival of type ${\textstyle i}$ during the time interval ${\textstyle [t,t+dt)}$ is

\lambda _{t}^{i}\,dt=\left(\mu _{i}(t)+\sum _{j=1}^{n}\sum _{t_{k}^{j}\,:\,t_{k}^{j}\,<\,t}\phi _{ij}(t-t_{k}^{j})\right)\,dt.

The interaction is now described by a matrix of functions ${\textstyle \mathbf {\Phi } =(\phi _{ij})_{1\leq i,j\leq n}}$ and the matrix integral $\int _{0}^{\infty }\mathbf {\Phi } (t)\,dt$ plays the role of branching ratio.

Here is an animation to visualize the infinitesimal probabilities $\lambda _{t}^{i}$ and associated events $t_{k}^{i}$ in a bivariate framework.

Nonlinear extension

Brémaud and Massoulié extended Hawkes processes in a nonlinear way. They considered an infinitesimal probability of arrival of the following form:

\lambda _{t}\,dt=f\left(\mu (t)+\sum _{t_{k}\,:\,t_{k}\,<\,t}\phi (t-t_{k})\right)\,dt,

for some nonlinear function $f$ .^[3] The original (linear) case thus corresponds to $f(x)=x$ . Of course, both nonlinear and multivariate extensions are possible at the same time.

Contrarily to the linear case, the nonlinear extension allows for negative functions $\phi$ . Hence it can model self-inhibition or mutual inhibition (in a multivariate context) which is essential in neuroscience applications for instance.^[4]

Applications

Hawkes processes are used for statistical modeling of events in mathematical finance,^[5] epidemiology,^[6] earthquake seismology,^[7] and other fields in which a random event exhibits self-exciting behavior.^[8]^[9]

References

^ Laub, Patrick J.; Lee, Young; Taimre, Thomas (2021). The Elements of Hawkes Processes. doi:10.1007/978-3-030-84639-8. ISBN 978-3-030-84638-1. S2CID 245682002.
^ ^a ^b Hawkes, Alan G. (1971). “Spectra of some self-exciting and mutually exciting point processes”. Biometrika. 58 (1): 83–90. doi:10.1093/biomet/58.1.83. ISSN 0006-3444.
^ Brémaud, Pierre; Massoulié, Laurent (1 July 1996). “Stability of nonlinear Hawkes processes”. The Annals of Probability. 24 (3). doi:10.1214/aop/1065725193.
^ Reynaud-Bouret, Patricia; Rivoirard, Vincent; Tuleau-Malot, Christine (December 2013). “Inference of functional connectivity in Neurosciences via Hawkes processes”. 2013 IEEE Global Conference on Signal and Information Processing: 317–320. doi:10.1109/GlobalSIP.2013.6736879.
^ Hawkes, Alan G. (2018). “Hawkes processes and their applications to finance: a review”. Quantitative Finance. 18 (2): 193–198. doi:10.1080/14697688.2017.1403131. ISSN 1469-7688. S2CID 158619662.
^ Rizoiu, Marian-Andrei; Mishra, Swapnil; Kong, Quyu; Carman, Mark; Xie, Lexing (2018). “SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations”. Proceedings of the 2018 World Wide Web Conference on World Wide Web – WWW ’18. pp. 419–428. arXiv:1711.01679. doi:10.1145/3178876.3186108. S2CID 195346881.
^ Kwon, Junhyeon; Zheng, Yingcai; Jun, Mikyoung (2023). “Flexible spatio-temporal Hawkes process models for earthquake occurrences”. Spatial Statistics. 54 100728. arXiv:2210.08053. Bibcode:2023SpaSt..5400728K. doi:10.1016/j.spasta.2023.100728. S2CID 252917746.
^ Tench, Stephen; Fry, Hannah; Gill, Paul (2016). “Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army”. European Journal of Applied Mathematics. 27 (3): 377–402. doi:10.1017/S0956792515000686. ISSN 0956-7925. S2CID 53692006.
^ Laub, Patrick J.; Taimre, Thomas; Pollett, Philip K. (2015). “Hawkes Processes”. arXiv:1507.02822 [math.PR].

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Multivariate extension

Nonlinear extension

Applications

See also

References

Further reading