Geodesic
Hawkes graphs
Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 1, pp. 163-193 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper introduces the Hawkes skeleton and the Hawkes graph. These objects summarize the branching structure of a multivariate Hawkes point process in a compact, yet meaningful way. We demonstrate how the graph-theoretic vocabulary (ancestor sets, parent sets, connectivity, walks, walk weights, etc.) is very convenient for the discussion of multivariate Hawkes processes. For example, we reformulate the classic eigenvalue-based subcriticality criterion of multitype branching processes in graph terms. Next to these more terminological contributions, we show how the graph view can be used for the specification and estimation of Hawkes models from large, multitype event streams. Based on earlier work, we give a nonparametric statistical procedure to estimate the Hawkes skeleton and the Hawkes graph from data. We show how the graph estimation can then be used for specifying and fitting parametric Hawkes models. Our estimation method avoids the a priori assumptions on the model from a straightforward MLE-approach and is numerically more flexible than the latter. Our method has two tuning parameters: one controlling numerical complexity, and the other controlling the sparseness of the estimated graph. A simulation study confirms that the presented procedure works as desired. We pay special attention to computational issues in the implementation. This makes our results applicable to high-dimensional event-stream data such as dozens of event streams and thousands of events per component.
Keywords: Hawkes processes, event streams, point process networks. 
@article{TVP_2017_62_1_a9,
     author = {P. Embrechts and M. Kirchner},
     title = {Hawkes graphs},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {163--193},
     year = {2017},
     volume = {62},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2017_62_1_a9/}
}
TY  - JOUR
AU  - P. Embrechts
AU  - M. Kirchner
TI  - Hawkes graphs
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2017
SP  - 163
EP  - 193
VL  - 62
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TVP_2017_62_1_a9/
LA  - en
ID  - TVP_2017_62_1_a9
ER  - 
%0 Journal Article
%A P. Embrechts
%A M. Kirchner
%T Hawkes graphs
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2017
%P 163-193
%V 62
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_2017_62_1_a9/
%G en
%F TVP_2017_62_1_a9
P. Embrechts; M. Kirchner. Hawkes graphs. Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 1, pp. 163-193. http://geodesic.mathdoc.fr/item/TVP_2017_62_1_a9/

[1] E. Bacry, S. Ga{ï}ffas, J. Muzy, A generalization error bound for sparse and low-rank multivariate Hawkes processes, arXiv: 1501.00725

[2] D. Bates, M. Maechler, Matrix: sparse and dense matrix classes and methods, 2015 http://CRAN.R-project.org/package=Matrix

[3] G. Csárdi, T. Nepusz, “The igraph software package for complex network research”, InterJournal Complex Systems, 1695 (2006)

[4] D. J. Daley, D. Vere-Jones, An introduction to the theory of point processes, v. I, Probab. Appl. (N. Y.), Elementary theory and methods, 2nd ed., Springer, New York, 2003, xxii+469 pp. ; v. II, General theory and structure, 2008, xviii+573 pp. | DOI | MR | Zbl | DOI | MR | Zbl

[5] S. Delattre, N. Fournier, M. Hoffmann, “Hawkes processes on large networks”, Ann. Appl. Probab., 26:1 (2015), 216–261 | DOI | MR | Zbl

[6] A. Gunawardana, C. Meek, Puyang Xu, A model for temporal dependencies in event streams, Microsoft Research, 2011 https://www.microsoft.com/en-us/research/publication/a-model-for-temporal-dependencies-in-event-streams/

[7] P. Haccou, P. Jagers, V. Vatutin, Branching processes: variation, growth and extinction of populations, Cambridge Stud. Adapt. Dyn., 5, Cambridge Univ. Press, Cambridge, 2005, xii+316 pp. | DOI | MR | Zbl

[8] E. C. Hall, R. M. Willett, “Tracking dynamic point processes on networks”, IEEE Trans. Inform. Theory, 62:7 (2016), 4327–4346 | DOI | MR

[9] A. G. Hawkes, “Point spectra of some mutually exciting point processes”, J. Roy. Statist. Soc. Ser. B, 33:3 (1971), 438–443 | MR | Zbl

[10] A. G. Hawkes, “Spectra of some self-exciting and mutually exciting point processes”, Biometrika, 58 (1971), 83–90 | DOI | MR | Zbl

[11] A. G. Hawkes, “A cluster process representation of a self-exciting process”, J. Appl. Probab., 11:3 (1974), 493–503 | DOI | MR | Zbl

[12] T. Hawkes, “On the class of Sylow tower groups”, Math. Z., 105:5 (1968), 393–398 | DOI | MR | Zbl

[13] M. Kirchner, “An estimation procedure for the Hawkes process”, Quant. Finance, 1–25 (to appear) | DOI

[14] M. Kirchner, “Hawkes and INAR($\infty$) processes”, Stochastic Process. Appl., 126:8 (2016), 2494–2525 | DOI | MR | Zbl

[15] T. Liniger, Multivariate Hawkes processes, PhD thesis, ETH, Zurich, 2009

[16] C. Meek, Toward learning graphical and causal process models, Microsoft Research, 2014

[17] Y. Ogata, “Statistical models for earthquake occurences and residual analysis for point processes”, J. Amer. Statist. Assoc., 83:401 (1988), 9–27 | DOI

[18] J. Pearl, Causality. Models, reasoning, and inference, 2nd ed., Cambridge Univ. Press, Cambridge, 2009, xx+464 pp. | DOI | MR | Zbl

[19] Zhan Shi, Branching random walks (École d'Été de Probabilités de Saint-Flour, 2012), Lecture Notes in Math., 2151, Springer, Cham, 2015, x+133 pp. | DOI | MR | Zbl

[20] Ke Zhou, Hongyuan Zha, Le Song, “Learning social infectivity in sparse low-rank networks using multi-dimensional Hawkes processes”, Proceedings of the 16th International Conference on Artificial Intelligence and Statistics (AISTATS), JMLR W, 31, 2013, 641–649

[21] C. Watson, The geometric series of a matrix, 2015, 5 pp. http://www.math.uvic.ca/~dcwatson/work/geometric.pdf