Uncovering Causality from Multivariate Hawkes Integrated Cumulants

We design a new nonparametric method that allows one to estimate the matrix of integrated kernels of a multivariate Hawkes process. This matrix not only encodes the mutual influences of each nodes of the process, but also disentangles the causality relationships between them. Our approach is the first that leads to an estimation of this matrix without any parametric modeling and estimation of the kernels themselves. A consequence is that it can give an estimation of causality relationships between nodes (or users), based on their activity timestamps (on a social network for instance), without knowing or estimating the shape of the activities lifetime. For that purpose, we introduce a moment matching method that fits the third-order integrated cumulants of the process. We show on numerical experiments that our approach is indeed very robust to the shape of the kernels, and gives appealing results on the MemeTracker database.

[1]  E. Bacry,et al.  Hawkes Processes in Finance , 2015, 1502.04592.

[2]  Shuang-Hong Yang,et al.  Mixture of Mutually Exciting Processes for Viral Diffusion , 2013, ICML.

[3]  Stefan Rotter,et al.  Cumulants of Hawkes point processes. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Le Song,et al.  Learning Triggering Kernels for Multi-dimensional Hawkes Processes , 2013, ICML.

[5]  Shuang Li,et al.  COEVOLVE: A Joint Point Process Model for Information Diffusion and Network Co-evolution , 2015, NIPS.

[6]  R. Dahlhaus,et al.  Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions , 2016, 1605.06759.

[7]  A. Hawkes.  Point Spectra of Some Mutually Exciting Point Processes , 1971 .

[8]  Yann LeCun,et al.  The Loss Surfaces of Multilayer Networks , 2014, AISTATS.

[9]  E. Bacry,et al.  Estimation of slowly decreasing Hawkes kernels: Application to high frequency order book modelling , 2014, 1412.7096.

[10]  M. Eichler.  Graphical modelling of multivariate time series , 2006, math/0610654.

[11]  Y. Ogata.  Space-Time Point-Process Models for Earthquake Occurrences , 1998 .

[12]  Le Song,et al.  Learning Social Infectivity in Sparse Low-rank Networks Using Multi-dimensional Hawkes Processes , 2013, AISTATS.

[13]  Emmanuel Bacry,et al.  Second order statistics characterization of Hawkes processes and non-parametric estimation , 2014, 1401.0903.

[14]  Jeffrey D. Scargle,et al.  An Introduction to the Theory of Point Processes, Vol. I: Elementary Theory and Methods , 2004, Technometrics.

[15]  Christopher R. Taber,et al.  Generalized Method of Moments , 2020, Time Series Analysis.

[16]  A. Hawkes,et al.  A cluster process representation of a self-exciting process , 1974, Journal of Applied Probability.

[17]  C. Granger.  Investigating Causal Relations by Econometric Models and Cross-Spectral Methods , 1969 .

[18]  P. Reynaud-Bouret,et al.  Adaptive estimation for Hawkes processes; application to genome analysis , 2009, 0903.2919.

[19]  Amos Azaria,et al.  The DARPA Twitter Bot Challenge , 2016, Computer.

[20]  Erik A. Lewis,et al.  RESEARCH ARTICLE A Nonparametric EM algorithm for Multiscale Hawkes Processes , 2011 .

[21]  Nicolas Vayatis,et al.  Nonparametric Markovian Learning of Triggering Kernels for Mutually Exciting and Mutually Inhibiting Multivariate Hawkes Processes , 2014, ECML/PKDD.

[22]  L. Hansen.  Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[23]  Hongyuan Zha,et al.  Learning Granger Causality for Hawkes Processes , 2016, ICML.

[24]  Ali Shojaie,et al.  Network granger causality with inherent grouping structure , 2012, J. Mach. Learn. Res..

[25]  Martín Abadi,et al.  TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems , 2016, ArXiv.

[26]  George E. Tita,et al.  Self-Exciting Point Process Modeling of Crime , 2011 .

[27]  Francis R. Bach,et al.  Rethinking LDA: Moment Matching for Discrete ICA , 2015, NIPS.

[28]  Yacine Ait-Sahalia,et al.  Modeling Financial Contagion Using Mutually Exciting Jump Processes , 2010 .

[29]  Emmanuel Bacry,et al.  Hawkes model for price and trades high-frequency dynamics , 2013, 1301.1135.

[30]  Jean-Philippe Bouchaud,et al.  Branching-ratio approximation for the self-exciting Hawkes process. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Lasso and probabilistic inequalities for multivariate point processes , 2015, 1208.0570.

[32]  Yosihiko Ogata,et al.  On Lewis' simulation method for point processes , 1981, IEEE Trans. Inf. Theory.

[33]  Riadh Zaatour,et al.  Hawkes Process: Fast Calibration, Application to Trade Clustering and Diffusive Limit , 2013, Journal of Futures Markets.

[34]  Yoram Singer,et al.  Adaptive Subgradient Methods for Online Learning and Stochastic Optimization , 2011, J. Mach. Learn. Res..

[35]  Didier Sornette,et al.  Robust dynamic classes revealed by measuring the response function of a social system , 2008, Proceedings of the National Academy of Sciences.

[36]  D. Rubin,et al.  Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction , 2016 .

[37]  E. Bacry,et al.  Estimation of slowly decreasing Hawkes kernels: application to high-frequency order book dynamics , 2016 .

[38]  Bernhard Schölkopf,et al.  Modeling Information Propagation with Survival Theory , 2013, ICML.

[39]  Yan Liu,et al.  Temporal causal modeling with graphical granger methods , 2007, KDD '07.

[40]  W. Newey,et al.  Large sample estimation and hypothesis testing , 1986 .