Mutually exciting point processes

See also Self-exciting point processes and spatiotemporal point processes.

• Eichler, M., Dahlhaus, R., & Dueck, J. (2017). Graphical modeling for multivariate Hawkes processes with nonparametric link functions. Journal of Time Series Analysis, 38(2), 225–242. doi:10.1111/jtsa.12213

  Gives Granger causality results for mutually exciting processes, showing that one process does not Granger-cause another if and only if the triggering function between the two is identically 0. This extends to Granger causality graphs, where we make directed graphs with no edges between variables where the triggering function is identically 0. A global Markov property applies. Also proposes a nonparametric estimator of the triggering function by discretizing the process and using time series analysis.

• Chen, S., Witten, D., & Shojaie, A. (2017). Nearly assumptionless screening for the mutually-exciting multivariate Hawkes process. Electronic Journal of Statistics, 11(1), 1207–1234. doi:10.1214/17-ejs1251

  Suppose we want to screen for edges in the Granger causality graph. (This paper doesn’t directly make the connection to Granger causality, but is seeking the edges where triggering functions have nonzero integrals.) Proposes a screening procedure which recovers a small superset of the true edge set and which “boils down to just screening pairs of nodes by thresholding an estimate of their cross-covariance”.

• Achab, M., Bacry, E., Gaı̈ffas, S., Mastromatteo, I., & Muzy, J.-F. (2018). Uncovering causality from multivariate Hawkes integrated cumulants. Journal of Machine Learning Research, 18(192). http://jmlr.org/papers/v18/17-284.html

  For a multivariate mutually exciting process, estimates the influence of each process on the others by estimating the integrals of the triggering functions directly, rather than needing parametric triggering functions or estimating the functions nonparametrically. Works by calculating the process cumulants and matching them to the matrix of integrals via a convoluted non-convex optimization.

• Chen, S., Shojaie, A., Shea-Brown, E., & Witten, D. (2017, July). The multivariate Hawkes process in high dimensions: Beyond mutual excitation. arXiv. https://arxiv.org/abs/1707.04928

  Extends mutually exciting processes to allow mutual inhibition as well, coming up with a replacement for the convenient cluster process representation that allows similar theory to be developed. Uses these results to do clustering on mutually exciting processes, along with a penalized regression scheme to recover the causal graph by estimating the triggering functions.

• Yuan, B., Li, H., Bertozzi, A. L., Brantingham, P. J., & Porter, M. A. (2019). Multivariate spatiotemporal Hawkes processes and network reconstruction. SIAM Journal on Mathematics of Data Science, 1(2), 356–382. doi:10.1137/18M1226993. https://arxiv.org/abs/1811.06321

  Considers mutually exciting spatiotemporal processes and estimates a nonparametric model where the triggering functions are piecewise constants, with the constants estimated as part of the fit. Thresholds the triggering function matrix to estimate the network of Granger causality, and since the model is spatiotemporal, spatial proximity informs the causal estimates. The model is fit with EM. Simulation studies show that the model performs well against alternatives that are purely temporal, such as the methods above.

• Apostolopoulou, I., Linderman, S. W., Miller, K., & Dubrawski, A. (2018). Multivariate mutually regressive point processes. In 1st Symposium on Advances in Approximate Bayesian Inference. http://approximateinference.org/2018/accepted/ApostolopoulouEtAl2018.pdf

  Very short paper, but introduces a mutually exciting and inhibiting process by making the model nonlinear, then inventing a clever Bayesian sampling scheme to make inference tractable. Applies to multi-neuron spike trains. I’d imagine diagnostics for these processes will be very difficult, and interpretation of the model somewhat intricate, but the sampling scheme seems to bypass many of the problems I’ve had.