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.12213Gives 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-ejs1251Suppose 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.htmlFor 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/1811.06321Considers 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.pdfVery 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.

Xu, L., Duan, J. A., & Whinston, A. (2014). Path to purchase: A mutually exciting point process model for online advertising and conversion.

*Management Science*,*60*(6), 1392–1412. doi:10.1287/mnsc.2014.1952Uses Bayesian mutually exciting point processes to model different types of ad clicks, as well as purchase events on a retail website, to see what kinds of ad clicks trigger purchases and what types of ad clicks excite future ad clicks.

Brantingham, P. J. et al. (2017).

*Does violence interruption work?*http://www.stat.ucla.edu/~frederic/papers/brantingham1.pdfConsiders a gang violence intervention program in LA. After some aggravated assaults an intervention is performed to prevent retaliation; after others, it isn’t. Modeling these as separate mutually exciting processes, they find the excitation after the intervention is lower than after assaults with no intervention. Uses Mohler’s nonparametric KDE background estimate.

Wang, H. (2018).

*Assessing the impact of interventions on retaliatory violent crimes using Hawkes models with covariates*(Master’s thesis). University of California, Los Angeles. https://escholarship.org/uc/item/44m965gkExtension of the previous paper to use background covariates in the intensity function instead of a nonparametric background. (Oddly, the intensity is linear in the covariates, so there’s nothing to prevent it becoming negative.) Finds this fits better, but the difference between the intervention and non-intervention excitation rates is no longer statistically significant.