Uncovering Causality from Multivariate Hawkes Integrated Cumulants

In many applications, one needs to deal with data containing a very large number of irregular timestamped events that are recorded in continuous time. These events can reflect, for instance, the activity of users on a social network, see Subrahmanian et al. (2016), the high-frequency variations of signals in finance, see Bacry et al. (2015), the earthquakes and aftershocks in geophysics, see Ogata (1998), the crime activity, see Mohler et al. (2011) or the position of genes in genomics, see Reynaud-Bouret and Schbath (2010). The succession of the precise timestamps carries a great deal of information about the dynamics of the underlying systems. In this context, multidimensional counting processes based models play a paramount role. Within this framework, an important task is to recover the mutual influence of the nodes (i.e., the different components of the counting process), by leveraging on their timestamp patterns, see, for instance, Bacry and Muzy (2016); Lemonnier and Vayatis (2014); Lewis and Mohler (2011); Zhou et al. (2013a); Gomez-Rodriguez et al. (2013); Farajtabar et al. (2015); Xu et al. (2016). Consider a set of nodes I {1,..., d}.