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High-dimensional autoregressive point processes model how current eventstrigger or inhibit future events, such as activity by one member of a socialnetwork can affect the future activity of his or her neighbors. While past workhas focused on estimating the underlying network structure based solely on thetimes at which events occur on each node of the network, this paper examinesthe more nuanced problem of estimating context-dependent networks that reflecthow features associated with an event (such as the content of a social mediapost) modulate the strength of influences among nodes. Specifically, weleverage ideas from compositional time series and regularization methods inmachine learning to conduct network estimation for high-dimensional markedpoint processes. Two models and corresponding estimators are considered indetail: an autoregressive multinomial model suited to categorical marks and alogistic-normal model suited to marks with mixed membership in differentcategories. Importantly, the logistic-normal model leads to a convex negativelog-likelihood objective and captures dependence across categories. We providetheoretical guarantees for both estimators, which we validate by simulationsand a synthetic data-generating model. We further validate our methods throughtwo real data examples and demonstrate the advantages and disadvantages of bothapproaches.

Context-dependent self-exciting point processes: models, methods, and risk bounds in high dimensions