Lasso‐type estimator and classification algorithm for high‐dimensional multivariate Hawkes processes

What the paper says

We propose to deal with high‐dimensional event‐based data using Hawkes processes. Focusing on the Multivariate Hawkes Processes (MHP) in high dimension, an estimation task, followed by a classification task, are addressed in this article. In both cases, we assume to have access to a large number of repeated observations of the process over the same short time interval. MHPs form a versatile class of point processes that model interactions among connected individuals within a network. In this work, we allow the network dimension to be large relative to the number of observations, which necessitates a sparsity assumption on the adjacency matrix. Furthermore, we assume that the observations belong to different classes, discriminated by both the exogenous intensity vector and the adjacency matrix, which encodes the strength of interactions. Specifically, the observed training data consist of labeled, repeated, and independent realizations over a fixed time interval. In this context, we propose a novel methodology comprising an initial interaction recovery step, conducted per class, followed by a refitting step guided by a suitable classification criterion. To recover the support of the adjacency matrix in each class, we introduce a Lasso‐type estimator and prove the consistency of the estimated supports under appropriate assumptions on the processes. Leveraging the estimated supports, we then construct a classification procedure based on empirical error minimization. Notably, we provide convergence rates for our classifier. An in‐depth numerical study, using both synthetic and real‐world datasets, supports our theoretical findings, both for support recovery and for supervised classification.