The Multivariate Generalized Linear Hawkes Process in High Dimensions with Applications in Neuroscience

The Hawkes process models have been recently become a popular tool for modeling and analysis of neural spike trains. In this article, motivated by neuronal spike trains study, we propose a novel multivariate generalized linear Hawkes process model, where covariates are included in the intensity function. We consider the problem of simultaneous variable selection and estimation for the multivariate generalized linear Hawkes process in the high-dimensional regime. Estimation of the intensity function of the high-dimensional point process is considered within a nonparametric framework, applying B-splines and the SCAD penalty for matters of sparsity. We apply the Doob-Kolmogorov inequality and the martingale central limit theory to establish the consistency and asymptotic normality of the resulting estimators. Finally, we illustrate the performance of our proposal through simulation and demonstrate its utility by applying it to the neuron spike train data set.