Robust Estimation of Self-Exciting Generalized Linear Models with Application to Neuronal Modeling

TL;DR

This paper develops methods for accurately estimating self-exciting generalized linear models from limited binary data, extending compressed sensing techniques to dependent covariates, with applications to neuronal spike data.

Contribution

It introduces analysis of $\, ext{l}_1$-regularized and greedy estimators for dependent covariates in self-exciting models, providing non-asymptotic sampling guarantees.

Findings

The estimators achieve stable recovery with limited samples.

Simulation results validate theoretical predictions.

Application to neuronal data demonstrates practical effectiveness.

Abstract

We consider the problem of estimating self-exciting generalized linear models from limited binary observations, where the history of the process serves as the covariate. We analyze the performance of two classes of estimators, namely the $\ell_1$-regularized maximum likelihood and greedy estimators, for a canonical self-exciting process and characterize the sampling tradeoffs required for stable recovery in the non-asymptotic regime. Our results extend those of compressed sensing for linear and generalized linear models with i.i.d. covariates to those with highly inter-dependent covariates. We further provide simulation studies as well as application to real spiking data from the mouse's lateral geniculate nucleus and the ferret's retinal ganglion cells which agree with our theoretical predictions.