Nonparametric likelihood based estimation of linear filters for point processes

TL;DR

This paper introduces a nonparametric likelihood-based method for estimating linear filters in multivariate point process models, utilizing reproducing kernel Hilbert spaces and efficient computational techniques, with applications to neuron network modeling.

Contribution

It develops a novel gradient representation for the nonparametric likelihood, enabling efficient approximation and optimization using sparse matrices, and provides an implementation in R for practical use.

Findings

Efficient approximation of the likelihood gradient via time discretization.

Implementation leveraging sparse matrices for computational efficiency.

Application to neuron network modeling demonstrating scalability.

Abstract

We consider models for multivariate point processes where the intensity is given nonparametrically in terms of functions in a reproducing kernel Hilbert space. The likelihood function involves a time integral and is consequently not given in terms of a finite number of kernel evaluations. The main result is a representation of the gradient of the log-likelihood, which we use to derive computable approximations of the log-likelihood and the gradient by time discretization. These approximations are then used to minimize the approximate penalized log-likelihood. For time and memory efficiency the implementation relies crucially on the use of sparse matrices. As an illustration we consider neuron network modeling, and we use this example to investigate how the computational costs of the approximations depend on the resolution of the time discretization. The implementation is available in the R package ppstat.