Nonparametric estimation of Hawkes processes with RKHSs

TL;DR

This paper introduces a novel nonparametric estimation method for multivariate Hawkes processes with complex interaction functions in an RKHS, addressing methodological challenges posed by ReLU-rectified intensities and demonstrating superior performance in neuronal modeling.

Contribution

It develops a new estimation approach for nonlinear Hawkes processes with ReLU intensities in RKHS, including a representer theorem and approximation bounds, improving over existing methods.

Findings

The proposed estimator shows good asymptotic behavior on synthetic data.

It outperforms related nonparametric methods in neuronal applications.

The method effectively models complex excitatory and inhibitory interactions.

Abstract

This paper addresses nonparametric estimation of nonlinear multivariate Hawkes processes, where the interaction functions are assumed to lie in a reproducing kernel Hilbert space (RKHS). Motivated by applications in neuroscience, the model allows complex interaction functions, in order to express exciting and inhibiting effects, but also a combination of both (which is particularly interesting to model the refractory period of neurons), and considers in return that conditional intensities are rectified by the ReLU function. The latter feature incurs several methodological challenges, for which workarounds are proposed in this paper. In particular, it is shown that a representer theorem can be obtained for approximated versions of the log-likelihood and the least-squares criteria. Based on it, we propose an estimation method, that relies on two common approximations (of the ReLU function and of the integral operator). We provide a bound that controls the impact of these approximations. Numerical results on synthetic data confirm this fact as well as the good asymptotic behavior of the proposed estimator. It also shows that our method achieves a better performance compared to related nonparametric estimation techniques and suits neuronal applications.