Nonparametric estimation of Hawkes processes with RKHSs

2024-11-01

arXiv.org

AI Summary

This paper addresses the nonparametric estimation of nonlinear multivariate Hawkes processes, where interaction functions reside in a reproducing kernel Hilbert space (RKHS) and the conditional intensity is truncated by the ReLU function to jointly model excitation, inhibition, and hybrid effects (e.g., neuronal refractoriness). To tackle the non-differentiability and non-convexity induced by ReLU, we establish— for the first time—the approximate representation theorem for ReLU-truncated Hawkes processes. We then propose a provably convergent two-step RKHS-based estimation method: first approximating the ReLU nonlinearity, then approximating the integral operator. Theoretical analysis provides upper bounds on statistical error and guarantees asymptotic consistency. Experiments on synthetic data and real neuronal spike trains demonstrate that our method achieves significantly higher accuracy than existing nonparametric Hawkes estimators, while maintaining statistical robustness and modeling flexibility.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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 simple approximations (of the ReLU function and of the integral operator). We provide an approximation bound, justifying the negligible statistical effect 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.

Problem

• Nonparametric estimation of nonlinear multivariate Hawkes processes
• Modeling complex neuron interactions with RKHS-based functions
• Addressing ReLU rectification challenges in conditional intensities

Innovation

• Nonparametric estimation using RKHS for Hawkes processes
• ReLU-rectified conditional intensities with approximation solutions
• Representer theorem for log-likelihood and least-squares criteria