Nonparametric estimation of Hawkes processes with RKHSs (2411.00621v2)

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.

• The paper introduces a novel nonparametric method utilizing Reproducing Kernel Hilbert Spaces (RKHSs) to estimate complex nonlinear multivariate Hawkes processes.
• Numerical experiments demonstrate that the RKHS-based approach effectively estimates nonlinear interactions and outperforms traditional parametric and other nonparametric methods on synthetic and neuronal datasets.
• This RKHS method provides a flexible tool with significant potential for modeling complex dynamics and understanding interactions in systems like neuronal networks.

Nonparametric Estimation of Nonlinear Multivariate Hawkes Processes with Reproducing Kernel Hilbert Spaces

The paper by Bonnet and Sangnier introduces a method for the nonparametric estimation of nonlinear multivariate Hawkes processes by leveraging the capabilities of Reproducing Kernel Hilbert Spaces (RKHSs). This approach addresses several challenges inherent in modeling complex interaction functions within multivariate point processes. This method is particularly motivated by applications in neuroscience, where phenomena such as neuronal interactions need to be precisely captured.

Summary of Contributions

The paper presents a novel estimation method for Hawkes processes, assuming that these processes' interaction functions reside within an RKHS. The model permits complex interactions that involve exciting and inhibiting effects, which are especially relevant for accurately modeling neuronal activity, including refractory periods. To manage methodological challenges posed by these processes, the authors propose approximations for the ReLU-based conditional intensities within the likelihood and least-squares estimation frameworks.

Key Theoretical Insights

The authors establish a representer theorem specific to these RKHS-based estimations, which forms the core theoretical contribution. They demonstrate the representational framework necessary to support these complex estimations. Specifically, they reveal that both the approximated likelihood and least-squares criteria can be addressed in a structured manner by proposing compact parameterizations of the interaction functions, enhancing computational tractability.

Numerical Experiments and Results

The method is experimentally validated using synthetic data and real neuronal datasets. The results illustrate that this RKHS-based method outperforms traditional parametric models and related nonparametric alternatives, such as Bernstein polynomial approximations and Gaussian basis function models. Notably, the RKHS method is effective in estimating nonlinear interactions due to its adaptability in capturing changes in interaction signs over time—a critical feature for neuronal data with both exciting and inhibiting interactions.

Implications and Future Directions

One prominent implication of this paper lies in its application in neuroscience, particularly for modeling and understanding complex neuronal interactions from neural spike train data. The proposed approach offers methodological flexibility that traditional linear or fixed-form models lack. Future work could consider extending this framework to spatiotemporal Hawkes processes and incorporating efficient estimation techniques, such as kernel approximation methods, to further reduce computational demands.

Additionally, adapting this method to work within the sparse learning framework or with operator-valued kernels could open new pathways for compressing data while preserving crucial structural properties of neuronal networks. Given the methodological novelty and the strong numerical performance on synthetic and real data, this approach provides a compelling foundation for future work seeking to understand the dynamics inherent in complex multivariate point processes, particularly in the context of biological systems.

Conclusion

Bonnet and Sangnier's work provides a substantial step forward in nonparametric estimation strategies for Hawkes processes, revealing the potency of RKHS-driven methods. The balance between methodological soundness and practical applicability demonstrated in this paper positions it as a useful tool for handling complex, interaction-rich process data across various fields, particularly those involving dynamical systems such as neuroscience and beyond.