Temporal point processes (TPP) are a natural tool for modeling event-based data. Among all TPP models, Hawkes processes have proven to be the most widely used, mainly due to their adequate modeling for various applications, particularly when considering exponential or non-parametric kernels. Although non-parametric kernels are an option, such models require large datasets. While exponential kernels are more data efficient and relevant for specific applications where events immediately trigger more events, they are ill-suited for applications where latencies need to be estimated, such as in neuroscience. This work aims to offer an efficient solution to TPP inference using general parametric kernels with finite support. The developed solution consists of a fast $\ell_2$ gradient-based solver leveraging a discretized version of the events. After theoretically supporting the use of discretization, the statistical and computational efficiency of the novel approach is demonstrated through various numerical experiments. Finally, the method's effectiveness is evaluated by modeling the occurrence of stimuli-induced patterns from brain signals recorded with magnetoencephalography (MEG). Given the use of general parametric kernels, results show that the proposed approach leads to an improved estimation of pattern latency than the state-of-the-art.

翻译：时间点过程（TPP）是建模基于事件数据的自然工具。在所有TPP模型中，霍克斯过程因其能充分建模多种应用场景而成为最广泛使用的模型，尤其是在考虑指数核或非参数核时。尽管非参数核是一种选择，但此类模型需要大规模数据集。而指数核虽在事件即时触发更多事件的特定应用中具有更高的数据效率且更具相关性，但它们在需要估计延迟的应用（如神经科学）中并不适用。本文旨在为使用有限支撑的一般参数核的TPP推理提供高效解决方案。所开发的解决方案基于事件的离散化版本，采用快速$\ell_2$梯度求解器。在从理论上支撑离散化方法的可行性后，通过多项数值实验展示了该新方法的统计与计算效率。最后，通过建模脑磁图（MEG）记录中刺激诱发模式的发生，评估了该方法的有效性。鉴于采用了通用参数核，结果表明所提出的方法在估计模式延迟方面优于现有技术。