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）是对基于事件的数据进行建模的自然工具。在所有TPP模型中，霍克斯过程已被证明是最广泛使用的，这主要归因于其在考虑指数核或非参数核时对各种应用场景的充分建模能力。尽管非参数核是一种选择，但此类模型需要大量数据集。虽然指数核在事件立即触发更多事件的应用中更具数据效率且适用性更强，但其并不适合需要估计延迟的应用（例如神经科学领域）。本研究旨在针对具有有限支撑的一般参数核，为TPP推理提供高效解决方案。所开发的方案是一个基于快速ℓ2梯度的求解器，利用了事件的离散化版本。在从理论上验证离散化方法的合理性后，通过多种数值实验证明了该新方法的统计与计算效率。最后，通过使用脑磁图（MEG）记录的脑信号对刺激诱发模式的发生进行建模，评估了该方法的有效性。鉴于使用了通用参数核，结果表明所提出的方法相比现有技术能够更准确地估计模式延迟。