The multivariate Hawkes process is a past-dependent point process used to model the relationship of event occurrences between different phenomena.Although the Hawkes process was originally introduced to describe excitation interactions, which means that one event increases the chances of another occurring, there has been a growing interest in modelling the opposite effect, known as inhibition.In this paper, we focus on how to infer the parameters of a multidimensional exponential Hawkes process with both excitation and inhibition effects. Our first result is to prove the identifiability of this model under a few sufficient assumptions. Then we propose a maximum likelihood approach to estimate the interaction functions, which is, to the best of our knowledge, the first exact inference procedure in the frequentist framework.Our method includes a variable selection step in order to recover the support of interactions and therefore to infer the connectivity graph.A benefit of our method is to provide an explicit computation of the log-likelihood, which enables in addition to perform a goodness-of-fit test for assessing the quality of estimations.We compare our method to standard approaches, which were developed in the linear framework and are not specifically designed for handling inhibiting effects.We show that the proposed estimator performs better on synthetic data than alternative approaches. We also illustrate the application of our procedure to a neuronal activity dataset, which highlights the presence of both exciting and inhibiting effects between neurons.

翻译：多元霍克斯过程是一种依赖于历史事件的点过程，用于刻画不同现象之间事件发生的关系。尽管霍克斯过程最初被提出时用于描述激发相互作用（即一个事件会增加另一事件发生的概率），但近年来对相反效应（即抑制效应）的建模兴趣日益增长。本文聚焦于如何推断同时具有激发与抑制效应的多维指数型霍克斯过程的参数。我们首先在若干充分假设下证明了该模型的可辨识性，随后提出一种基于最大似然的估计方法以估计交互函数——据我们所知，这是频率学派框架下首个精确推断流程。该方法包含变量选择步骤以恢复交互作用的支撑集，进而推断连接图。其优势在于能够显式计算对数似然，从而可额外进行拟合优度检验以评估估计质量。我们将该方法与线性框架下开发的、未专门设计以处理抑制效应的标准方法进行比较。实验表明，本文提出的估计器在合成数据上的表现优于替代方法。我们还将该流程应用于神经元活动数据集，结果揭示了神经元之间同时存在激发与抑制效应。