具有抑制效应的多变量指数Hawkes过程的推断及其在神经元活动中的应用
Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity
AI总结：
本文研究同时具有激发和抑制效应的多维指数Hawkes过程的参数推断，提出频率学派框架下首个基于最大似然的精确推断方法，证明了模型可识别性，并在合成与神经元数据上验证了其优越性。
摘要：
多变量Hawkes过程是一种依赖于历史的点过程，用于建模不同现象之间事件发生的关系。尽管Hawkes过程最初被引入是为了描述激发效应，即一个事件增加另一个事件发生的概率，但人们对建模相反的效应（即抑制效应）的兴趣日益浓厚。在本文中，我们关注如何推断同时具有激发和抑制效应的多维指数Hawkes过程的参数。我们的第一个结果是在少数充分假设下证明了该模型的可识别性。然后，我们提出了一种最大似然方法来估计交互函数，据我们所知，这是频率学派框架下的首次精确推断过程。我们的方法包括一个变量选择步骤，以恢复交互的支持集，从而推断连接图。该方法的一个优势是能够显式计算对数似然，这额外使得能够执行拟合优度检验以评估估计的质量。我们将该方法与在线性框架下开发且并非专为处理抑制效应而设计的标准方法进行了比较。结果表明，所提出的估计量在合成数据上的表现优于替代方法。我们还展示了该方法在神经元活动数据集上的应用，突出了神经元之间同时存在激发和抑制效应。
英文摘要：
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 effects, 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.
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