Flexible estimation in temporal point processes and graphs

时间点过程和图表中的灵活估计

• 批准号: 2095161
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2095161
• 关键词: Flexible; estimation; temporal; point; processes

Multivariate point processes with history dependence such as Hawkes processes naturally admit a graph representation. Nodes are the dimensions of the process and edges appear between dimensions having a non-null interaction function. In the non-negative case - accounting for excitation phenomena - nonparametric estimation of those functions have been studied in the Bayesian framework in Donnet et al. [1]. In this context, estimation of the graph and the non-positive case - accounting for inhibition phenomena - are open topics. In fact, these problems encounter a lot of interest in the neuroscience context. Neurons interact through action potentials which are generally treated as identical events (spike trains). Multivariate Hawkes Processes can thus model functional connectivity between neurons and identify excitation/inhibition relationships. To analyse the graph obtained from Multivariate Hawkes processes, weighted edges can be obtained by associating to each of them a norm of the interaction functions. In the case of non-linear Hawkes processes, those functions are allowed to be non-positive and lead to negatively weighted edges. The resulting graph is said to be signed, and requires special methods of graph analysis. In particular, clustering is a popular task which aims at identifying communities of nodes having similar features within a network. For the signed case, spectral algorithms have been adapted but are still considered suboptimal. Cucuringu et al. [2] developed a new method based on the combination of regularized graph Laplacian matrices. However, theoretical results on random graph models have only been obtained in the case of two disjoint communities. Extension to a larger number of communities and the context of sparse networks are still unanswered questions.[1] Sophie Donnet, Vincent Rivoirard, and Judith Rousseau. Nonparametric Bayesian Estimation of Multivariate Hawkes Processes. arXiv:1802.05975v2, 2018.[2] Mihai Cucuringu, Peter Davies and Hemant Tyagi. SPONGE: A generalized eigenproblem for clustering signed networks. AISTATS, 2018.

具有历史依赖性的多变量点过程，如Hawkes过程，自然允许用图表示。节点是过程的维度，边出现在具有非空交互函数的维度之间。在非负的情况下-考虑激励现象-这些函数的非参数估计已经在Donnet等人的贝叶斯框架中进行了研究。[1]。在这种情况下，估计的图形和非积极的情况下-占抑制现象-是开放的话题。事实上，这些问题在神经科学的背景下遇到了很多兴趣。神经元通过动作电位相互作用，动作电位通常被视为相同的事件（尖峰序列）。因此，多变量霍克斯过程可以模拟神经元之间的功能连接，并确定兴奋/抑制关系。 为了分析从多元霍克斯过程获得的图，可以通过将每个边与相互作用函数的范数关联来获得加权边。在非线性Hawkes过程的情况下，允许这些函数是非正的，并导致负加权边。所得到的图被称为是有符号的，并且需要特殊的图分析方法。特别是，聚类是一个流行的任务，其目的是识别网络中具有相似特征的节点社区。对于有符号的情况，谱算法已被改编，但仍然被认为是次优的。Cucuringu等人[2]开发了一种基于正则化图拉普拉斯矩阵组合的新方法。然而，随机图模型的理论结果只在两个不相交社区的情况下得到。扩展到更大数量的社区和稀疏网络的上下文仍然是未回答的问题。[1]索菲·唐内，文森特·里瓦拉德，朱迪思·卢梭。多变量Hawkes过程的非参数贝叶斯估计。arXiv：1802.05975v2，2018. [2]Mihai Cucuringu，Peter Davies和Hemant Tyagi。SPONGE：A Generalized eigenproblem for clustering signed networks. AISTATS，2018年。