高维Hawkes过程的统计推断

如何科学地刻画粒子系统中点与点之间在时间推演下事件发生的相互作用的统计规律是一个重要的科学问题。Hawkes过程是研究这一科学问题的主要工具，但是目前的研究主要局限于低维系统的情形，而对于高维系统的研究有待开展。另一方面，高维Hawkes过程作为一种典型的与时间相依的非独立的高维数据类型在经济学、社会学、神经科学等领域频繁出现，这为当前仅适用于独立的高维数据类型的统计推断方法带来挑战，需要发展新的统计方法。基于以上理论方法与实际应用的需要，本项目创造性地从有向单边、无向双边、同质聚集以及同质散发四个不同的角度对高维Hawkes过程的点与点的作用方式深入地进行研究，采用和发展现代统计理论和技巧度量时序相依性，系统地建立点与点之间连接结构的识别与评估的统计推断方法，为实际应用提供理论基础和方法指导。

In a particle system, it is essential to describe the statistical law of interaction among points as time evolves, which is often primarily interested in some scientific fields and otherwise statistical applications. The Hawkes processes is one of the effective tools for addressing these scientific issues, however, the sophisticated statistical methods are mainly restrictive themselves to the low-dimensional particle system. On the other hand, as a typical example with dependent observations, temporally dependent high-dimensional Hawkes processes are emerging in application of fields such as economy, society and neural science so on, rendering the current high-dimensional inference for independent observations inapplicable. Therefore, it is imperative to develop new statistical methodologies for inferring the high-dimensional Hawkes processes to bridge the gap between the theory and new applications. This proposal will investigate four important ways of point interactions, which are the single directed edge, two undirected edges, homogeneous assembling and homogeneous emitting. This proposal will offer deep and comprehensive insight into interaction mechanism among points of high-dimensional Hawkes processes by employing and developing modern statistical theory and techniques to quantify the temporal dependence. It aims to systematically establish the high-dimensional inference methodologies for identifying and evaluating the connectivity structure among points. Remarkably, the output of this proposal will afford investigators the solid theoretical basis and methodological guidance, based on which the conclusions drawn from real examples can thus be scientifically achieved and reliable.