Point-to-Point Process Models for Spatio-temporal Networks

实时数据收集系统使研究人员能够监控和分析数十亿个流媒体事件。 交通数据通常由从一个位置到另一个位置的行程（流事件）组成。 这就形成了一个时空网络，在这个网络中，连接发生在位置之间。 这种时空网络在计量经济学、交通和基础设施应用、互联网安全数据、神经学数据和流行病学网络中很普遍。 大多数建模网络的统计方法都假设观察到一个静态网络。 相比之下，此设置假设每个网络连接都是在特定时间发生的事件，因此必须调整网络模型以适应这种新颖的数据形态。 预测这些事件特别具有挑战性，因为事件发生率可能取决于时间和两个相关的空间位置，导致参数数量激增。 人们可以做出假设，这将取决于应用，以数据自适应的方式减少参数的有效数量。 例如，本研究将研究交通规划应用中的空间社区结构、互联网安全数据的时间趋势以及金融交易事件之间的复杂依赖关系。 这就需要一个广泛的概率框架，可以容纳这样的假设，和计算上易于处理的统计方法来预测时空网络中的连接，这是自然的模型，这些边缘事件在这样的动态网络作为点过程与条件强度，和整个系统的结果模型将被称为点对点过程。 这种新颖的方法具有在连续时间内对网络进行建模的优点，因此基于自然似然性的过程可以直接作用于事务性的嵌套框架，其中每行对应于边缘事件。 研究重点分为三个主题和应用领域。 首先，对于交通网络，使用低秩方法来分割空间位置是很自然的，对该方法的研究将需要使随机块模型适应点对点过程框架。 其次，动态网络中的时间变化点的检测和定位可以通过使用组融合惩罚和趋势滤波来完成。 第三，对于金融交易网络和流行病网络，可以通过将Hawkes模型调整到点对点过程框架来检查复杂的依赖关系。 该研究将研究这些统计问题的理论，并开发计算上易于处理的算法。

Real-time data collection systems allow researchers to monitor and analyze billions of streaming events. Transportation data often consists of trips, the streaming events, from one location to another. This forms a spatio-temporal network, in which the connections occur between locations. Such spatio-temporal networks are prevalent in econometrics, transportation and infrastructure applications, internet security data, neurology data, and epidemiological networks. The majority of statistical methodology for modelling networks assumes that a single static network is observed. In contrast, this setting presumes that each network connection is an event that happens at specific times, so one must adapt network models to accommodate this novel data modality. Predicting these events is particularly challenging because the event rate can depend on time and the two associated spatial locations, resulting in an explosion of the number of parameters. One can make assumptions, which will depend on the application, that reduce the effective number of parameters in a data adaptive fashion. For example, this research will study spatial community structure in transportation planning applications, temporal trends in internet security data, and complex dependencies between events for financial transactions. This will require a broad probabilistic framework that can accommodate such assumptions, and computationally tractable statistical methodology for predicting connections in spatio-temporal networks.It is natural to model these edge events in such dynamic networks as point processes with conditional intensities, and the resulting model for the entire system will be called a point-to-point process. This novel approach has the advantage of modelling the network in continuous time, so that natural likelihood-based procedures can work directly on transactional dataframes where each row corresponds to an edge event. The research focus is divided into three topics and areas of application. First, for transportation networks, it is natural to segment spatial locations using low rank methods, the study of which will require adapting the stochastic block model to the point-to-point process framework. Secondly, detecting and localizing temporal changepoints in dynamic networks can be accomplished by employing group fusion penalties and trend filtering. Thirdly, for financial transaction networks and epidemiological networks, complex dependencies can be examined by adapting Hawkes models to the point-to-point process framework. The research will examine the theory of these statistical problems and develop computationally tractable algorithms.