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事件。交通数据通常包括从一个地点到另一个地点的行程和流事件。这形成了一个时空网络，其中的连接发生在不同的地点之间。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非常普遍。大多数用于网络建模的统计方法都假设观察到一个单一的静态网络。相反，此设置假定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模式。预测这些事件尤其具有挑战性，因为事件发生率可能取决于时间和两个相关的空间位置，从而导致参数数量的爆炸性增长。人们可以做出假设，这将取决于应用程序，以数据自适应的方式减少参数的有效数量。例如，这项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。这将需要一个能够适应这些假设的广泛的概率框架，以及用于预测时空网络中的连接的易于计算的统计方法。在这样的动态网络中，将这些边缘事件建模为具有条件强度的点过程是很自然的，整个系统的结果模型将被称为点对点过程。这种新的方法具有在连续时间内对网络建模的优点，因此基于自然似然的过程可以直接在事务数据帧上工作，其中每行对应于一个边缘事件。本文的研究重点分为三个主题和应用领域。首先，对于交通网络来说，使用低秩法对空间位置进行分割是很自然的，这需要将随机区块模型应用于点对点过程框架。其次，利用分组融合惩罚和趋势滤波实现动态网络中时间变化点的检测和定位。第三，对于金融交易网络和流行病网络，可以通过将霍克斯模型应用于点对点过程框架来检验复杂的依赖关系。这项研究将检验这些统计问题的理论，并开发易于计算的算法。

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.