Change-point analysis in high dimensions

现代应用程序通常会生成按时间排序的数据集，其中许多协变量是随着时间的推移同时测量的。例如，可穿戴技术通过多传感器反馈记录个人的健康状态，数以万计的路由器收集的互联网流量数据，以及记录大脑不同区域特定化学对比演变的功能磁共振成像(FMRI)扫描。这种高维数据流数量的爆炸性增长要求对它们的分析方法进行改进。变点分析是一种基本的统计技术，用于识别此类数据流中的突变。被识别的“变点”通常是有趣或异常事件的信号，可以用来将数据流分割成更短的更容易分析的片段。经典的变点分析方法识别单个变量随时间的变化。然而，当以组件方式应用时，它们在高维数据集中经常遭受显著的性能损失。高维变点分析领域源于应对高维数据流带来的挑战的需要。在这个相对较新的领域，已经提出了几种方法。然而，它们通常需要简化假设，这限制了它们在许多应用中的实用性。在这个提案中，我将开发能够处理更现实的数据设置的新方法。具体地说，我将开发(1)一个算法，当一个接一个的数据点被观察到时，它可以在线监控数据流，以便它在保持低误警率的同时尽可能快地响应变化；(2)可以处理高度相关的分量序列的变点程序，这种情况在多传感器测量中非常常见；(3)在存在丢失或污染数据的情况下，用于变点估计的稳健方法。我将为开发的方法提供理论上的性能保证，并在开源的R包中实现它们。

Modern applications routinely generate time-ordered datasets, where many covariates are simultaneously measured over time. Examples include wearable technologies recording the health state of individuals from multi-sensor feedbacks, internet traffic data collected by tens of thousands of routers and functional magnetic resonance imaging (fMRI) scans that record the evolution of certain chemical contrast in different areas of the brain. The explosion in number of such high-dimensional data streams calls for methodological advances for their analysis. Change-point analysis is an essential statistical technique used in identifying abrupt changes in such data streams. The identified 'change-points' often signal interesting or abnormal events, and can be used to carve up the data streams into shorter segments that are easier to analyse.Classical change-point analysis methods identify changes in a single variable over time. However, they often suffer from significant performance loss in high-dimensional datasets when applied componentwise. The area of high-dimensional change-point analysis grew out of the need to respond to the challenge created by high-dimensional data streams. A few methods have been proposed in this relatively new area. However, they often require simplifying assumptions that restrict their usefulness in many applications. In this proposal, I will develop new methods that can handle more realistic data settings. Specifically, I will develop (1) an algorithm that can monitor the data stream 'online' as data points are observed one after another, so that it responds to changes as quickly as possible while maintaining a low rate of false alarms; (2) a change-point procedure that can handle highly correlated component series, a situation that is very common in multi-sensor measurements; (3) a robust method for change-point estimation in the presence of missing or contaminated data. I will provide theoretical performance guarantees for the developed methods and implement them in open-source R packages.