Sequential Monitoring of the Location and CovarianceBehaviour of High-Dimensional Time Series

由于近年来计算机技术的快速发展，现在可以存储和分析庞大的数据集。在这种情况下，潜在的随机模型的维度变得相当大，必须开发新的方法，以便在这种情况下提供统计推断。当数据维数较大时，经典的极限定理不再适用，传统的估计量在高维渐近情况下会严重偏离最优估计量。这个项目的主题是高维过程的监控。这是一个全新的领域。大多数已发表的文献处理的是自变量的分析，但这是一个戏剧性的简化，并经常不能在应用中实现。在这里，我们将假设底层过程遵循高维时间序列，即数据具有一定的内存。我们的目标是分别快速检测均值行为和协方差行为的变化。为此，有必要对多元统计过程控制的控制程序进行扩展和调整，使其适用于高维过程。我们想介绍几种新的基于似然比方法、广义似然比方法、Shiryaev-Roberts过程、广义Shiryaev-Roberts方法的控制图。我们将区分用于位置行为的控制方案和用于协方差行为的图表。此外，还将考虑同步方案。它们提供了检测位置或协方差结构变化的可能性。所介绍的控制方案将使用各种性能标准相互比较，例如，平均运行长度和平均延迟。在我们的研究中，我们想讨论几个应用。这些问题可以在工业应用中观察到，这是工业4.0中新一代数字化生产的结果。新兴技术（如增材制造、微制造）与新的检测解决方案（如非接触式系统、x射线计算机断层扫描）和快速多流高速传感器（如声学、温度、压力信号）相结合，为新一代工业大数据铺平了道路，需要新的建模和监测方法来实现零缺陷制造。为了降低生产成本，保证高质量的生产，有必要快速检测与目标生产过程的偏差。另一个重要的应用领域是对投资组合的监控。通常情况下，投资组合中的股票数量相对于观察的数量来说是很大的。对于投资者来说，尽快发现风险行为或平均回报的任何变化以重新分配他的投资组合是很重要的。

Due to the fast development of computer technology in recent years, it is nowadays possible to store and analyze huge data sets. In these cases, the dimension of the underlying stochastic model becomes quite large and new approaches must be developed in order to provide statistical inferences in such a situation. Whenever the dimension of the data is large, the classical limit theorems are no longer suitable and the traditional estimators will result in serious departures from the optimal ones under high-dimensional asymptotics. The subject of this project is the monitoring of high-dimensional processes. This is a completely new field. Most of the published literature deals with the analysis of independent variables but this is a dramatical simplification and it is frequently not fulfilled in applications. Here we will assume that the underlying process follows a high-dimensional time series, i.e. that the data have a certain memory. Our aim is to rapidly detect a change in the mean behavior and the covariance behavior, respectively. In order to do this, it is necessary to extend and to adapt the control procedures of multivariate statistical process control to high-dimensional processes. We want to introduce several types of new control charts based on, e.g., the likelihood ratio approach, the generalized likelihood ratio method, the Shiryaev-Roberts procedure, the generalized Shiryaev-Roberts approach. We will distinguish between control schemes for the location behavior and charts for the covariance behavior. Moreover, simultaneous schemes will be considered as well. They provide the possibility to detect changes in the location or the covariance structure. The introduced control schemes will be compared with each other using various performance criteria as, e.g., the average run length and the average delay.In our study we want to discuss several applications. Such problems can be observed in industrial applications as a consequence of the new generation of digital production in Industry 4.0. Emerging technologies (e.g., additive manufacturing, micro-manufacturing) combined with new inspection solutions (e.g., non-contact systems, X-ray computer tomography) and fast multi-stream high-speed sensors (e.g., acoustic, temperature, pressure signals) are paving the way to a new generation of industrial big data requiring novel modeling and monitoring approaches for zero-defect manufacturing. In order to reduce the production costs and to guaranty a high-quality production it is necessary to rapidly detect deviations from the target production process. A further important field of application is the monitoring of a portfolio. Frequently the number of stocks in the portfolio is large with respect to the number of observations. For an investor it is important to detect any change in the risk behavior or in the average returns as soon as possible to reallocate his portfolio.