Sequential Monitoring of the Location and Covariance Behaviour of High-Dimensional Time Series

• Applicant: Professor Dr. Wolfgang Schmid
• Subject Area: Statistics and Econometrics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 428472210

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.

• DFG Programme: Research Grants