Inference for High-Dimensional Covariance Matrices: Testing and Estimating Changes for Time Series
The problem to test for a change in the covariance structure of a high-dimensional time series arises in diverse fields such as environmetrics, statistical genetics, finance or industrial quality control. This talk presents a methodology based on bilinear forms with respect to weighting vectors, which allows for analyses based on fixed bases (such as wavelets), random projections (have in mind the Johnson-Lindenstrauss theorem) or eigenstructures. The assumed linear framework covers wide classes of time series. Especially, classes of VARMA models with colored noise, heavily used in econometrics as well as environmetrics, and spiked covariance models, which received considerable interest to study statistical methods for high-dimensional data, can be treated.
The statistical procedures employ unweighted and weighted cumulated sum (CUSUM) statistics and associated change-point estimators, multivariate CUSUM transforms of growing dimension, in order to handle analyses in subspaces, and 2 -tests. The asymptotic results are based on large sample approximations of partial sums and hold for a linear process framework without constraints on the growths of the dimension of the vector time series and of the CUSUM transform. The asymptotic variances and covariances of the cumulated sums can be estimated consistently by a class of homogenous estimators. By studying consistency for sequential versions of these estimators, we are in a position to establish consistency of a stopped-sample estimator.
Finite sample properties are studied by simulations. As a real-world application and to illustrate the method, we analyze monitoring spatial-temporal data from ozone sensors, where sensor data is compressed by projecting it onto sparse directions.
Short Bio:
Ansgar Steland received the M.Sc. and Ph.D. degrees in mathematics from the University of Göttingen. After positions at Technische Universität Berlin, as a consultant, at the European University Viadrina and Ruhr-University of Bochum, he joined the faculty at RWTH Aachen University, where he was appointed Full Professor at the Institute of Statistics in 2006. He is an Elected Member of the International Statistical Institute and acts as the Chair of the Society for Reliability, Quality and Safety and as the Chair of the Section on Statistics in Natural Sciences and Technology of the German Statistical Society. Ansgar Steland organized and co-organized several international workshops and invited sessions, was member of program committees of several conferences and of a couple of prize committees. His main research interests are in change detection and quality control, high-dimensional statistics, time series analysis, nonparametric statistics, image analysis and applications to econometrics, natural sciences and engineering, especially photovoltaics.