Weighted norms in subspace-based methods for time series analysis

• Published in: Numerical linear algebra with applications Vol. 23; no. 5; pp. 947 - 967
• Main Authors: Gillard, J. W., Zhigljavsky, A. A.
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.10.2016; Wiley Subscription Services, Inc
• Subjects: Approximation; Construction methods; Hankel matrices; low-rank approximation; Mathematical analysis; missing values and forecasting; Norms; Time series; Time series analysis; Vectors (mathematics)
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2062

Summary Many modern approaches of time series analysis belong to the class of methods based on approximating high‐dimensional spaces by low‐dimensional subspaces. A typical method would embed a given time series into a structured matrix and find a low‐dimensional approximation to this structured matrix. The purpose of this paper is twofold: (i) to establish a correspondence between a class of SVD‐compatible matrix norms on the space of Hankel matrices and weighted vector norms (and provide methods to construct this correspondence) and (ii) to motivate the importance of this for problems in time series analysis. Examples are provided to demonstrate the merits of judiciously selecting weights on imputing missing data and forecasting in time series. Copyright © 2016 John Wiley & Sons, Ltd.