Statistics and Its Interface

Volume 11 (2018)

Number 4

A sequential majorization method for approximating weighted time series of finite rank

Pages: 615 – 630



Hou-Duo Qi (School of Mathematics, University of Southampton, Highfield, Southampton, United Kingdom)

Jian Shen (School of Mathematics, University of Southampton, Highfield, Southampton, United Kingdom)

Naihua Xiu (Department of Applied Mathematics, Beijing Jiaotong University, Beijing, China)


The low-rank Hankel matrix optimization has become one of the main approaches to the signal extraction from noisy time series of finite rank. The approach is particularly effective if different weights are enforced to the data points to reflect their relative importance. Two guiding principles for developing such an approach are (i) the Hankel matrix optimization should be computationally tractable, and (ii) the objective in the optimization should be a close approximation to the original weighted least-squares. In this paper, we introduce a sequential approximation that satisfies (i) and (ii) based on the technique of majorization. A new approximation is constructed as soon as a new iterate is computed from the previous approximation and it makes use of the latest gradient information of the objective, leading to more accurate an approximation to the objective. The resulting subproblem bears a similar structure to an existing scheme and hence can be efficiently solved. Convergence of the sequential majorization method (SMM) is guaranteed provided that the solution of the subproblem satisfies a sandwich inequality. We also compare SMM with two leading methods in literature on real-life problems. Significant improvement is observed in some cases.


singular spectrum analysis, time series of finite rank, Hankel matrix, majorization method, alternating projection

2010 Mathematics Subject Classification

Primary 62M10, 62M15. Secondary 62P99.

This research was supported by the National Science Foundation of China (11728101) and the 111 Project of China (B16002).

Received 15 May 2017

Published 19 September 2018