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Mathematical Problems in Engineering
Volume 2011, Article ID 793429, 14 pages
http://dx.doi.org/10.1155/2011/793429
Research Article

Symplectic Principal Component Analysis: A New Method for Time Series Analysis

Institute of Vibration, Shock & Noise and State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200030, China

Received 9 July 2011; Accepted 22 September 2011

Academic Editor: Mahmoud T. Yassen

Copyright © 2011 Min Lei and Guang Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Experimental data are often very complex since the underlying dynamical system may be unknown and the data may heavily be corrupted by noise. It is a crucial task to properly analyze data to get maximal information of the underlying dynamical system. This paper presents a novel principal component analysis (PCA) method based on symplectic geometry, called symplectic PCA (SPCA), to study nonlinear time series. Being nonlinear, it is different from the traditional PCA method based on linear singular value decomposition (SVD). It is thus perceived to be able to better represent nonlinear, especially chaotic data, than PCA. Using the chaotic Lorenz time series data, we show that this is indeed the case. Furthermore, we show that SPCA can conveniently reduce measurement noise.