Singular Spectrum Analysis A New Tool in Time Series Analysis
The term singular spectrum comes from the spectral (eigenvalue) decomposition of a matrix A into its set (spectrum) of eigenvalues. These eigenvalues, A, are the numbers that make the matrix A -AI singular. The term singular spectrum analysis· is unfortun
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Singular Spectrum Analysis A New Tool in Time Series Analysis
James B. Elsner Florida State University Tallahassee. Florida
and
Anastasios A. Tsonis University of Wisconsin-Milwaukee Milwaukee. Wisconsin
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data On file
ISBN 978-1-4757-2514-8 (eBook) ISBN 978-1-4419-3266-2 DOI 10.1007/978-1-4757-2514-8 © 1996 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1996 Sof'tcover reprint of the hardcover 1st edition 1996 All rights reserved 1098765432 I No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher
With gratitude to our parents Roger and Diane Elsner Antonios and Isidora Tsonis
Preface The term singular spectrum comes from the spectral (eigenvalue) decomposition of a matrix A into its set (spectrum) of eigenvalues. These eigenvalues, A, are the numbers that make the matrix A - AI singular. The term singular spectrum analysis· is unfortunate since the traditional eigenvalue decomposition involving multivariate data is also an analysis of the singular spectrum. More properly, singular spectrum analysis (SSA) should be called the analysis of time series using the singular spectrum. Spectral decomposition of matrices is fundamental to much theory of linear algebra and it has many applications to problems in the natural and related sciences. Its widespread use as a tool for timeseries analysis is fairly recent, however, emerging to a large extent from applications of dynamical systems theory (sometimes called chaos theory). SSA was introduced into chaos theory by Fraedrich (1986) and Broomhead and King (l986a). Prior to this, SSA was used in biological oceanography by Colebrook (1978). In the digital signal processing community, the approach is also known as the Karhunen-Loeve (K-L) expansion (Pike et aI., 1984). Like other techniques based on spectral decomposition, SSA is attractive in that it holds a promise for a reduction in the dimen• Singular spectrum analysis is sometimes called singular systems analysis or singular spectrum approach. vii
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Preface
sionality. This reduction in dimensionality is often accompanied by a simpler explanation of the underlying physics. SSA is a linear approach to analysis and prediction of time series. The data-adaptive nature of the basis functions used in SSA gives it particular strength over classical spectral methods and makes the approach suitable for analysis of some nonlinear dynamics. But this strength comes at a price, namely, a difficulty in assigning statistical significance to the results. When carefully done, however, SSA is capable of providing useful insights into a range of systems and can be used to make predictions even when data amounts are modest. Throughout scientific research, measured time series are essential for describing and characterizing
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