Table Of ContentSingular Spectrum
Analysis
A New Tool in
Time Series Analysis
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 978-1-4419-3266-2 ISBN 978-1-4757-2514-8 (eBook)
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 the
ory of linear algebra and it has many applications to problems in the
natural and related sciences. Its widespread use as a tool for time
series 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 digi
tal 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
viii 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 sta
tistical 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 essen
tial for describing and characterizing a physical system. Adequate
descriptions, in tum, can lead to useful forecasts of future behav
ior. When prognostic equations governing the physical system are
known, and are insensitive to the initial data, forecasting is straight
forward and there is little requirement for extensive time-series anal
ysis. This is often not the case, however. Typically it turns out that
even for systems for which the governing equations are known pre
cisely, accurate predictions are limited to the short term. This is the
situation when the dynamical system has instabilities and nonlinear
ities that give rise to chaos.
All is not hopeless since for longer time scales the system may
exhibit regularities or near periodicities that can be examined and
possibly used for predictions. The Earth's climate is an example. In
these cases time-series analysis on observables from the system is
valuable. Another class of systems requiring time-series analysis are
those for which the governing equations are unknown. Often such
systems have a large number of interacting subsystems with feed
backs and which can be described as complex. Examples include
cell growth and division, the human brain, and the stock market, to
mention but a few.
The purpose of this book is to provide a useful introduction to
SSA for time-series analysis, providing background materials, the
ory, and practical examples along the way. The level of presenta
tion is undergraduate science majors. The primary target audience
Preface ix
is graduate students and scientists uninitiated in the area of spectral
decomposition, but with an interest in applying SSA to particular
problems and with the knowledge that SSA extracts reliable infor
mation from short and noisy time series without relying on prior
knowledge about the underlying physics or biology of the system.
The exclusive emphasis on this new approach should not be
interpreted to mean that it is a sufficient method for understanding
all there is to know about a particular time record. As is usually
the case in data analysis and statistics, SSA works best when it is
done in concert with other independent techniques.
In writing this book we have borrowed liberally from other
sources. In particular we mention the meritorious work of Dr. Myles
Allen. His writings and our extensive discussions with him have
resulted in a more concise and thorough understanding of SSA, from
which we hope the reader will benefit. Finally, one of the authors
(JBE) is grateful to Svetoslava for her spiritual and editorial support
throughout this project.
Partial financial support for this work came from the National
Science Foundation through grants ATM 93-10715, ATM 93-10959,
and ATM 94-17528.
James B. Elsner
Anastasios A. Tsonis
Contents
I. Mathematical Notes
1. Review of Linear Algebra . . . . . . . . 3
1.1. Introduction. . . . . . . . . . . . . . 3
1.2. Matrix Notation and Multiplication . 3
1.3. Matrix Factorization . . . . 9
1.4. Inverses and Transposes . . 12
1.5. Properties of Determinants 15
1.6. Round-off Error ..... 17
2. Eigenvalues and Eigenvectors 19
2.1. Physical Intetpretation . . 19
2.2. Finding the Eigenmodes . 21
2.3. Diagonal Form of a Matrix 25
2.4. Spectral Decomposition 26
3. Multivariate Statistics . . 29
3.1. Introduction ..... 29
3.2. Mean and Variance. 30
3.3. Covariance and Correlation 30
xi
xii Contents
II. Theory and Methods
4. Foundations of SSA . . . . . . . 39
4.1. Trajectory Matrix . . . . . . 39
4.2. Lagged-Covariance Matrix . 45
4.3. The Singular Spectrum .. 45
4.4. Recovering the Time Series . 48
4.5. Comparison with Principal Component Analysis 50
5. Details ......... 51
5.1. Trends ...... 52
5.2. Window Length 57
5.3. Toeplitz Structure . 59
5.4. Filtering. 65
5.5. Centering 65
6. Noise ...... 69
6.1. White Noise 70
6.2. Autocorrelated Noise. 71
6.3. Dominant but Not Significant . 77
6.4. The Null Hypothesis . . . . . . 83
m.
Applications
. .....
7. Signal Detection 89
7.1. Parameter Estimation . 90
7.2. Type-One Errors . .. 94
7.3. Using the Eigenvector Shape 94
7.4. Using the Eigenvalues .... 97
7.5. Significant Oscillations. . . . 105
7.6. A Comparison with Fourier Analysis . 109
8. Filtering ........... 113
8.1. Statistical Dimension . 114
Contents xiii
8.2. Eigenvector and PC Pairs . 118
8.3. Effect of SSA on Spectra . 120
8.4. Nonlinear Trend Removal 127
9. Prediction ......... . 133
9.1. AR Model Approach. 133
9.2. Iterative Approach .. 138
9.3. Whole-Field Predictions 139
10. Phase Space Reconstruction 143
10.1. Method of Delays versus SSA 144
10.2. Estimating Dimensions . 147
10.3. Limitations .... 149
10.4. Multichannel SSA 152
References 157
Index ... 161
