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关于奇异谱分析的教材.pdf
Singular Spectrum Analysisfor Time Series
Contents
1 Introduction
1.1 Preliminaries
1.2 SSA Methodology and the Structure of the Book
1.3 SSA Topics Outside the Scope of This Book
1.4 Common Symbols and Acronyms
References
2 Basic SSA
2.1 The Main Algorithm
2.1.1 Description of the Algorithm
2.1.2 Analysis of the Four Steps in Basic SSA
2.2 Potential of Basic SSA
2.2.1 Extraction of Trends and Smoothing
2.2.2 Extraction of Periodic Components
2.2.3 Complex Trends and Periodicities with Varying Amplitudes
2.2.4 Finding Structure in Short Time Series
2.2.5 Envelopes of Oscillating Signals and Estimation of Volatility
2.3 Models of Time Series and SSA Objectives
2.3.1 SSA and Models of Time Series
2.3.2 Classification of the Main SSA Tasks
2.3.3 Separability of Components of Time Series
2.4 Choice of Parameters in Basic SSA
2.4.1 General Issues
2.4.2 Grouping for Given Window Length
2.4.3 Window Length
2.4.4 Signal Extraction
2.4.5 Automatic Identification of SSA Components
2.5 Some Variations of Basic SSA
2.5.1 Preprocessing
2.5.2 Centering in SSA
2.5.3 Stationary Series and Toeplitz SSA
2.5.4 Rotations for Separability: SSA--ICA
2.5.5 Sequential SSA
2.5.6 Computer Implementation of SSA
2.5.7 Replacing the SVD with Other Procedures
References
3 SSA for Forecasting, Interpolation, Filtration and Estimation
3.1 SSA Forecasting Algorithms
3.1.1 Main Ideas and Notation
3.1.2 Formal Description of the Algorithms
3.1.3 SSA Forecasting Algorithms: Similarities and Dissimilarities
3.1.4 Appendix: Vectors in a Subspace
3.2 LRR and Associated Characteristic Polynomials
3.2.1 Basic Facts
3.2.2 Roots of the Characteristic Polynomials
3.2.3 Min-Norm LRR
3.3 Recurrent Forecasting as Approximate Continuation
3.3.1 Approximate Separability and Forecasting Errors
3.3.2 Approximate Continuation and the Characteristic Polynomials
3.4 Confidence Bounds for the Forecast
3.4.1 Monte Carlo and Bootstrap Confidence Intervals
3.4.2 Confidence Intervals: Comparison of Forecasting Methods
3.5 Summary and Recommendations on Forecasting Parameters
3.6 Case Study: `Fortified Wine'
3.6.1 Linear Recurrence Relation Governing the Time Series
3.6.2 Choice of Forecasting Methods and Parameters
3.7 Missing Value Imputation
3.7.1 SSA for Time Series with Missing Data: Algorithm
3.7.2 Discussion
3.7.3 Example
3.8 Subspace-Based Methods and Estimation of Signal Parameters
3.8.1 Basic Facts
3.8.2 ESPRIT
3.8.3 Overview of Other Subspace-Based Methods
3.8.4 Cadzow Iterations
3.9 SSA and Filters
3.9.1 Linear Filters and Their Characteristics
3.9.2 SSA Reconstruction as a Linear Filter
3.9.3 Middle Point Filter
3.9.4 Last Point Filter and Forecasting
3.9.5 Causal SSA (Last-Point SSA)
References
SpringerBriefs in Statistics
For further volumes:
http://www.springer.com/series/8921
Nina Golyandina • Anatoly Zhigljavsky
Singular Spectrum Analysis
for Time Series
123
Nina Golyandina
Department of Mathematics
St. Petersburg University
St. Petersburg
Russia
Anatoly Zhigljavsky
School of Mathematics
Cardiff University
Cardiff
UK
ISSN 2191-544X
ISBN 978-3-642-34912-6
DOI 10.1007/978-3-642-34913-3
Springer Heidelberg New York Dordrecht London
ISSN 2191-5458 (electronic)
ISBN 978-3-642-34913-3
(eBook)
Library of Congress Control Number: 2012953018
Ó The Author(s) 2013
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 SSA Methodology and the Structure of the Book . . . . . . . . . . .
1.3 SSA Topics Outside the Scope of This Book . . . . . . . . . . . . . .
1.4 Common Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Basic SSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 The Main Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Description of the Algorithm . . . . . . . . . . . . . . . . . . . .
2.1.2 Analysis of the Four Steps in Basic SSA . . . . . . . . . . . .
2.2 Potential of Basic SSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Extraction of Trends and Smoothing . . . . . . . . . . . . . . .
2.2.2 Extraction of Periodic Components . . . . . . . . . . . . . . . .
2.2.3 Complex Trends and Periodicities
with Varying Amplitudes . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Finding Structure in Short Time Series . . . . . . . . . . . . .
2.2.5 Envelopes of Oscillating Signals and Estimation
of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Models of Time Series and SSA Objectives . . . . . . . . . . . . . . .
2.3.1 SSA and Models of Time Series . . . . . . . . . . . . . . . . . .
2.3.2 Classification of the Main SSA Tasks . . . . . . . . . . . . . .
2.3.3 Separability of Components of Time Series . . . . . . . . . .
2.4 Choice of Parameters in Basic SSA . . . . . . . . . . . . . . . . . . . . .
2.4.1 General Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Grouping for Given Window Length . . . . . . . . . . . . . . .
2.4.3 Window Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Signal Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.5 Automatic Identification of SSA Components. . . . . . . . .
1
1
3
6
8
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Contents
2.5 Some Variations of Basic SSA . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Centering in SSA . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Stationary Series and Toeplitz SSA. . . . . . . . . . . . . . . .
2.5.4 Rotations for Separability: SSA–ICA. . . . . . . . . . . . . . .
2.5.5 Sequential SSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.6 Computer Implementation of SSA. . . . . . . . . . . . . . . . .
2.5.7 Replacing the SVD with Other Procedures. . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 SSA for Forecasting, Interpolation, Filtration and Estimation . . . .
3.1 SSA Forecasting Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Main Ideas and Notation . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Formal Description of the Algorithms . . . . . . . . . . . . . .
3.1.3 SSA Forecasting Algorithms: Similarities
and Dissimilarities. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Appendix: Vectors in a Subspace . . . . . . . . . . . . . . . . .
3.2 LRR and Associated Characteristic Polynomials . . . . . . . . . . . .
3.2.1 Basic Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Roots of the Characteristic Polynomials. . . . . . . . . . . . .
3.2.3 Min-Norm LRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Recurrent Forecasting as Approximate Continuation . . . . . . . . .
3.3.1 Approximate Separability and Forecasting Errors . . . . . .
3.3.2 Approximate Continuation and the Characteristic
3.4 Confidence Bounds for the Forecast
Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
3.4.1 Monte Carlo and Bootstrap Confidence Intervals . . . . . .
3.4.2 Confidence Intervals: Comparison
of Forecasting Methods . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Summary and Recommendations on Forecasting Parameters. . . .
3.6 Case Study: ‘Fortified Wine’ . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Linear Recurrence Relation Governing
the Time Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Choice of Forecasting Methods and Parameters . . . . . . .
3.7 Missing Value Imputation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.1 SSA for Time Series with Missing Data: Algorithm . . . .
3.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Subspace-Based Methods and Estimation
of Signal Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1 Basic Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2 ESPRIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.3 Overview of Other Subspace-Based Methods . . . . . . . . .
3.8.4 Cadzow Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
3.9 SSA and Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9.1 Linear Filters and Their Characteristics . . . . . . . . . . . . .
3.9.2 SSA Reconstruction as a Linear Filter . . . . . . . . . . . . . .
3.9.3 Middle Point Filter . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9.4 Last Point Filter and Forecasting . . . . . . . . . . . . . . . . .
3.9.5 Causal SSA (Last-Point SSA). . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
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118
Chapter 1
Introduction
1.1 Preliminaries
Singular spectrum analysis (SSA) is a technique of time series analysis and
forecasting. It combines elements of classical time series analysis, multivariate sta-
tistics, multivariate geometry, dynamical systems and signal processing. SSA aims at
decomposing the original series into a sum of a small number of interpretable com-
ponents such as a slowly varying trend, oscillatory components and a ‘structureless’
noise. It is based on the singular value decomposition (SVD) of a specific matrix
constructed upon the time series. Neither a parametric model nor stationarity-type
conditions have to be assumed for the time series. This makes SSA a model-free
method and hence enables SSA to have a very wide range of applicability.
The present book is fully devoted to the methodology of SSA. It exhibits the huge
potential of SSA and shows how to use SSA both safely and with maximum effect.
Potential readers of the book. (a) Professional statisticians and econometricians;
(b) specialists in any discipline where problems of time series analysis and forecast-
ing occur; (c) specialists in signal processing and those needed to extract signals
from noisy data; (d) PhD students working on topics related to time series analy-
sis; (e) students taking appropriate MSc courses on applied time series analysis;
(f) anyone interested in the interdisciplinarity of statistics and mathematics.
Historical remarks. The first publication, which can be considered as one of
the origins of SSA (and more generally of the subspace-based methods of signal
processing), can be traced back to the eighteenth century [28].
The commencement of SSA is usually associated with publication in 1986 of the
papers [4, 5] by Broomhead and King. Since then SSA has received a fair amount
of attention in literature. Additionally to [4, 5] the list of most cited papers on SSA
published in the 1980s and 1990s includes [2, 10, 32, 33].
There are three books fully devoted to SSA, [8, 9, 14]. The book [9] is well
written but it only provides a very elementary introduction to SSA. The volume
[8] is a collection of papers written entirely by statisticians based at that time at
St.Petersburg university. All these papers are devoted to the so-called ‘Caterpillar’
N. Golyandina and A. Zhigljavsky, Singular Spectrum Analysis for Time Series,
SpringerBriefs in Statistics, DOI: 10.1007/978-3-642-34913-3_1,
© The Author(s) 2013
1
2
1 Introduction
methodology (the words ‘Caterpillar’ or ‘Gusenitsa’ is due to the association with
the moving window). This methodology is a version of SSA that was developed in
the former Soviet Union independently (the ‘iron curtain effect’) of the mainstream
SSA. The work on the ‘Caterpillar’ methodology has started long after publication
of [28] but well before 1986, the year of publication of [4] and [5].
The main difference between the main-stream SSA of [2, 4, 5, 10, 32, 33] and
the ‘Caterpillar’ SSA is not in the algorithmic details but rather in the assumptions
and in the emphasis in the study of SSA properties. To apply the mainstream SSA,
one often needs to assume some kind of stationarity of the time series and think in
terms of the ‘signal plus noise’ model (where the noise is often assumed to be ‘red’).
In the ‘Caterpillar’ SSA, the main methodological stress is on separability (of one
component of the series from another one) and neither the assumption of stationarity
nor the model in the form ‘signal plus noise’ are required.
The main methodological principles described in [8] have been further developed
in the monograph [14]. The publication of [14] has helped to attract much wider atten-
tion to SSA from the statistical circles as well as many other scientific communities.
During the last 10 years much new SSA-related research has been done and many
new successful applications of SSA have been reported. A recent special issue of
‘Statistics and Its Interface’ [35] gives an indication of how much progress in the-
oretical and methodological developments of SSA, as well as its applications, has
been achieved in recent years. The SSA community regularly organizes international
workshops on SSA. The latest SSA workshop was held in Beijing in May 2012, see
http://www.cefs.ac.cn/express/SSA.html.
The research on the theory and methodology of SSA performed in the last two
decades has resulted in a rather pleasing state of affairs: (i) the existence of an
active SSA community and (ii) the existence of a general methodology of SSA
rather than simply a collection of many different SSA algorithms. This methodology
unifies different versions of SSA into a very powerful tool of time series analysis
and forecasting. Description of SSA methodology is the sole purpose of the present
book.
Correspondence between the present book and [14]. Some entirely new topics
are included (for example, Sect. 3.7–3.9) but a few topics thoroughly described in [14]
are not considered at all (see, for example, [14, Chap. 3]). This volume is fully devoted
to the methodology of SSA unlike [14], where many theoretical issues were also
considered. The material is correspondingly revised in view of the new objectives.
The main aim of [14] is to establish SSA as a serious subject. There is no need to do
it now and the aspiration of this book is to show the power and beauty of SSA to as
wide audience as possible.
Several reasons why SSA is still not very popular among statisticians. First
reason is tradition: SSA is not a classical statistical method, and therefore many
people are simply not aware of it. Second, SSA demands more computing power than
the traditional methods.Third, many people prefer model-based statistical techniques
where calculations are automatic and do not require the computer-analyst interaction.
Finally, SSA is sometimes too flexible (especially when analyzing multivariate series)
and therefore has too many options which are difficult to formalize.
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