Singular Spectrum Analysis for Time Series

Singular Spectrum Analysis for Time Series

Nina Golyandina
Anatoly Zhigljavsky

Springer, 2013 (Chapter 2 is available as a sample chapter)

Description

  • Presents the methodology of SSA
  • Shows how to use SSA both safely and with maximum effect
  • For professional statisticians, econometricians and specialists in any discipline
  • For students taking courses on applied time series analysis

Correspondence between the present book and "Analysis of time series structure: SSA and related techniques" (2001):

Some entirely new topics are included (for example, SSA as a filter, SSA and subspace-based methods, SSA and Independent Component Analysis, missing data imputation) but a few topics thoroughly described in the book (2001) are not considered at all (e.g. change point detection). This volume is fully devoted to the methodology of SSA unlike the book (2001), where many theoretical issues were also considered. The material is correspondingly revised in view of the new objectives. The main aim of the book (2001) 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.

Contents

1Introduction1
 1.1    Preliminaries 1
 1.2    SSA Methodology and the Structure of the Book3
 1.3    SSA Topics Outside the Scope of This Book 6
 1.4    Common Symbols and Acronyms 8
2Basic SSA11
 2.1    The Main Algorithm 11
           2.1.1 Description of the Algorithm 11
           2.1.2 Analysis of the Four Steps in Basic SSA 13
 2.2    Potential of Basic SSA19
           2.2.1  Extraction of Trends and Smoothing 19
           2.2.2  Extraction of Periodic Components 21
           2.2.3  Complex Trends and Periodicities with Varying Amplitudes 22
           2.2.4  Finding Structure in Short Time Series 23
           2.2.5  Envelopes of Oscillating Signals and Estimation of Volatility 24
 2.3    Models of Time Series and SSA Objectives 25
           2.3.1  SSA and Models of Time Series 25
           2.3.2  Classification of the Main SSA Tasks 35
           2.3.3  Separability of Components of Time Series 37
 2.4    Choice of Parameters in Basic SSA 39
           2.4.1  General Issues 39
           2.4.2  Grouping for Given Window Length 43
           2.4.3  Window Length 47
           2.4.4  Signal Extraction 53
           2.4.5  Automatic Identification of SSA Components 54
 2.5    Some Variations of Basic SSA 58
           2.5.1  Preprocessing 58
           2.5.2  Centering in SSA 59
           2.5.3  Stationary Series and Toeplitz SSA 60
           2.5.4  Rotations for Separability: SSA–ICA 61
           2.5.5  Sequential SSA 65
           2.5.6  Computer Implementation of SSA 67
           2.5.7  Replacing the SVD with Other Procedures 68
3SSA for Forecasting, Interpolation, Filtration and Estimation71
 3.1    SSA Forecasting Algorithms71
           3.1.1  Main Ideas and Notation 71
           3.1.2  Formal Description of the Algorithms 73
           3.1.3  SSA Forecasting Algorithms: Similarities and Dissimilarities 75
           3.1.4  Appendix: Vectors in a Subspace 77
 3.2    LRR and Associated Characteristic Polynomials 78
           3.2.1  Basic Facts 78
           3.2.2  Roots of the Characteristic Polynomials 79
           3.2.3  Min-Norm LRR 80
 3.3    Recurrent Forecasting as Approximate Continuation 83
           3.3.1  Approximate Separability and Forecasting Errors 83
           3.3.2  Approximate Continuation and the Characteristic Polynomials 84
 3.4    Confidence Bounds for the Forecast 86
           3.4.1  Monte Carlo and Bootstrap Confidence Intervals 87
           3.4.2  Confidence Intervals: Comparison of Forecasting Methods 89
 3.5    Summary and Recommendations on Forecasting Parameters90
 3.6    Case Study: ‘Fortified Wine’ 94
           3.6.1  Linear Recurrence Relation Governing the Time Series 94
           3.6.2  Choice of Forecasting Methods and Parameters 96
 3.7    Missing Value Imputation98
           3.7.1  SSA for Time Series with Missing Data: Algorithm 99
           3.7.2  Discussion 102
           3.7.3  Example 102
 3.8    Subspace-Based Methods and Estimation of Signal Parameters 104
           3.8.1  Basic Facts 105
           3.8.2  ESPRIT 106
           3.8.3  Overview of Other Subspace-Based Methods 108
           3.8.4  Cadzow Iterations 110
 3.9    SSA and Filters111
           3.9.1 Linear Filters and Their Characteristics 111
           3.9.2 SSA Reconstruction as a Linear Filter 112
           3.9.3 Middle Point Filter 113
           3.9.4 Last Point Filter and Forecasting 116
           3.9.5 Causal SSA (Last-Point SSA) 116

You can buy the book on Amazon.com or on the Springer site and in eBook format.


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