Over the last 15 years, singular spectrum analysis (SSA) has proven very successful. It has already become a standard tool in climatic and meteorological time series analysis and well known in nonlinear physics and signal processing. However, despite the promise it holds for time series applications in other disciplines, SSA is not widely known among statisticians and econometrists, and although the basic SSA algorithm looks simple, understanding what it does and where its pitfalls lay is by no means simple.Analysis of Time Series Structure: SSA and Related Techniques provides a careful, lucid description of its general theory and methodology. Part I introduces the basic concepts, and sets forth the main findings and results, then presents a detailed treatment of the methodology. After introducing the basic SSA algorithm, the authors explore forecasting and apply SSA ideas to change-point detection algorithms. Part II is devoted to the theory of SSA. Here the authors formulate and prove the statements of Part I. They address the singular value decomposition (SVD) of real matrices, time series of finite rank, and SVD of trajectory matrices.Based on the authors' original work and filled with applications illustrated with real data sets, this book offers an outstanding opportunity to obtain a working knowledge of why, when, and how SSA works. It builds a strong foundation for successfully using the technique in applications ranging from mathematics and nonlinear physics to economics, biology, oceanology, social science, engineering, financial econometrics, and market research.
Author(s): Nina Golyandina, Vladimir Nekrutkin, Anatoly A Zhigljavsky
Edition: 1
Year: 2001
Language: English
Pages: 309
Cover Page......Page 1
Title Page......Page 2
Analysis of Time Series Structure: SSA and Related Techniques......Page 3
Preface......Page 10
Notation......Page 12
SSA analysis of time series......Page 14
SSA forecasting of time series......Page 17
SSA detection of structural changes in time series......Page 20
Composition of the book......Page 22
1. Multichannel SSA......Page 23
4. SSA for sequential detection of structural changes......Page 24
Contents......Page 8
PART I SSA: Methodology......Page 26
CHAPTER 1: Basic SSA......Page 27
1st step: Embedding......Page 28
3rd step. Grouping......Page 29
1.2.1 Embedding......Page 30
1.2.2 Singular value decomposition......Page 31
1.2.3 Grouping......Page 34
1.2.4 Diagonal averaging......Page 35
1.3 Basic SSA: basic capabilities......Page 36
1.3.2 Smoothing......Page 37
1.3.3 Extraction of seasonality components......Page 39
1.3.4 Extraction of cycles with small and large periods......Page 40
1.3.6 Complex trends and periodicities......Page 41
1.3.7 Finding structure in short time series......Page 43
1.4.1 Models of time series and the periodograms......Page 44
(a) Stationary series......Page 45
(b) Amplitude-modulated periodicities......Page 48
(c) Trends......Page 50
1.4.2 Basic SSA: Classification of the main tasks......Page 52
1. Trend extraction and smoothing......Page 53
2. Extraction of oscillatory components......Page 54
3. Obtaining the refined structure of a series......Page 55
1.5 Separability......Page 56
1.5.1 Weak and strong separability......Page 57
1.5.2 Approximate and asymptotic separability......Page 59
1.5.3 Separability and Fourier expansions......Page 60
1.5.4 Strong separability......Page 63
(a) General issues......Page 65
1. Trends......Page 67
2. Smoothing......Page 68
1. Harmonic series......Page 69
2. Grouping for identification of a general periodic component......Page 72
3. Modulated periodicities......Page 74
(d) Grouping for finding a refined decomposition of a series......Page 77
1. Singular values......Page 78
2. w-Correlations......Page 79
1.6.2 Window length effects......Page 80
(a) General effects......Page 81
(i) Trends: reliable separation......Page 82
(ii) Trends: difficult case......Page 83
2. Smoothing......Page 84
(c) Window length for periodicities......Page 86
(d) Refined structure......Page 88
(e) Hints......Page 89
1.7 Supplementary SSA techniques......Page 90
(a) Single and Double centring......Page 91
(b) Double centring and linear regression......Page 93
1.7.2 Stationary series and Toeplitz SSA......Page 96
1.7.3 Close singular values......Page 99
(a) Series modification......Page 100
(b) Sequential SSA......Page 101
1.7.4 Envelopes of highly oscillating signals......Page 103
CHAPTER 2: SSA forecasting......Page 105
SSA recurrent forecasting algorithm......Page 107
2.2.1 Linear recurrent formulae and their characteristic polynomials......Page 108
(a) Series governed by linear recurrent formulae......Page 109
(b) Characteristic polynomials and their roots......Page 110
(a) L-continuation......Page 113
(b) Recurrent continuation and Basic SSA forecasting......Page 114
(a) Approximate separability and forecasting errors......Page 115
(b) Approximate continuation and the characteristic polynomials......Page 118
2.3.1 SSA vector forecasting......Page 119
Preliminaries......Page 120
(c) Details......Page 121
2.3.2 Toeplitz SSA forecasting......Page 122
2.3.3 Centring in SSA forecasting......Page 123
(a) Minimal recurrent formula: Schubert and reduction methods......Page 125
(b) The nearest subspace......Page 126
2.4 Forecast confidence bounds......Page 127
2.4.1 Empirical confidence intervals for the forecast of the initial series......Page 128
2.4.2 Bootstrap confidence bounds for the forecast of a signal......Page 129
2.4.3 Confidence intervals: comparison of forecasting variants......Page 130
(a) Periodic signal: recurrent and vector forecasting......Page 131
(b) Periodic signal: Basic and Toeplitz recurrent forecasting......Page 133
(c) Separability and forecasting......Page 134
(d) Confidence intervals of different kinds......Page 136
4. Specifics and dangers......Page 139
6. The role of the initial data......Page 140
7. Reconstructed series and LRFs......Page 141
9. Confidence intervals......Page 142
2.6.1 ‘Wages’: Forecast of the exponential tendency......Page 143
2.6.2 ‘Eggs’: Minimal LRF......Page 147
2.6.3 ‘Precipitation’: Toeplitz forecasting......Page 148
2.6.4 ‘Fortified wine’: Vector and recurrent forecasting......Page 150
2.6.5 ‘Gold price’: Confidence intervals and forecast stability......Page 154
3.1 Main definitions and concepts......Page 160
(a) Heterogeneity matrix......Page 161
1. Row heterogeneity functions......Page 162
4. Symmetric heterogeneity function......Page 163
(a) Structural changes and heterogeneity functions......Page 164
(b) Types of detection function......Page 165
4. Symmetric detection function......Page 166
3.2 Homogeneity and heterogeneity......Page 167
3.2.1 Types of single heterogeneity......Page 168
(a) Permanent violation (‘tile-structure’ matrices)......Page 169
(b) Temporary violation (‘cross-structure’ matrices)......Page 175
3.2.2 Multiple heterogeneity......Page 178
3.3 Heterogeneity and separability......Page 180
(a) Stable separability......Page 181
(i) Deviations from weak separability.......Page 183
(ii) Rearrangement of the eigentriples.......Page 187
(a) Stable separability......Page 192
(b) Deviations from stable separability......Page 196
3.4 Choice of detection parameters......Page 200
(b) Nonzero F(2) N : identification......Page 201
(c) Small noisy-like F(2) N......Page 202
(d) General F(2) N......Page 203
3.4.3 Multiple heterogeneity......Page 204
(a) Trends......Page 205
(b) Periodicities......Page 206
3.5.1 Renormalized heterogeneity matrices......Page 207
3.5.2 Roots of characteristic polynomials......Page 210
3.5.3 Moving periodograms......Page 213
3.6.1 ‘Coal sales’: detection of outliers......Page 215
3.6.2 ‘Petroleum sales’: detection of trend changes......Page 217
3.6.3 ‘Sunspots’: detection of changes in amplitudes and frequencies......Page 220
3.6.4 ‘Demands’: rearrangement or heterogeneity?......Page 222
PART II SSA: Theory......Page 227
4.1 Existence and uniqueness......Page 228
4.2 SVD matrices......Page 231
4.2.1 Matrix orthogonal decompositions and SVD......Page 232
4.3 Optimality of SVDs......Page 236
4.4.1 Single centring......Page 241
4.4.2 Double centring......Page 244
5.1 General properties......Page 246
3. A series that does not have a finite rank......Page 247
5.2 Series of finite rank and recurrent formulae......Page 252
5.3 Time series continuation......Page 261
6.1.1 Definition and examples......Page 266
6.1.2 Approximate and asymptotic separability......Page 269
6.1.3 Separation of a signal from noise......Page 273
6.2 Hankelization......Page 275
6.3.1 Single centring SSA and the constant series......Page 277
(a) Separation of a constant series......Page 278
(b) Asymptotic separation of a constant series......Page 279
(c) Stochastic separability......Page 280
6.3.2 Double centring and linear series......Page 281
(a) Separation of a linear series......Page 282
(c) Stochastic separability......Page 283
6.4 SSA for stationary series......Page 285
6.4.1 Spectral representation of stationary series......Page 286
(a) Periodic series......Page 293
(b) Almost periodic sequences......Page 294
6.4.3 SVD for stationary series......Page 297
(a) SVD for almost periodic sequences......Page 298
(b) SVD for aperiodic sequences......Page 299
(c) Summary......Page 301
6.4.4 Separability of stationary series......Page 302
6.4.5 Periodograms......Page 303
List of data sets and their sources......Page 305
References......Page 306
