Climate Time Series Analysis (eBook)

Manfred Mudelsee received his diploma in Physics from the University of Heidelberg and his doctoral degree in Geology from the University of Kiel. He was then postdoc in Statistics at the University of Kent at Canterbury, research scientist in Meteorology at the University of Leipzig and visiting scholar in Earth Sciences at Boston University; currently he does climate research at the Alfred Wegener Institute for Polar and Marine Research, Bremerhaven. His science focuses on climate extremes, time series analysis and mathematical simulation methods. He has authored over 50 peer-reviewed articles. In his 2003 Nature paper, Mudelsee introduced the bootstrap method to flood risk analysis. In 2005, he founded the company Climate Risk Analysis.

Preface 6
Acknowledgements 10
List of Algorithms 19
List of Figures 22
List of Tables 27
Part I Fundamental Concepts 31
1. Introduction 32
1.1 Climate archives, variables and dating 34
1.2 Noise and statistical distribution 35
1.3 Persistence 40
1.4 Spacing 43
1.5 Aim and structure of this book 53
1.6 Background material 55
2. Persistence Models 62
2.1 First-order autoregressive model 62
2.1.1 Even spacing 63
2.1.1.1 Effective data size 65
2.1.2 Uneven spacing 66
2.1.2.1 Embedding in continuous time 67
2.2 Second-order autoregressive model 68
2.3 Mixed autoregressive moving average model 70
2.4 Other models 71
2.4.1 Long-memory processes 71
2.4.2 Nonlinear and non-Gaussian models 72
2.5 Climate theory 73
2.5.1 Stochastic climate models 74
2.5.2 Long memory of temperature fluctuations? 76
2.5.3 Long memory of river runoff 80
2.6 Background material 83
2.7 Technical issues 92
3. Bootstrap Confidence Intervals 94
3.1 Error bars and confidence intervals 95
3.1.1 Theoretical example: mean estimation of Gaussian white noise 97
3.1.2 Theoretical example: standard deviation estimation of Gaussian white noise 98
3.1.3 Real world 100
3.2 Bootstrap principle 103
3.3 Bootstrap resampling 105
3.3.1 Nonparametric: moving block bootstrap 107
3.3.1.1 Block length selection 107
3.3.1.2 Uneven spacing 109
3.3.1.3 Systematic model parts and nonstationarity 110
3.3.2 Parametric: autoregressive bootstrap 112
3.3.2.1 Even spacing 112
3.3.2.2 Uneven spacing 112
3.3.3 Parametric: surrogate data 115
3.4 Bootstrap confidence intervals 115
3.4.1 Normal confidence interval 116
3.4.2 Student's t confidence interval 117
3.4.3 Percentile confidence interval 117
3.4.4 BCa confidence interval 117
3.5 Examples 118
3.6 Bootstrap hypothesis tests 120
3.7 Notation 123
3.8 Background material 128
3.9 Technical issues 135
Part II Univariate Time Series 140
4. Regression I 141
4.1 Linear regression 142
4.1.1 Weighted least-squares and ordinary least-squares estimation 142
4.1.1.1 Example: Arctic river runoff 143
4.1.2 Generalized least-squares estimation 144
4.1.3 Other estimation types 146
4.1.4 Classical confidence intervals 147
4.1.4.1 Prais--Winsten procedure 149
4.1.4.2 Cochrane--Orcutt transformation 149
4.1.4.3 Approach via effective data size 151
4.1.5 Bootstrap confidence intervals 152
4.1.6 Monte Carlo experiments: ordinary least-squares estimation 152
4.1.7 Timescale errors 157
4.1.7.1 Nonparametric: pairwise-moving block bootstrap 159
4.1.7.2 Parametric: timescale-autoregressive bootstrap 159
4.1.7.3 Hybrid: timescale-moving block bootstrap 164
4.1.7.4 Monte Carlo experiments 164
4.2 Nonlinear regression 169
4.2.1 Climate transition model: ramp 170
4.2.1.1 Estimation 171
4.2.1.2 Example: Northern Hemisphere Glaciation 173
4.2.1.3 Bootstrap confidence intervals 174
4.2.1.4 Example: onset of Dansgaard--Oeschger event 5 174
4.2.2 Trend-change model: break 178
4.2.2.1 Estimation 178
4.2.2.2 Example: Arctic river runoff (continued) 180
4.2.2.3 Bootstrap confidence intervals 180
4.3 Nonparametric regression or smoothing 181
4.3.1 Kernel estimation 181
4.3.2 Bootstrap confidence intervals and bands 184
4.3.3 Extremes or outlier detection 185
4.3.3.1 Example: volcanic peaks in the NGRIP sulfate record 187
4.3.3.2 Example: hurricane peaks in the Lower Mystic Lake varve thickness record 187
4.4 Background material 189
4.5 Technical issues 201
5. Spectral Analysis 205
5.1 Spectrum 205
5.1.1 Example: AR(1) process, discrete time 208
5.1.2 Example: AR(2) process, discrete time 208
5.1.3 Physical meaning 209
5.2 Spectral estimation 211
5.2.1 Periodogram 211
5.2.2 Welch's Overlapped Segment Averaging 214
5.2.3 Multitaper estimation 216
5.2.3.1 F test 218
5.2.3.2 Weighted eigenspectra 219
5.2.3.3 Zero padding 220
5.2.3.4 Jackknife 220
5.2.3.5 Advanced topics: CI coverage accuracy and uneven spacing 222
5.2.3.6 Example: radiocarbon spectrum 223
5.2.4 Lomb--Scargle estimation 224
5.2.4.1 Bias correction 225
5.2.4.2 Covariance 227
5.2.4.3 Harmonic filter 227
5.2.4.4 Advanced topics: degrees of freedom, bandwidth, oversampling and highest frequency 229
5.2.5 Peak detection: red-noise hypothesis 230
5.2.5.1 Multiple tests 230
5.2.6 Example: peaks in monsoon spectrum 233
5.2.7 Aliasing 233
5.2.8 Timescale errors 236
5.2.9 Example: peaks in monsoon spectrum (continued) 237
5.3 Background material 243
5.4 Technical issues 253
6. Extreme Value Time Series 256
6.1 Data types 256
6.1.1 Event times 257
6.1.1.1 Example: Elbe winter floods 257
6.1.2 Peaks over threshold 257
6.1.2.1 Example: volcanic peaks in the NGRIP sulfate record (continued) 258
6.1.3 Block extremes 258
6.1.4 Remarks on data selection 259
6.2 Stationary models 259
6.2.1 Generalized Extreme Value distribution 259
6.2.1.1 Model 260
6.2.1.2 Maximum likelihood estimation 260
6.2.2 Generalized Pareto distribution 262
6.2.2.1 Model 262
6.2.2.2 Maximum likelihood estimation 262
6.2.2.3 Model suitability 264
6.2.2.4 Return period 265
6.2.2.5 Probability weighted moment estimation 266
6.2.3 Bootstrap confidence intervals 267
6.2.4 Example: Elbe summer floods, 1852 to 2002 268
6.2.5 Persistence 270
6.2.5.1 Condition D(un) 270
6.2.5.2 Extremal index 270
6.2.5.3 Long memory 271
6.2.6 Remark: tail estimation 271
6.2.7 Remark: optimal estimation 273
6.3 Nonstationary models 273
6.3.1 Time-dependent Generalized Extreme Value distribution 274
6.3.2 Inhomogeneous Poisson process 275
6.3.2.1 Model 275
6.3.2.2 Nonparametric occurrence rate estimation 276
6.3.2.3 Boundary bias reduction 277
6.3.2.4 Bandwidth selection 278
6.3.2.5 Example: Elbe winter floods (continued) 279
6.3.2.6 Bootstrap confidence band 280
6.3.2.7 Example: Elbe winter floods (continued) 283
6.3.2.8 Example: volcanic peaks in the NGRIP sulfate record (continued) 284
6.3.2.9 Example: hurricane peaks in the Lower Mystic Lake varve thickness record (continued) 284
6.3.2.10 Parametric Poisson models and hypothesis tests 285
6.3.2.11 Monte Carlo experiment: Cox--Lewis test versus Mann--Kendall test 287
6.3.3 Hybrid: Poisson--extreme value distribution 291
6.4 Sampling and time spacing 293
6.5 Background material 296
6.6 Technical issues 306
Part III Bivariate Time Series 310
7. Correlation 311
7.1 Pearson's correlation coefficient 312
7.1.1 Remark: alternative correlation measures 313
7.1.2 Classical confidence intervals, non-persistent processes 313
7.1.3 Bivariate time series models 315
7.1.3.1 Bivariate white noise 315
7.1.3.2 Bivariate first-order autoregressive process 316
7.1.4 Classical confidence intervals, persistent processes 317
7.1.5 Bootstrap confidence intervals 319
7.1.5.1 Pairwise-moving block bootstrap 319
7.1.5.2 Pairwise-autoregressive bootstrap 321
7.2 Spearman's rank correlation coefficient 321
7.2.1 Classical confidence intervals, non-persistent processes 324
7.2.2 Classical confidence intervals, persistent processes 326
7.2.3 Bootstrap confidence intervals 327
7.2.3.1 Pairwise-moving block bootstrap 327
7.2.3.2 Pairwise-autoregressive bootstrap 328
7.3 Monte Carlo experiments 328
7.4 Example: Elbe runoff variations 335
7.5 Unequal timescales 337
7.5.1 Binned correlation 338
7.5.2 Synchrony correlation 340
7.5.3 Monte Carlo experiments 342
7.5.3.1 Optimal estimation 347
7.5.4 Example: Vostok ice core records 348
7.6 Background material 349
7.7 Technical issues 364
8. Regression II 365
8.1 Linear regression 366
8.1.1 Ordinary least-squares estimation 366
8.1.1.1 Bias correction 367
8.1.1.2 Prior knowledge about standard deviations 367
8.1.2 Weighted least-squares for both variables estimation 369
8.1.2.1 Prior knowledge about standard deviation ratio 369
8.1.2.2 Geometric interpretation 370
8.1.3 Wald--Bartlett procedure 371
8.2 Bootstrap confidence intervals 372
8.2.1 Simulating incomplete prior knowledge 374
8.3 Monte Carlo experiments 376
8.3.1 Easy setting 376
8.3.2 Realistic setting: incomplete prior knowledge 379
8.3.3 Dependence on accuracy of prior knowledge 381
8.3.4 Mis-specified prior knowledge 383
8.4 Example: climate sensitivity 385
8.5 Prediction 388
8.5.1 Example: calibration of a proxy variable 390
8.6 Lagged regression 393
8.6.1 Example: CO2 and temperature variations in the Pleistocene 394
8.7 Background material 399
8.8 Technical issues 405
Part IV Outlook 407
9. Future Directions 408
9.1 Timescale modelling 408
9.2 Novel estimation problems 409
9.3 Higher dimensions 410
9.4 Climate models 410
9.4.1 Fitting climate models to observations 412
9.4.2 Forecasting with climate models 413
9.4.3 Design of the cost function 414
9.4.4 Climate model bias 415
9.5 Optimal estimation 416
References 418
Subject Index 478
Author Index 489