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Climate is a paradigm of a complex system. Analysing climate data is an exciting challenge, which is increased by non-normal distributional shape, serial dependence, uneven spacing and timescale uncertainties. This book presents bootstrap resampling as a computing-intensive method able to meet the challenge. It shows the bootstrap to perform reliably in the most important statistical estimation techniques: regression, spectral analysis, extreme values and correlation. This book is written for climatologists and applied statisticians. It explains step by step the bootstrap algorithms (including novel adaptions) and methods for confidence interval construction. It tests the accuracy of the algorithms by means of Monte Carlo experiments. It analyses a large array of climate time series, giving a detailed account on the data and the associated climatological questions. This makes the book self-contained for graduate students and researchers.

1;Preface;6 2;Acknowledgements ;10 3;List of Algorithms ;19 4;List of Figures ;22 5;List of Tables ;27 6;Part I Fundamental Concepts;31 6.1;1. Introduction;32 6.1.1;1.1 Climate archives, variables and dating ;34 6.1.2;1.2 Noise and statistical distribution ;35 6.1.3;1.3 Persistence ;40 6.1.4;1.4 Spacing ;43 6.1.5;1.5 Aim and structure of this book ;53 6.1.6;1.6 Background material ;55 6.2;2. Persistence Models;62 6.2.1;2.1 First-order autoregressive model ;62 6.2.1.1;2.1.1 Even spacing ;63 6.2.1.1.1;2.1.1.1 Effective data size ;65 6.2.1.2;2.1.2 Uneven spacing ;66 6.2.1.2.1;2.1.2.1 Embedding in continuous time ;67 6.2.2;2.2 Second-order autoregressive model ;68 6.2.3;2.3 Mixed autoregressive moving average model ;70 6.2.4;2.4 Other models ;71 6.2.4.1;2.4.1 Long-memory processes ;71 6.2.4.2;2.4.2 Nonlinear and non-Gaussian models ;72 6.2.5;2.5 Climate theory ;73 6.2.5.1;2.5.1 Stochastic climate models ;74 6.2.5.2;2.5.2 Long memory of temperature fluctuations? ;76 6.2.5.3;2.5.3 Long memory of river runoff ;80 6.2.6;2.6 Background material ;83 6.2.7;2.7 Technical issues ;92 6.3;3. Bootstrap Confidence Intervals;94 6.3.1;3.1 Error bars and confidence intervals ;95 6.3.1.1;3.1.1 Theoretical example: mean estimation of Gaussian white noise ;97 6.3.1.2;3.1.2 Theoretical example: standard deviation estimation of Gaussian white noise ;98 6.3.1.3;3.1.3 Real world ;100 6.3.2;3.2 Bootstrap principle ;103 6.3.3;3.3 Bootstrap resampling ;105 6.3.3.1;3.3.1 Nonparametric: moving block bootstrap ;107 6.3.3.1.1;3.3.1.1 Block length selection ;107 6.3.3.1.2;3.3.1.2 Uneven spacing ;109 6.3.3.1.3;3.3.1.3 Systematic model parts and nonstationarity ;110 6.3.3.2;3.3.2 Parametric: autoregressive bootstrap ;112 6.3.3.2.1;3.3.2.1 Even spacing ;112 6.3.3.2.2;3.3.2.2 Uneven spacing ;112 6.3.3.3;3.3.3 Parametric: surrogate data ;115 6.3.4;3.4 Bootstrap confidence intervals ;115 6.3.4.1;3.4.1 Normal confidence interval ;116 6.3.4.2;3.4.2 Student's t confidence interval ;117 6.3.4.3;3.4.3 Percentil
e confidence interval ;117 6.3.4.4;3.4.4 BCa confidence interval ;117 6.3.5;3.5 Examples ;118 6.3.6;3.6 Bootstrap hypothesis tests ;120 6.3.7;3.7 Notation ;123 6.3.8;3.8 Background material ;128 6.3.9;3.9 Technical issues ;135 7;Part II Univariate Time Series;140 7.1;4. Regression I;141 7.1.1;4.1 Linear regression ;142 7.1.1.1;4.1.1 Weighted least-squares and ordinary least-squares estimation ;142 7.1.1.1.1;4.1.1.1 Example: Arctic river runoff ;143 7.1.1.2;4.1.2 Generalized least-squares estimation ;144 7.1.1.3;4.1.3 Other estimation types ;146 7.1.1.4;4.1.4 Classical confidence intervals ;147 7.1.1.4.1;4.1.4.1 Prais--Winsten procedure ;149 7.1.1.4.2;4.1.4.2 Cochrane--Orcutt transformation ;149 7.1.1.4.3;4.1.4.3 Approach via effective data size ;151 7.1.1.5;4.1.5 Bootstrap confidence intervals ;152 7.1.1.6;4.1.6 Monte Carlo experiments: ordinary least-squares estimation ;152 7.1.1.7;4.1.7 Timescale errors ;157 7.1.1.7.1;4.1.7.1 Nonparametric: pairwise-moving block bootstrap ;159 7.1.1.7.2;4.1.7.2 Parametric: timescale-autoregressive bootstrap ;159 7.1.1.7.3;4.1.7.3 Hybrid: timescale-moving block bootstrap ;164 7.1.1.7.4;4.1.7.4 Monte Carlo experiments ;164 7.1.2;4.2 Nonlinear regression ;169 7.1.2.1;4.2.1 Climate transition model: ramp ;170 7.1.2.1.1;4.2.1.1 Estimation ;171 7.1.2.1.2;4.2.1.2 Example: Northern Hemisphere Glaciation ;173 7.1.2.1.3;4.2.1.3 Bootstrap confidence intervals ;174 7.1.2.1.4;4.2.1.4 Example: onset of Dansgaard--Oeschger event 5 ;174 7.1.2.2;4.2.2 Trend-change model: break ;178 7.1.2.2.1;4.2.2.1 Estimation ;178 7.1.2.2.2;4.2.2.2 Example: Arctic river runoff (continued) ;180 7.1.2.2.3;4.2.2.3 Bootstrap confidence intervals ;180 7.1.3;4.3 Nonparametric regression or smoothing ;181 7.1.3.1;4.3.1 Kernel estimation ;181 7.1.3.2;4.3.2 Bootstrap confidence intervals and bands ;184 7.1.3.3;4.3.3 Extremes or outlier detection ;185 7.1.3.3.1;4.3.3.1 Example: volcanic peaks in the NGRIP sulfate record ;187 7.1.3.3.2;4.3.3.2 Example: hurricane peaks in th
e Lower Mystic Lake varve thickness record ;187 7.1.4;4.4 Background material ;189 7.1.5;4.5 Technical issues ;201 7.2;5. Spectral Analysis;205 7.2.1;5.1 Spectrum ;205 7.2.1.1;5.1.1 Example: AR(1) process, discrete time ;208 7.2.1.2;5.1.2 Example: AR(2) process, discrete time ;208 7.2.1.3;5.1.3 Physical meaning ;209 7.2.2;5.2 Spectral estimation ;211 7.2.2.1;5.2.1 Periodogram ;211 7.2.2.2;5.2.2 Welch's Overlapped Segment Averaging ;214 7.2.2.3;5.2.3 Multitaper estimation ;216 7.2.2.3.1;5.2.3.1 F test ;218 7.2.2.3.2;5.2.3.2 Weighted eigenspectra ;219 7.2.2.3.3;5.2.3.3 Zero padding ;220 7.2.2.3.4;5.2.3.4 Jackknife ;220 7.2.2.3.5;5.2.3.5 Advanced topics: CI coverage accuracy and uneven spacing ;222 7.2.2.3.6;5.2.3.6 Example: radiocarbon spectrum ;223 7.2.2.4;5.2.4 Lomb--Scargle estimation ;224 7.2.2.4.1;5.2.4.1 Bias correction ;225 7.2.2.4.2;5.2.4.2 Covariance ;227 7.2.2.4.3;5.2.4.3 Harmonic filter ;227 7.2.2.4.4;5.2.4.4 Advanced topics: degrees of freedom, bandwidth, oversampling and highest frequency ;229 7.2.2.5;5.2.5 Peak detection: red-noise hypothesis ;230 7.2.2.5.1;5.2.5.1 Multiple tests ;230 7.2.2.6;5.2.6 Example: peaks in monsoon spectrum ;233 7.2.2.7;5.2.7 Aliasing ;233 7.2.2.8;5.2.8 Timescale errors ;236 7.2.2.9;5.2.9 Example: peaks in monsoon spectrum (continued) ;237 7.2.3;5.3 Background material ;243 7.2.4;5.4 Technical issues ;253 7.3;6. Extreme Value Time Series;256 7.3.1;6.1 Data types ;256 7.3.1.1;6.1.1 Event times ;257 7.3.1.1.1;6.1.1.1 Example: Elbe winter floods ;257 7.3.1.2;6.1.2 Peaks over threshold ;257 7.3.1.2.1;6.1.2.1 Example: volcanic peaks in the NGRIP sulfate record (continued) ;258 7.3.1.3;6.1.3 Block extremes ;258 7.3.1.4;6.1.4 Remarks on data selection ;259 7.3.2;6.2 Stationary models ;259 7.3.2.1;6.2.1 Generalized Extreme Value distribution ;259 7.3.2.1.1;6.2.1.1 Model ;260 7.3.2.1.2;6.2.1.2 Maximum likelihood estimation ;260 7.3.2.2;6.2.2 Generalized Pareto distribution ;262 7.3.2.2.1;6.2.2.1 Model ;262 7.3.2.2.2;6.2.2.2 Maximum likel
ihood estimation ;262 7.3.2.2.3;6.2.2.3 Model suitability ;264 7.3.2.2.4;6.2.2.4 Return period ;265 7.3.2.2.5;6.2.2.5 Probability weighted moment estimation ;266 7.3.2.3;6.2.3 Bootstrap confidence intervals ;267 7.3.2.4;6.2.4 Example: Elbe summer floods, 1852 to 2002 ;268 7.3.2.5;6.2.5 Persistence ;270 7.3.2.5.1;6.2.5.1 Condition D(un) ;270 7.3.2.5.2;6.2.5.2 Extremal index ;270 7.3.2.5.3;6.2.5.3 Long memory ;271 7.3.2.6;6.2.6 Remark: tail estimation ;271 7.3.2.7;6.2.7 Remark: optimal estimation ;273 7.3.3;6.3 Nonstationary models ;273 7.3.3.1;6.3.1 Time-dependent Generalized Extreme Value distribution ;274 7.3.3.2;6.3.2 Inhomogeneous Poisson process ;275 7.3.3.2.1;6.3.2.1 Model ;275 7.3.3.2.2;6.3.2.2 Nonparametric occurrence rate estimation ;276 7.3.3.2.3;6.3.2.3 Boundary bias reduction ;277 7.3.3.2.4;6.3.2.4 Bandwidth selection ;278 7.3.3.2.5;6.3.2.5 Example: Elbe winter floods (continued) ;279 7.3.3.2.6;6.3.2.6 Bootstrap confidence band ;280 7.3.3.2.7;6.3.2.7 Example: Elbe winter floods (continued) ;283 7.3.3.2.8;6.3.2.8 Example: volcanic peaks in the NGRIP sulfate record (continued) ;284 7.3.3.2.9;6.3.2.9 Example: hurricane peaks in the Lower Mystic Lake varve thickness record (continued) ;284 7.3.3.2.10;6.3.2.10 Parametric Poisson models and hypothesis tests ;285 7.3.3.2.11;6.3.2.11 Monte Carlo experiment: Cox--Lewis test versus Mann--Kendall test ;287 7.3.3.3;6.3.3 Hybrid: Poisson--extreme value distribution ;291 7.3.4;6.4 Sampling and time spacing ;293 7.3.5;6.5 Background material ;296 7.3.6;6.6 Technical issues ;306 8;Part III Bivariate Time Series;310 8.1;7. Correlation;311 8.1.1;7.1 Pearson's correlation coefficient ;312 8.1.1.1;7.1.1 Remark: alternative correlation measures ;313 8.1.1.2;7.1.2 Classical confidence intervals, non-persistent processes ;313 8.1.1.3;7.1.3 Bivariate time series models ;315 8.1.1.3.1;7.1.3.1 Bivariate white noise ;315 8.1.1.3.2;7.1.3.2 Bivariate first-order autoregressive process ;316 8.1.1.4;7.1.4 Classical confidence intervals
, persistent processes ;317 8.1.1.5;7.1.5 Bootstrap confidence intervals ;319 8.1.1.5.1;7.1.5.1 Pairwise-moving block bootstrap ;319 8.1.1.5.2;7.1.5.2 Pairwise-autoregressive bootstrap ;321 8.1.2;7.2 Spearman's rank correlation coefficient ;321 8.1.2.1;7.2.1 Classical confidence intervals, non-persistent processes ;324 8.1.2.2;7.2.2 Classical confidence intervals, persistent processes ;326 8.1.2.3;7.2.3 Bootstrap confidence intervals ;327 8.1.2.3.1;7.2.3.1 Pairwise-moving block bootstrap ;327 8.1.2.3.2;7.2.3.2 Pairwise-autoregressive bootstrap ;328 8.1.3;7.3 Monte Carlo experiments ;328 8.1.4;7.4 Example: Elbe runoff variations ;335 8.1.5;7.5 Unequal timescales ;337 8.1.5.1;7.5.1 Binned correlation ;338 8.1.5.2;7.5.2 Synchrony correlation ;340 8.1.5.3;7.5.3 Monte Carlo experiments ;342 8.1.5.3.1;7.5.3.1 Optimal estimation ;347 8.1.5.4;7.5.4 Example: Vostok ice core records ;348 8.1.6;7.6 Background material ;349 8.1.7;7.7 Technical issues ;364 8.2;8. Regression II;365 8.2.1;8.1 Linear regression ;366 8.2.1.1;8.1.1 Ordinary least-squares estimation ;366 8.2.1.1.1;8.1.1.1 Bias correction ;367 8.2.1.1.2;8.1.1.2 Prior knowledge about standard deviations ;367 8.2.1.2;8.1.2 Weighted least-squares for both variables estimation ;369 8.2.1.2.1;8.1.2.1 Prior knowledge about standard deviation ratio ;369 8.2.1.2.2;8.1.2.2 Geometric interpretation ;370 8.2.1.3;8.1.3 Wald--Bartlett procedure ;371 8.2.2;8.2 Bootstrap confidence intervals ;372 8.2.2.1;8.2.1 Simulating incomplete prior knowledge ;374 8.2.3;8.3 Monte Carlo experiments ;376 8.2.3.1;8.3.1 Easy setting ;376 8.2.3.2;8.3.2 Realistic setting: incomplete prior knowledge ;379 8.2.3.3;8.3.3 Dependence on accuracy of prior knowledge ;381 8.2.3.4;8.3.4 Mis-specified prior knowledge ;383 8.2.4;8.4 Example: climate sensitivity ;385 8.2.5;8.5 Prediction ;388 8.2.5.1;8.5.1 Example: calibration of a proxy variable ;390 8.2.6;8.6 Lagged regression ;393 8.2.6.1;8.6.1 Example: CO2 and temperature variations in the Pleistocene ;394 8.2
.7;8.7 Background material ;399 8.2.8;8.8 Technical issues ;405 9;Part IV Outlook;407 9.1;9. Future Directions;408 9.1.1;9.1 Timescale modelling ;408 9.1.2;9.2 Novel estimation problems ;409 9.1.3;9.3 Higher dimensions ;410 9.1.4;9.4 Climate models ;410 9.1.4.1;9.4.1 Fitting climate models to observations ;412 9.1.4.2;9.4.2 Forecasting with climate models ;413 9.1.4.3;9.4.3 Design of the cost function ;414 9.1.4.4;9.4.4 Climate model bias ;415 9.1.5;9.5 Optimal estimation ;416 9.2;References;418 9.3;Subject Index;478 9.4;Author Index ;489