Univariate Time Series in Geosciences: Theory and Examples

Cover......Page 1
Contents......Page 13
1.1 Data in Geosciences: for Example, Surface Solar Radiation Records......Page 21
1.2 R......Page 28
1.3.1 Univariate Analysis of the Pyranometer Data......Page 30
1.3.2 Unbiased and Consistent Estimators, Central Limit Theorem......Page 33
1.3.3 Bivariate Analysis of the Pyranometer Data......Page 37
1.3.4 Are the Pyranometer Daily Values Iid.?......Page 42
1.4.1 Time Series Plots......Page 43
1.4.2 A Model for the Error of the Pyranometer Measurements......Page 50
1.5.1 Formulas for Calculating the Moments of Random Variables......Page 51
1.5.3 Independent and Normally Distributed Random Variables......Page 52
1.5.4 Bivariate (two-dimensional) Normal Density......Page 54
1.5.5 Multivariate Normal Distribution......Page 55
1.6 Problems......Page 56
2.1 Empirical Moments of a Wind Speed Time Series......Page 59
2.2.1 Stochastic Processes......Page 64
2.2.2 1st and 2nd Moment Functions......Page 70
2.2.3 Stationarity Assumptions......Page 72
2.3.1 White Noise Process......Page 76
2.3.2 First Order Autoregressive (AR[1]) Process......Page 80
2.3.3 Random Walk......Page 84
2.3.4 First Order Moving Average Process......Page 86
2.3.5 Linear Process......Page 90
2.4.1 Linear and Time-invariant Transformations......Page 94
2.4.2 Existence and Calculation of Convolution Sums......Page 98
2.4.3 Inverse Sequences......Page 102
2.4.4 Examples: AR[1] and AR[2] Models......Page 104
2.5.1 Diagnostics for the Stationarity of a Time Series......Page 108
2.5.2 Locally Stationary Time Series......Page 112
2.6.1 Properties of the Empirical Moment Functions of a Stationary Time Series......Page 117
2.6.2 Ergodic Theorems: a Cautionary Remark......Page 121
2.7 Optimal Linear Predictions......Page 122
2.8.1 First and/or second Moment Functions of some Processes......Page 124
2.8.2 Moments of the Mean of a Stationary Time Series......Page 126
2.8.3 Properties of the Optimal Linear Prediction......Page 128
2.8.4 The Conditional Expectation as Optimal Prediction......Page 131
2.9 Problems......Page 133
3 Linear Models for the Expectation Function......Page 141
3.1.1 An Introductory Example......Page 142
3.1.2 Simple Linear Regression......Page 145
3.2.1 A Linear Model for the Expectation Function......Page 150
3.2.2 Least Squares......Page 152
3.2.4 Generalised least squares......Page 153
3.3.1 Plots......Page 156
3.3.2 Model Selection......Page 162
3.4.1 Seasonal Models......Page 165
3.4.2 Component Models......Page 168
3.5 Trends and Station Effects of Shortwave Incoming Radiation......Page 171
3.6 Trend Surfaces for the Tree-line in the Valais Alps......Page 181
3.7 Problems......Page 187
4 Interpolation......Page 191
4.1.1 Distance Weighted Interpolation......Page 193
4.1.2 Tessellation of the Area under Study......Page 198
4.1.3 Discussion......Page 200
4.2 The Spatial Random Function and its Variogram......Page 201
4.3 Estimating the Variogram......Page 205
4.3.1 Empirical Variograms I: the Isotropic Case......Page 206
4.3.2 Variogram Models......Page 213
4.3.3 Empirical Variograms II: the Anisotropic Case......Page 216
4.3.4 Empirical Variograms III: the Case with Non-constant Drift......Page 220
4.4 Optimal Linear Predictors for Spatial Random Functions......Page 226
4.5.1 The Stationary Case: Simple Kriging......Page 229
4.5.2 The Intrinsic Case: Ordinary Kriging......Page 235
4.5.3 The Case with a Non-constant Drift: Universal Kriging......Page 236
4.5.4 Summary......Page 242
4.6.1 Isotropic Empirical Variograms......Page 245
4.6.2 Support......Page 247
4.6.3 Measurement Errors......Page 249
4.6.4 Local Interpolation......Page 256
4.6.5 Spatial Averages......Page 259
4.6.6 Optimal Interpolation: Summary......Page 263
4.7 Problems......Page 264
5 Linear Processes......Page 269
5.1 AR[p] Models......Page 270
5.1.1 The Yule-Walker-Equations of an AR[p] Model......Page 271
5.1.2 Levinson-Durbin Recursion, Partial Correlation Function......Page 275
5.1.3 Examples......Page 277
5.2.1 Burg’s Algorithm......Page 282
5.2.2 Regression Estimates......Page 284
5.2.3 Maximum Likelihood Estimates......Page 286
5.2.4 Summary of Sects. 5.1 and 5.2......Page 289
5.3.1 MA[q] Model......Page 291
5.3.2 ARMA[p, q] Model......Page 295
5.3.3 ARIMA[p, d, q] Model......Page 299
5.3.4 Estimators for an ARMA[p, q] Model......Page 300
5.4.1 Removing Trends or other Fluctuations from a Time Series......Page 302
5.4.2 Identification......Page 309
5.4.3 Estimation......Page 315
5.4.4 Diagnostics......Page 317
5.4.5 Selection and Summary......Page 324
5.5.1 Predictions Using a Theoretical ARMA[p, q] Model......Page 326
5.5.2 Predictions Using an Estimated ARMA[p, q] Model......Page 331
5.5.3 Summary......Page 333
5.6.1 Levinson-Durbin Recursion......Page 334
5.6.2 Partial Correlation Function......Page 336
5.6.3 Predictions Derived from the AR[∞] and MA[∞] representations of an ARMA[p, q] Model......Page 338
5.6.4 Predictions Derived from the Difference Equation of an ARMA[p, q] Model......Page 340
5.7 Problems......Page 342
6 Fourier Transforms of Deterministic Functions......Page 349
6.1.1 Representation of a Sequence using Trigonometric Functions......Page 350
6.1.2 Fouriertransforms: a Preview to Chap. 6......Page 357
6.1.3 Spectra of Time Series: a Preview to Chaps. 7, 9 and 10......Page 358
6.2.1 Linear Vector Spaces......Page 360
6.2.2 Orthogonal Functions......Page 363
6.2.3 Orthogonal Trigonometric Functions......Page 365
6.3.1 Case 1: Discrete Fourier Transform......Page 367
6.3.2 Case 2: Fourier Series......Page 373
6.3.3 Case 3: Fourier Integrals......Page 377
6.3.4 Case 4: Fourier Series with t and s Reversed......Page 383
6.3.5 Delta Function......Page 387
6.4 Spectra......Page 392
6.5 Aliasing and Leakage......Page 394
6.5.1 Measuring a Function f(t) with Real Argument t......Page 395
6.5.2 Aliasing......Page 400
6.5.3 Leakage......Page 403
6.6 Width of Functions in a Fourier Transform Pair......Page 409
6.6.1 Dynamic Range and Width at Half Height......Page 410
6.6.2 Equivalent Width and σ-Width......Page 412
6.6.3 Autocorrelation Width and Uncertainty Relationship......Page 414
6.6.4 Time-limited and Band-limited Sequences......Page 417
6.7 Using a Data Taper to Reduce Leakage......Page 419
6.7.1 Data Taper......Page 420
6.7.2 Aliasing and Leakage Circumvented......Page 424
6.8 Convolution II......Page 428
6.8.1 Linear and Time-invariant (LTI) Systems......Page 429
6.8.2 First Order Differences and Moving Average......Page 432
6.8.3 Low-pass, High-pass and Band-pass Filters......Page 436
6.9.1 Orthogonal Trigonometric Functions......Page 438
6.9.3 Sums of Trigonometric Functions......Page 440
6.9.4 Properties of Fourier Transform Pairs......Page 441
6.9.5 Bessel’s Inequality and Parseval’s Identity......Page 444
6.9.6 Properties of the Autocorrelation......Page 445
6.9.7 Demonstration of the Convolution Theorem......Page 448
6.9.8 Autocorrelation and Discrete Fourier Transform in R......Page 449
6.10 Problems......Page 453
7 Fourier Representation of a Stationary Stochastic Process......Page 461
7.1.1 Poisson Process......Page 462
7.1.2 Exponential Waiting Times......Page 464
7.1.3 On-road Measurements of Aerosols......Page 467
7.2.1 Random Variables in L[sup(2)](Ω, F)......Page 471
7.2.2 Derivatives and Integrals of Random Functions......Page 472
7.2.3 Examples......Page 476
7.3.1 The Harmonic Process and its Fourier Representation......Page 480
7.3.2 Fourier Representation of a Stationary Stochastic Process......Page 487
7.3.3 Covariance Function and Spectrum......Page 489
7.3.4 Types of Spectra......Page 492
7.3.5 Bandwidth of a Spectrum......Page 495
7.4.1 Stochastic Filters......Page 497
7.4.2 Examples......Page 500
7.5.1 Gamma Distribution......Page 505
7.5.2 Integrating with Respect to an Orthogonal Increment Process......Page 506
7.5.3 Moments of the Harmonic Process......Page 509
7.5.4 Spectrum of a Discrete-time Stochastic Process......Page 511
7.5.5 Spectrum and/or Covariance Function?......Page 513
7.6 Problems......Page 515
8 Does a Periodogram Estimate a Spectrum?......Page 519
8.1 The Periodogram......Page 520
8.2.1 Its Properties......Page 523
8.2.2 Tests......Page 525
8.3 Its Expectation Function......Page 531
8.4 Its Covariance Function and its Distribution......Page 538
8.5.1 Expectation Function of the Periodogram......Page 540
8.5.2 The Periodogram of a Linear Process......Page 543
8.6 Problems......Page 545
9 Estimators for a Continuous Spectrum......Page 547
9.1 The Model......Page 548
9.2 Direct Spectral Estimator......Page 549
9.2.1 Definition and Expectation......Page 550
9.2.2 Variance and Covariance Functions......Page 559
9.2.3 Probability Distribution......Page 566
9.2.4 Calculated for Non-Fourier Frequencies......Page 569
9.2.5 Alternatives: Parametric Estimation, Pre-Whitening......Page 572
9.2.6 Summary......Page 576
9.3 Smoothed Direct Spectral Estimator......Page 577
9.3.1 Discretely Smoothed Direct Spectral Estimator......Page 578
9.3.2 Lag Window Spectral Estimator......Page 588
9.3.3 Moment Functions of the Lag Window Estimator......Page 594
9.3.4 Distribution of the Lag Window Estimator......Page 606
9.3.5 Estimating a Spectrum with Unknown Bandwidth......Page 611
9.3.6 Summary and Alternatives......Page 618
9.4.1 Microseisms......Page 619
9.4.2 Turbulent Atmospheric Flow......Page 630
9.5.1 Direct Spectral Estimator and Autocorrelation of Tapered Observations......Page 640
9.5.2 Expectation Function of a Direct Spectral Estimator......Page 641
9.5.3 Covariance Function of a Direct Spectral Estimator......Page 642
9.5.4 Widths of Smoothing Kernels......Page 645
9.5.5 Variance Function of a Lag Window Estimator......Page 647
9.5.6 Spectral Estimation in R......Page 649
9.6 Problems......Page 654
10 Estimators for a Spectrum Having a Discrete Part......Page 661
10.1.1 The Model......Page 662
10.1.2 Diagnostics......Page 665
10.1.3 Example and Summary......Page 666
10.2.1 Sum of a Harmonic Process and White Noise......Page 670
10.2.2 Sum of a Harmonic Process and Coloured Noise......Page 680
10.2.3 Summary......Page 694
10.3.1 Estimating the Model defined in Sect. 10.1......Page 695
10.3.2 Detrending and Spectral Estimation: an Example......Page 696
10.4 Problems......Page 699
Appendix: Answers to Problems......Page 703
References......Page 717
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