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| Percival D.B., Walden A.T. — Wavelet methods for time series analysis |
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| Предметный указатель |
Discrete Fourier transform (DFT), odd indexed subsequence 39
Discrete Fourier transform (DFT), orthonormal 41
Discrete Fourier transform (DFT), periodicity for finite sequence 28 36
Discrete Fourier transform (DFT), periodicity for infinite sequence 22 35
Discrete Fourier transform (DFT), periodized filter 33—34 37 39
Discrete Fourier transform (DFT), periodized scaling and wavelet filters of jth level 100
Discrete Fourier transform (DFT), periodized scaling and wavelet filters of second level 92
Discrete fractional Brownian motion (DFBM) 279
Discrete fractional Brownian motion (DFBM), parameter and properties 286
Discrete fractional Brownian motion (DFBM), random walk 288
Discrete fractional Brownian motion (DFBM), SDF 280
Discrete parameter process 266
Discrete parameter process, pure power law process 281
Discrete RV 256
Discrete wavelet empirical power spectrum, DWT 62—63 66—67 156 160 297
Discrete wavelet empirical power spectrum, DWT, lack of invariance to shifts 63
Discrete wavelet empirical power spectrum, MODWT 160 181
Discrete wavelet packet transform (DWP) 15 206 278
Discrete wavelet packet transform (DWP) for solar magnetic field 220
Discrete wavelet packet transform (DWP), advanced LA(8) coefficients for solar magnetic field 221—222
Discrete wavelet packet transform (DWP), ANOVA 208 250
Discrete wavelet packet transform (DWP), defined 209
Discrete wavelet packet transform (DWP), energy decomposition 208 250
Discrete wavelet packet transform (DWP), Haar filter and Walsh transform 217—218
Discrete wavelet packet transform (DWP), orthonormality 206 210 221
Discrete wavelet packet transform (DWP), parent node 210
Discrete wavelet packet transform (DWP), regular DWT 218
Discrete wavelet packet transform (DWP), second level 208
Discrete wavelet packet transform (DWP), table 209 212
Discrete wavelet packet transform (DWP), time/frequency decomposition 206
Discrete wavelet packet transform (DWP), tree 209 212
Discrete wavelet packet transform (DWP), vector for node (j,n) 250
Discrete wavelet packet transform (DWP), vectors of jth level 249
Discrete wavelet packet transform (DWP), via modification of DWT 207
discrete wavelet transform (DWT) 1 41 56
Discrete wavelet transform (DWT) and its time series 15
Discrete wavelet transform (DWT) and thresholding/shrinkage 417
Discrete wavelet transform (DWT), alternative definitions 149—150
Discrete wavelet transform (DWT), analysis 94
Discrete wavelet transform (DWT), approximate covariance matrix of coefficients 355 389
Discrete wavelet transform (DWT), atomic clock data 14
Discrete wavelet transform (DWT), autocorrelation implementation 149
Discrete wavelet transform (DWT), clustering of coefficients 450—451 455
Discrete wavelet transform (DWT), coefficients 13 57
Discrete wavelet transform (DWT), coefficients via subsampling of MODWT 174
Discrete wavelet transform (DWT), coefficients, vector 61 67
Discrete wavelet transform (DWT), convolutional implementation 149
Discrete wavelet transform (DWT), correlation matrix from FD process 349
Discrete wavelet transform (DWT), correlation matrix from FD process, D(4) 351
Discrete wavelet transform (DWT), correlation matrix from FD process, Haar 350
Discrete wavelet transform (DWT), correlation matrix from FD process, LA(8) 351
Discrete wavelet transform (DWT), covariance matrix of coefficients 349 355
Discrete wavelet transform (DWT), D(4) matrix 59—60
Discrete wavelet transform (DWT), decomposition of sample variance 62 67
Discrete wavelet transform (DWT), decorrelation of FD process 340 345 350
Discrete wavelet transform (DWT), energy decomposition 19 62
Discrete wavelet transform (DWT), energy preservation 57
Discrete wavelet transform (DWT), Haar 62—63 156 441
Discrete wavelet transform (DWT), Haar matrix 57—58
Discrete wavelet transform (DWT), initialization 464
Discrete wavelet transform (DWT), inverse 422
Discrete wavelet transform (DWT), localization 59
Discrete wavelet transform (DWT), matrix, 57 63 99 156
Discrete wavelet transform (DWT), non-decimated 159
Discrete wavelet transform (DWT), orthonormality 57 67 72
Discrete wavelet transform (DWT), parameter estimation for FD process 341
Discrete wavelet transform (DWT), parameter estimation for FD process, applied to trend 368—369
Discrete wavelet transform (DWT), partial 55 341 350 385 418
Discrete wavelet transform (DWT), persistence of coefficients 450 451 455
Discrete wavelet transform (DWT), power of two length 57 67 141 150
Discrete wavelet transform (DWT), pyramid algorithm see "Pyramid algorithm for DWT"
Discrete wavelet transform (DWT), reconstruction of a time series 16
Discrete wavelet transform (DWT), scale 58
Discrete wavelet transform (DWT), sensitivity due to downsampling 164
Discrete wavelet transform (DWT), shift invariant 159
Discrete wavelet transform (DWT), stationary 159
Discrete wavelet transform (DWT), synthesis 83 94
Discrete wavelet transform (DWT), time invariant 159
Discrete wavelet transform (DWT), time/scale decomposition 206
Discrete wavelet transform (DWT), translation invariant 159 174
Discrete wavelet transform (DWT), undecimated 159
Discrete wavelet transform (DWT), versus CWT 1 13
Disjoint dyadic decomposition and WP table 213 250
Disjoint dyadic decomposition, best basis algorithm 225
Disjoint dyadic decomposition, best basis from WP table 223 226 251
Disjoint dyadic decomposition, best basis from WP table, example 224
Disjoint dyadic decomposition, best shift basis from WP table 228
Donoho, D.L. 144 159 204 374 400 418 420 429 456 553—555
Doroslovacki, M.I. 119 555
Downsampling 70 74 80 90 92 156 164 185 211 221 463
Downsampling and bandwidth, scaling and wavelet coefficients 84—86
Downsampling and filtering, magnitude squared DFTs 209 211
Downsampling and frequency reversal, DWPT 211
Downsampling and frequency reversal, DWT wavelet coefficients 85—86
Downsampling, energy preservation 70
Downsampling, retaining even-indexed values 75 149
Downsampling, retaining odd-indexed values 70 149 152
Draper, N.R. 376 378 555
Duhamel, P. 103 561
Durbin, J. 553
DWPT see "Discrete wavelet packet transform"
DWT see "Discrete wavelet transform"
Dyadic dilation 458
Dyadic dilation, inverse 459
Dyadic scales 13 15
Dynamic range 273 437
ECG see "Electrocardiogram"
EDOF see "Equivalent degrees of freedom"
Efficiency (statistical) 17 309
Eigenspectra 274—275 444
El Nino 192
Electrocardiogram (ECG) 125 56 59 456
Electrocardiogram (ECG), advanced LA(8) coefficients, MODWT 182—183
Electrocardiogram (ECG), advanced LA(8) coefficients, partial DWT 127—128
Electrocardiogram (ECG), clustering and persistence in DWT coefficients 450
Electrocardiogram (ECG), illustration of DWT tree structure 451
Electrocardiogram (ECG), instrumentation effects 134
Electrocardiogram (ECG), MRA for LA(8) MODWT 182—184
Electrocardiogram (ECG), MRA for partial DWT, C(6) wavelet 129 132
Electrocardiogram (ECG), MRA for partial DWT, D(4) wavelet 129 131
Electrocardiogram (ECG), MRA for partial DWT, Haar wavelet 129 130
Electrocardiogram (ECG), MRA for partial DWT, LA(8) wavelet 129 133 142
Electrocardiogram (ECG), MRA using circular boundary conditions 133 140
Electrocardiogram (ECG), MRA using reflecting boundary conditions 140 142
Electrocardiogram (ECG), P waves 125 129 135
Electrocardiogram (ECG), partial DWTs 126
Electrocardiogram (ECG), R waves 125 129 134
Electrocardiogram (ECG), T waves 125 129 135
Electrocardiogram (ECG), wavelet smooths of 6th level 196—197
Eltahir, E.A.B. 192 555
Embedded backward difference filters 306
Embedded backward difference filters, L/2 for Daubechies wavelet filter 106 123—124 153 188
Embedded backward difference filters, L/3 for coiflet wavelet filter 124 156 188
Energy decomposition of finite energy function (signal) 481 497
Energy decomposition via CWT 11
Energy decomposition via DWPT 208 250
Energy decomposition via DWT 19 62
Energy decomposition via matching pursuit 240 252
Energy decomposition via MODWPT 233 251
Energy decomposition via MODWT 16—17 166 168—169 171 173 200
Energy decomposition via ODFT 48 54
Energy decomposition via orthonormal transform 43
Energy decomposition via partial DWT 15—16 95 104 151
Energy decomposition via smooths and details 95 104
Energy decomposition via WP table disjoint dyadic decomposition 214
Energy decomposition, across time 43
Energy decomposition, frequency by frequency 48 54
Energy of time series 42
Energy preservation and downsampling 70
Energy preservation via DWT 57
| Energy preservation via ODFT 47
Energy preservation via orthonormal transform 43 53
Energy preservation via unitary transform 46 53
Energy spectrum 72—73 268
Entropy cost functional 223 225
Entropy cost functional, best basis for solar magnetic field 227—228
Entropy cost functional, best shift basis for solar magnetic field 228
Entropy cost functional, cost table example 225
Eom, K.-B. 364 557
Equal in distribution 257
Equality in the mean square sense 38
Equivalent degrees of freedom (EDOF) 313—314 336 338—339 376 390
Equivalent filter for a cascade of filters 28 39
Equivalent width 68
Euler relationship 21—22 48
Euler — Maclaurin summation 280
Euler's constant 270 432
Evans, J.M. 553
Even indexed subsequence and DFT 23
Events, intersection 435
Events, union 435
Expected value see "Mean"
Exponential distribution (prior) 413 426—427
Exponential PDF 264—265
Extending/padding a sample for the DWT 141 422—423
Extremal phase (minimum delay) filter 106 117 155 493 see
False alarm probability 435
Fast Fourier transform (FFT) 17 28 68 432 440 290
FBM see "Fractional Brownian motion"
FD process see "Fractionally differenced process"
FFT see "Fast Fourier transform"
fGn see "Fractional Gaussian noise"
Fidelity condition 398
Filter 267
Filter cascade 27 89
Filter cascade, D(4) wavelet filter 68
Filter cascade, Daubechies wavelet filter 105 107
Filter cascade, flow diagram 27—28
Filter cascade, for wavelet coefficients of second level 90
Filter width 90
Filter with inserted zeros, transfer function 91 96—97
Filter, backward difference 107 304
Filter, band-pass 5
Filter, center of energy 118 231
Filter, circular 31—32 55
Filter, dipole 488
Filter, extremal phase (minimum delay) 106 117 155 488 493
Filter, first backward difference 105
Filter, frequency response function 25 33—34
Filter, gain function 25
Filter, generalized linear phase 117
Filter, half-band 76 152
Filter, high-pass 27
Filter, impulse response sequence 25
Filter, infinite width 79
Filter, least asymmetric see "LA and least asymmetric"
Filter, linear phase 108 111 155 158
Filter, linear time-invariant 25
Filter, low-pass 26 185
Filter, maximum delay 488
Filter, maximum phase 488
Filter, minimum delay (extremal phase) 106 117 155 488 493
Filter, minimum phase 488
Filter, periodized 20 32—33 35 37 39—40
Filter, phase function 25
Filter, squared gain function 25
Filter, transfer function 25 33—34
Filter, transfer function for reversed form 117
Filter, two-stage cascade 304
Filter, width 28 39
Filter, zero phase 108 110 158 179 234 237 252
Filtering 20
Filtering and convolution 25
Filtering and downsampling, magnitude squared DFTs 209 211
Filtering, circular 136 139 145 197 268
Filtering, finite sequences 29
Filtering, flow diagram 25
Filtering, high-pass example 27
Filtering, infinite sequences 25
Filtering, low-pass example 26
Finite Fourier transform 28
Finite sequences, circular autocorrelation 30 37
Finite sequences, circular convolution 29 37
Finite sequences, circular cross-correlation 30 37
Finite sequences, discrete Fourier transform of 28 36
Finite support 465 472 474 496 498
Firm thresholding 400 411 453
First backward difference 60
First backward difference, filter 105
First derivative of Gaussian CWT 10
First derivative of Gaussian wavelet function 3 10
First derivative of Gaussian wavelet function, autocorrelation width 12
First order Markov chain 451
Fitzgerald, W.J. 560
Flandrin, P. 280 295 321 340 354 464 552 555
Flannery, B.P. 560
Fodor, I.K. 558
Forsyth, R.J. 552 556
Foufoula-Georgiou, E. 4 5 6 12 295 555 558 560
Fourier analysis 1 20 22 35—37 47
Fourier coefficient 46
Fourier details 50—51 54—55
Fourier details, circularly shifted series 55
Fourier details, synthesis of time series 50
Fourier frequencies 271 330
Fourier roughs 50—51 54
Fourier smooths 50—51 54
Fourier synthesis 22 29 35—37 49 51
Fourier transform and derivatives of wavelet function 483—484
Fourier transform of autocorrelation 38
Fourier transform of convolution 38
Fourier transform of cross-correlation 38
Fourier transform of finite sequence see "DFT"
Fourier transform of function 37
Fourier transform of infinite sequence see "DFT"
Fourier transform of scaling function 465 468—469 496
Fourier transform of time reversed scaling filter 465 484 496
Fourier transform of time reversed wavelet filter 476 483 498
Fourier transform of wavelet function 476 498
Fourier transform pair 23—24 29 35—38 307 312
Fourier transform, inverse for function 37
Fourier transform, short time 206
Fractional Brownian motion (FBM) 279 285 354
Fractional Brownian motion (FBM), SDF 280
Fractional Brownian motion (FBM), simulation 361
Fractional difference parameter 286
Fractional frequency deviates see "Atomic clock average fractional frequency deviates"
Fractional Gaussian noise (FGN) 191 279 299 300 354
Fractional Gaussian noise (FGN), approximate SDF 280
Fractional Gaussian noise (FGN), autocovariance sequence 279
Fractional Gaussian noise (FGN), long memory property 280
Fractional Gaussian noise (FGN), parameter and properties 285—286
Fractional Gaussian noise (FGN), sample interval 285
Fractional Gaussian noise (FGN), SDF 280 282
Fractional Gaussian noise (FGN), simulation 283
Fractionally differenced (FD) process 281 307 349
Fractionally differenced (FD) process, ACS 345—346
Fractionally differenced (FD) process, ACS of nonboundary wavelet coefficients 345—346
Fractionally differenced (FD) process, ACVS 284 341
Fractionally differenced (FD) process, correlation between nonboundary wavelet coefficients 347
Fractionally differenced (FD) process, covariance matrix 355
Fractionally differenced (FD) process, decorrelation via DWT 340 345 350 380 388
Fractionally differenced (FD) process, DWT-based and exact MLEs 363—364 390
Fractionally differenced (FD) process, DWT-based and exact MLEs, with unknown mean 366
Fractionally differenced (FD) process, DWT-based log likelihood 363
Fractionally differenced (FD) process, DWT-based log likelihood, with unknown mean 365
Fractionally differenced (FD) process, DWT-based MLEs 364
Fractionally differenced (FD) process, DWT-based MLEs, and scaling coefficient 366
Fractionally differenced (FD) process, DWT-based MLEs, with unknown mean 365
Fractionally differenced (FD) process, exact MLE 367
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