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书名：Time series：data analysis and theory
作者：David R. Brillinger
页数：561
格式：djvu
目录：
Preface to the Classics Edition xiii
Preface to the Expanded Edition xvii
Preface to the First Edition hxix
1 TheNature of Time Seriesand
Their Frequency Analysis 1
1.1 Introduction 1
1.2 A Reason for Harmonic Analysis 7
1.3 Mixing 8
1.4 Historical Development 9
1.5 The Uses of the Frequency Analysis 10
1.6 Inference on Time Series 12
1.7 Exercises 13
2 Foundations 16
2.1 Introduction 16
2.2 Stochastics 17
2.3 Cumulants 19
2.4 Stationarity 22
2.5 Second-Order Spectra 23
2.6 Cumulant Spectra of Order k 25
2.7 Filters 27
2.8 Invariance Properties of Cumulant Spectra 34
2.9 Examples of Stationary Time Series 35
2.10 Examples of Cumulant Spectra 39
2.11 The Functional and Stochastic Approaches to Time Series Analysis
41
2.12 Trends 43
2.13 Exercises 44
ix
X CONTENTS
3 Analytic Properties of Fourier Transforms
and Complex Matrices 49
3.1 Introduction 49
3.2 Fourier Series 49
3.3 Convergence Factors 52
3.4 Finite Fourier Transforms and Their Properties 60
3.5 The Fast Fourier Transform 64
3.6 Applications of Discrete Fourier Transforms 67
3.7 Complex Matrices and Their Extremal Values 70
3.8 Functions of Fourier Transforms 75
3.9 Spectral Representations in the Functional Approach to Time
Series 80
3.10 Exercises 82
4 Stochastic Properties of Finite Fourier Transforms 88
4.1 Introduction 88
4.2 The Complex Normal Distribution 89
4.3 Stochastic Properties of the Finite Fourier Transform 90
4.4 Asymptotic Distribution of the Finite Fourier Transform 94
4.5 Probability 1 Bounds 98
4.6 The Cramer Representation 100
4.7 Principal Component Analysis and its Relation to the Cramer
Representation 106
4.8 Exercises 109
5 The Estimation of Power Spectra 116
5.1 Power Spectra and Their Interpretation 116
5.2 The Periodogram 120
5.3 Further Aspects of the Periodogram 128
5.4 The Smoothed Periodogram 131
5.5 A General Class of Spectral Estimates 142
5.6 A Class of Consistent Estimates 146
5.7 Confidence Intervals 151
5.8 Bias and Prefiltering 154
5.9 Alternate Estimates 160
5.10 Estimating the Spectral Measure and Autocovariance Function 166
5.11 Departures from Assumptions 172
5.12 The Uses of Power Spectrum Analysis 179
5.13 Exercises 181
CONTENTS xi
6 Analysisof A Linear Time Invariant RelationBetween
A Stochastic Series and Several DeterministicSeries 186
6.1 Introduction 186
6.2 Least Squares and Regression Theory 188
6.3 Heuristic Construction of Estimates 192
6.4 A Form of Asymptotic Distribution 194
6.5 Expected Values of Estimates of the Transfer Function and Error
Spectrum 196
6.6 Asymptotic Covariances of the Proposed Estimates 200
6.7 Asymptotic Normality of the Estimates 203
6.8 Estimating the Impulse Response 204
6.9 Confidence Regions 206
6.10 A Worked Example 209
6.11 Further Considerations 219
6.12 A Comparison of Three Estimates of the Impulse Response 223
6.13 Uses of the Proposed Technique 225
6.14 Exercises 227
7 Estimating TheSecond-Order Spectra
of Vector-Valued Series 232
7.1 The Spectral Density Matrix and its Interpretation 232
7.2 Second-Order Periodograms 235
7.3 Estimating the Spectral Density Matrix by Smoothing 242
7.4 Consistent Estimates of the Spectral Density Matrix 247
7.5 Construction of Confidence Limits 252
7.6 The Estimation of Related Parameters 254
7.7 Further Considerations in the Estimation of Second-Order Spectra
260
7.8 A Worked Example 267
7.9 The Analysis of Series Collected in an Experimental Design 276
7.10 Exercises 279
8 Analysisof A Linear TimeInvariant RelationBetween
Two Vector-Valued Stochastic Series 286
8.1 Introduction 286
8.2 Analogous Multivariate Results 287
8.3 Determination of an Optimum Linear Filter 295
8.4 HeuristicInterpretation of Parameters and Construction of Estimates
299
8.5 A LimitingDistribution for Estimates 304
8.6 A Class of Consistent Estimates 306
8.7 Second-Order AsymptoticMoments of the Estimates 309
xii CONTENTS
8.8 Asymptotic Distribution of the Estimates 313
8.9 Confidence Regions for the Proposed Estimates 314
8.10 Estimation of the Filter Coefficients 317
8.11 Probability 1Bounds 321
8.12 Further Considerations 322
8.13 Alternate Forms of Estimates 325
8.14 A Worked Example 330
8.15 Uses of the Analysis of this Chapter 331
8.16 Exercises 332
9 Principal Components in The Frequency Domain 337
9.1 Introduction 337
9.2 Principal Component Analysis of Vector-ValuedVariates 339
9.3 The Principal Component Series 344
9.4 The Construction of Estimates and Asymptotic Properties 348
9.5 Further Aspects of Principal Components 353
9.6 A Worked Example 355
9.7 Exercises 364
10 The Canonical Analysis of Time Series 367
10.1 Introduction 367
10.2 The Canonical Analysis of Vector-ValuedVariates 368
10.3 The Canonical Variate Series 379
10.4 The Construction of Estimates and Asymptotic Properties 384
10.5 Further Aspects of Canonical Variates 388
10.6 Exercises 390
Proofs of Theorems 392
References 461
Notation Index 488
Author Index 490
Subject Index 496
Addendum: FourierAnalysis of StationaryProcesses 501
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