Fourier analysis of time series : an introduction

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Title:

Personal Author:

Bloomfield, Peter

Publication Information:

New York : J Wiley, 1976

ISBN:

9780471082569

Subject Term:

Time-series analysis

Fourier analysis

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000000744593	QA280 B466 1976	Open Access Book	Book	Searching... Unknown

A new, revised edition of a yet unrivaled work on frequency domain analysis Long recognized for his unique focus on frequency domain methods for the analysis of time series data as well as for his applied, easy-to-understand approach, Peter Bloomfield brings his well-known 1976 work thoroughly up to date. With a minimum of mathematics and an engaging, highly rewarding style, Bloomfield provides in-depth discussions of harmonic regression, harmonic analysis, complex demodulation, and spectrum analysis. All methods are clearly illustrated using examples of specific data sets, while ample exercises acquaint readers with Fourier analysis and its applications. The Second Edition: * Devotes an entire chapter to complex demodulation * Treats harmonic regression in two separate chapters * Features a more succinct discussion of the fast Fourier transform * Uses S-PLUS commands (replacing FORTRAN) to accommodate programming needs and graphic flexibility * Includes Web addresses for all time series data used in the examples An invaluable reference for statisticians seeking to expand their understanding of frequency domain methods, Fourier Analysis of Time Series, Second Edition also provides easy access to sophisticated statistical tools for scientists and professionals in such areas as atmospheric science, oceanography, climatology, and biology.

1 Introduction	p. 1
1.1 Fourier Analysis	p. 2
1.2 Historical Development of Fourier Methods	p. 5
1.3 Why Use Trigonometric Functions?	p. 7
2 Fitting Sinusoids	p. 9
2.1 Curve-Fitting Approach	p. 9
2.2 Least Squares Fitting of Sinusoids	p. 11
2.3 Multiple Periodicities	p. 17
2.4 Orthogonality of Sinusoids	p. 19
2.5 Effect of Discrete Time: Aliasing	p. 21
2.6 Some Statistical Results	p. 23
Appendix	p. 24
3 The Search for Periodicity	p. 25
3.1 Fitting the Frequency	p. 25
3.2 Fitting Multiple Frequencies	p. 28
3.3 Some More Statistical Results	p. 30
Appendix	p. 34
4 Harmonic Analysis	p. 37
4.1 Fourier Frequencies	p. 37
4.2 Discrete Fourier Transform	p. 40
4.3 Decomposing the Sum of Squares	p. 44
4.4 Special Functions	p. 45
4.5 Smooth Functions	p. 53
5 The Fast Fourier Transform	p. 57
5.1 Computational Cost of Fourier Transforms	p. 57
5.2 Two-Factor Case	p. 58
5.3 Application to Harmonic Analysis of Data	p. 61
6 Examples of Harmonic Analysis	p. 63
6.1 Variable Star Data	p. 63
6.2 Leakage Reduction by Data Windows	p. 66
6.3 Tapering the Variable Star Data	p. 72
6.4 Wolf's Sunspot Numbers	p. 76
6.5 Nonsinusoidal Oscillations	p. 78
6.6 Amplitude and Phase Fluctuations	p. 81
6.7 Transformations	p. 83
6.8 Periodogram of a Noise Series	p. 87
6.9 Fisher's Test for Periodicity	p. 91
Appendix	p. 95
7 Complex Demodulation	p. 97
7.1 Introduction	p. 97
7.2 Smoothing: Linear Filtering	p. 100
7.3 Designing a Filter	p. 105
7.4 Least Squares Filter Design	p. 110
7.5 Demodulating the Sunspot Series	p. 118
7.6 Complex Time Series	p. 124
7.7 Sunspots: The Complex Series	p. 126
Appendix	p. 130
8 The Spectrum	p. 133
8.1 Periodogram Analysis of Wheat Prices	p. 133
8.2 Analysis of Segments of a Series	p. 140
8.3 Smoothing the Periodogram	p. 142
8.4 Autocovariances and Spectrum Estimates	p. 147
8.5 Alternative Representations	p. 149
8.6 Choice of a Spectral Window	p. 155
8.7 Examples of Smoothing the Periodogram	p. 157
8.8 Reroughing the Spectrum	p. 160
Appendix	p. 164
9 Some Stationary Time Series Theory	p. 167
9.1 Stationary Time Series	p. 167
9.2 Continuous Spectra	p. 173
9.3 Time Averaging and Ensemble Averaging	p. 175
9.4 Periodogram and Continuous Spectra	p. 176
9.5 Approximate Mean and Variance	p. 177
9.6 Properties of Spectral Windows	p. 190
9.7 Aliasing and the Spectrum	p. 195
10 Analysis of Multiple Series	p. 201
10.1 Cross Periodogram	p. 202
10.2 Estimating the Cross Spectrum	p. 204
10.3 Theoretical Cross Spectrum	p. 211
10.4 Distribution of the Cross Periodogram	p. 214
10.5 Distribution of Estimated Cross Spectra	p. 218
10.6 Alignment	p. 226
Appendix	p. 230
11 Further Topics	p. 233
11.1 Time Domain Analysis	p. 233
11.2 Spatial Series	p. 234
11.3 Multiple Series	p. 236
11.4 Higher Order Spectra	p. 238
11.5 Nonquadratic Spectrum Estimates	p. 239
11.6 Incomplete and Irregular Data	p. 242
References	p. 247
Author Index	p. 255
Subject Index	p. 257

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