Abstract
Using Bayesian probability theory we demonstrate that the Lomb-Scargle periodogram may be generalized in a straightforward manner to nonuniformly nonsimultaneously sampled quadrature data when the sinusoid has a arbitrary amplitude modulation. This generalized Lomb-Scargle periodogram is the sufficient statistic for single frequency estimation in real uniformly sampled data, to frequency estimation for a single sinusoid having exponential, Gaussian, or arbitrary amplitude modulation. In addition we define the bandwidth of a nonuniformly sampled data set and show that the phenomenon of aliases exists in both uniformly and nonuniformly sampled data and that the phenomenon has the same cause in both types of data. Finally, we show that nonuniform sampling does not affect the accuracy of the frequency estimates; although it may affect the accuracy of the amplitude estimates.
This paper is followed by a commentary by Thomas J. Loredo.
This paper is followed by a commentary by Thomas J. Loredo.
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References
Bretthorst, G. Larry (1988), “Bayesian Spectrum Analysis and Parameter Estimation,” in Lecture Notes in Statistics, 48, J. Berger, S. Fienberg, J. Gani, K. Krickenberg, and B. Singer (eds), Springer-Verlag, New York, New York.
Bretthorst, G. Larry, (2000) “Nonuniform Sampling: Aliasing and Bandwidth,” Maximum Entropy and Bayesian Methods, G. Erickson ed., Kluwer Academic press, the Netherlands.
Gilks, W. R., S. Richardson and D. J. Spiegelhalter (1996), “Markov Chain Monte Carlo in Practice,” Chapman & Hall, London.
Lomb, N. R. (1976) “Least-Squares Frequency Analysis of Unevenly Spaced Data,” Astrophysical and Space Science, 39, pp. 447–462.
Marple, S. L. (1987) Digital spectral Analysis with applications, Prentice-Hall, Inc., Englewood Clifts. New Jersey.
Neal, Radford M. (1993), “Probabilistic Inference Using Markov Chain Monte Carlo Methods,” technical report CRG-TR-93-1, Dept. of Computer Science, University of Toronto.
Priestley, M. B. (1981), Spectral Analysis and Time Series, 2 Vols., Academic Press, Inc., Orlando FL, Combined paperback edition with corrections (1983).
Scargle, J. D. (1982) “Studies in Astronomical Time Series Analysis II. Statistical Aspects of Spectral Analysis of Unevenly Sampled Data,” Astrophysical Journal, 263, pp. 835–853.
Scargle, J. D. (1989) “Studies in Astronomical Time Series Analysis. III. Fourier Transforms, Autocorrelation and Cross-correlation Functions of Unevenly Spaced Data,” Astrophysical Journal, 343, pp. 874–887.
Schuster, A., (1905), “The Periodogram and its Optical Analogy,” Proceedings of the Royal Society of London, 77, p. 136.
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© 2003 Springer-Verlag New York, Inc.
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Bretthorst, G.L. (2003). Frequency Estimation and Generalized Lomb-Scargle Periodograms. In: Statistical Challenges in Astronomy. Springer, New York, NY. https://doi.org/10.1007/0-387-21529-8_21
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DOI: https://doi.org/10.1007/0-387-21529-8_21
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-95546-9
Online ISBN: 978-0-387-21529-7
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