Excerpts from Bayesian Spectrum Analysis and Parameter Estimation

  • G. Larry Bretthorst
Part of the Fundamental Theories of Physics book series (FTPH, volume 31-32)


Bayesian spectrum analysis is still in its infancy. It was born when E. T. Jaynes derived the periodogram2 as a sufficient statistic for determining the spectrum of a time sampled data set containing a single stationary frequency. Here we extend that analysis and explicitly calculate the joint posterior probability that multiple frequencies are present, independent of their amplitude and phase, and the noise level. This is then generalized to include other parameters such as decay and chirp. Results are given for computer simulated data and for real data ranging from magnetic resonance to astronomy to economic cycles. We find substantial improvements in resolution over Fourier transform methods.


Power Spectral Density Discrete Fourier Transform Sunspot Number Harmonic Frequency Nuisance Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    G. Larry Bretthrost, Bayesian Spectrum Analysis and Parameter Estimation, Ph.D. thesis, University Microfilms Inc., Washington University, St. Louis, MO, Aug. 1987.Google Scholar
  2. 2.
    E. T. Jaynes, “Bayesian Spectrum and Chirp Analysis,” in Proceedings of the Third Workshop on Maximum-Entropy and Bayesian Methods (1983), ed. C. Ray Simth, D. Reidel, Boston, 1987. (the third workshop was held in Laramie, Wyoming)Google Scholar
  3. 3.
    R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, Dover Publications, Inc., New York, 1959.zbMATHGoogle Scholar
  4. 4.
    E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics, D. Reidel, Boston, 1983.zbMATHGoogle Scholar
  5. 5.
    A. Schuster, “The Periodogram an its Optical Analogy,” Proceedings of the Royal Society of London, vol. 77, p. 136, 1905.Google Scholar
  6. 6.
    Lord Rayleigh, Philosophical Magazine, vol. (5) 8, p. 261, 1879.Google Scholar
  7. 7.
    J. W. Tukey, several conversations with E. T. Jaynes, in the period 1980–1983.Google Scholar
  8. 8.
    Sir Harold Jeffreys, Theory of Probability, Oxford University press, London, 1939. (Later editions, 1948, 1961)Google Scholar
  9. 9.
    E. T. Jaynes, “Prior Probabilities,” in Papers on Probability, Statistics and Statistical Physics, ed. R. D. Rosenkrantz, D. Reidel, Boston, 1983.Google Scholar
  10. 10.
    E. T. Jaynes, “Marginalization and Prior Probabilities,” in Papers Oil Probability, Statistics and Statistical Physics, ed. R. D. Rosenkrantz, D. Reidel, Boston, 1983.Google Scholar
  11. 11.
    M. Waldmeier, in The Sunspot Activity ill the Years 1610–1960, Schulthes, Zurich, 1961.Google Scholar
  12. 12.
    Robert Hooke and T. A. Jeeves, “‘Direct Search’ solution of Numerical and Statistical Problems,” J. Assoc. Compo Mach., p. 212, March 1962.Google Scholar
  13. 13.
    Derek Shaw, in Fourier Transform NMR Spectroscopy, Elsevier Scientific Pub. Co., New York, 1976.Google Scholar
  14. 14.
    Joseph W. Ganem and R. E. Norberg, Private Communicatioll, 1987.Google Scholar
  15. 15.
    A. Abragam, in Principles of Nuclear Magnetism, p. 187, Oxford Science Publications, London, 1961. reprint (1985)Google Scholar
  16. 16.
    Richard James Beckett, The Temperature and Density Dependence of Nuclear Spin-Spin Interactions in Hydrogen-Deuteride Gas and Fluid., Rutgers University Ph.D. Thesis. New Brunswick, New Jersey, 1979. (unpublished)Google Scholar
  17. 17.
    Robert Guinn Currie. Private Communication, 1985.Google Scholar
  18. 18.
    Robert Guinn Currie and Sultan Hameed, “Climatically Induced Cyclic Variations in United States Corn Yield and Possible Economic Implications,” presented at the Canadian Hydrology Symposium, Regina, Saskatchewan, June 3, 1986.Google Scholar
  19. 19.
    John Parker Burg, Maximum Entropy Spectral Analysis, Ph.D. Dissertation, (University Microfilms No. 75–25), Stanford University, 1975.Google Scholar
  20. 20.
    T. J. Cohen and P. R. Lintz, “Long Term Periodic ties in the Sunspot Cycle,” Nature. vol. 250, p. 398, 1974.CrossRefGoogle Scholar
  21. 21.
    C. P. Sonnet, “Sunspot Time Series: Spectrum From Square Law Modulation of the Half Cycle,” Geophysical Research Letters, vol. 9 NO 12, pp. 1313–1316. 1982.CrossRefGoogle Scholar
  22. 22.
    R. N. Bracewell. “Simulating the Sunspot Cycle,” Nature, vol. 323, p. 516, Oct. 9). 1986.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • G. Larry Bretthorst
    • 1
  1. 1.Department of PhysicsWashington UniversitySt. LouisUSA

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