Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Combined and Comparative Analysis of Power Spectra

Abstract

In solar physics, especially in exploratory stages of research, it is often necessary to compare the power spectra of two or more time series. One may, for instance, wish to estimate what the power spectrum of the combined data sets might have been, or one may wish to estimate the significance of a particular peak that shows up in two or more power spectra. One may also on occasion need to search for a complex of peaks in a single power spectrum, such as a fundamental and one or more harmonics, or a fundamental plus sidebands, etc. Visual inspection can be revealing, but it can also be misleading. This leads one to look for one or more ways of forming statistics, which readily lend themselves to significance estimation, from two or more power spectra. We derive formulas for statistics formed from the sum, the minimum, and the product of two or more power spectra. A distinguishing feature of our formulae is that, if each power spectrum has an exponential distribution, each statistic also has an exponential distribution. The statistic formed from the minimum power of two or more power spectra is well known and has an exponential distribution. The sum of two or more powers also has a well-known distribution that is not exponential, but a simple operation does lead to an exponential distribution. Concerning the product of two or more power spectra, we find an analytical expression for the case n = 2, and a procedure for computing the statistic for n > 2. We also show that some quite simple expressions give surprisingly good approximations.

This is a preview of subscription content, log in to check access.

References

  1. Bahcall, J. N. and Press, W. H.: 1991, Astrophys. J. 370, 730.

  2. Bahcall, J. N., Field, G. B. and Press, W. H.: 1987, Astrophys. J. (Letters) 320, L69.

  3. Bai, T.: 1992, Astrophys. J. 397, 584.

  4. Bai, T. A.: 2003, Astrophys. J. 591, 406.

  5. Bai, T. and Cliver, E. W.: 1990, Astrophys. J. 363, 299.

  6. Bartoszynki, R. and Niewiadomska-Bugaj, M.: 1996, Probability and Statistical Inference, Wiley, New York.

  7. Bickel, P. J. and Doksum, K. A.: 1977, Mathematical Statistics. Basic Ideas and Selected Topics, Holden-Day, Oakland.

  8. Bieber, J. W., Seckel, D., Stanev, T. and Steigman, G.: 1990, Nature 348, 407.

  9. Cleveland, B. T. et~al.: 1995, Proc. Nucl. Phys. B (Proc. Suppl), 38, 47.

  10. Cleveland, B. T. et~al.: 1998, Astrophys. J., 496, 505.

  11. Davis Jr, R. and Cox, A. N.: 1991, in A. N. Cox, W. C. Livingston, and M. S. Matthews (eds.), Solar Interior and Atmosphere, Arizona University Press, Tucson, p. 51.

  12. Dorman, L. I. and Wolfendale, A. W.: 1991, J. Phys. G: Nucl. Part. Phys. 17, 789.

  13. Horne, J. H. and Baliunas, S. L.: 1986, Astrophys. J. 302, 757.

  14. Lande, K. et~al.: 1992, in W. T. H. van Oers (ed.), AIP Conf. Proc., No. 243. Particles and Nuclear Physics, American Institute of Physics, New York, p. 1122.

  15. Lomnicki, Z. A.: 1967, Proc. Roy. Stat. Soc. B 29, 513.

  16. Massetti, S. and Storini, M.: 1996, Astrophys. J. 472, 827.

  17. McNutt, R. L., Jr.: 1995, Science 270, 1635.

  18. Oakley, D. S., Snodgrass, H. B., Ulrich, R. K. and VanDeKop, T. L.: 1994, Astrophys. J. (Letters) 437, L63.

  19. Oppenheim, A. V., Schafer, R. W. and Buck, J. R.: 1999, Discrete-Time Signal Processing 2nd ed., Prentice Hall, New York.

  20. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P.: 1992, Numerical Recipes, Cambridge University Press, Cambridge, p. 570.

  21. Rayleigh, Lord: 1903, Phil. Mag. 41, 238 (reprinted in Scientific Papers of Lord Rayleigh, Vol. V, Article 285, pp. 98–102 (New York: Dover, 1964)).

  22. Rice, J. A.: 1988, Mathematical Statistics and Data Analysis, Wadsworth & Brooks, Pacific Grove, California.

  23. Rieger, E., Share. G. H., Forrest, D. J., Kanbach, G., Reppin, C. and Chupp, E. L.: 1984, Nature 312, 623.

  24. Saio, H.: 1982, Astrophys. J. 256, 717.

  25. Scargle, J. D.: 1982, Astrophys. J., 263, 835.

  26. Schuster, A.: 1905, Proc. Royal Soc. 77, 136.

  27. Sturrock, P. A.: 2003, hep-ph/0304106.

  28. Sturrock, P. A.: 2004, Astrophys. J. 605, 568.

  29. Sturrock, P. A. and Weber, M. A.: 2002, Astrophys. J. 565, 1366.

  30. Sturrock, P. A., Walther, G. and Wheatland, M. S.: 1997, Astrophys. J. 491, 409.

  31. Sturrock, P. A., Walther, G. and Wheatland, M. S.: 1998, Astrophys. J. 507, 978.

  32. Thomson, D. J.: 1977, Bell System Tech. J. 56, 1769 and 1983.

  33. Thomson, D. J.: 2000, Proc. Tenth IEEE Signal Processing Workshop, IEEE Press, Pocono Manor, PA, p. 414.

  34. Thomson, D. J., Lanzerotti, L. J. and Maclennan, C. G.: 2001, J. Geophys. Res. 106, 15941.

  35. Walther, G.: 1999, Astrophys. J. 513, 990.

  36. Wolff, C. L.: 1976, Astrophys. J. 205, 612.

  37. Wolff, C. L.: 2002, Astrophys. J. (Letters) 580, L181.

Download references

Author information

Correspondence to P. A. Sturrock.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sturrock, P.A., Scargle, J.D., Walther, G. et al. Combined and Comparative Analysis of Power Spectra. Sol Phys 227, 137–153 (2005). https://doi.org/10.1007/s11207-005-7424-x

Download citation

Keywords

  • Time Series
  • Comparative Analysis
  • Power Spectrum
  • Visual Inspection
  • Exponential Distribution