On smoothing techniques for the removal of periodic noise of known period

  • R. S. Anderssen
  • E. Seneta


This paper is concerned with the analysis of the linear modely(n)=Xβ(n)+S(n)+γ(n) for the data sequencey(n) (n=1, 2, ..., N) whereX={xIJ} is a knownJ × M matrix of full rankM. Here, theβ(n) are unknown vectors, which we wish to estimate for eachn; S(n) (n=1, 2, ..., N) is a periodic component (which we wish to estimate or remove) superimposed on the linear structureXβ(n); andγ(n) is an error vector which is specified as having zero expectation (with possible further properties). Such models commonly occur in geophysical data analysis.

Key words

data processing Fourier analysis moving average time series 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aitken, A. C., 1957, Statistical mathematics (8th ed.): Oliver and Boyd, Edinburgh, 153 p.Google Scholar
  2. Anderssen, R. S., 1969a, On the solution of certain overdetermined systems of linear equations which arise in geophysics: Jour. Geophys. Res., v. 74, no. 4, p. 1045–1051.Google Scholar
  3. Anderssen, R. S., 1969b, Computational considerations for linear least squares methods,in Least squares methods in data analysis: Australian Nat. Univ. Computer Centre Publ. CC2/69, p. 19–31.Google Scholar
  4. Anderssen, R. S., and Seneta, E., 1969, New analysis for the geomagneticD st field of the magnetic storm on June 18–19, 1936: Jour. Geophys. Res., v. 74, no. 10, p. 2768–2773.Google Scholar
  5. Anderssen, R. S., Doyle, H. A., Petersons, H. F., and Seneta, E., 1970, On the smoothing and spherical harmonic analysis of the storm of September 25, 1958: Jour. Geophys. Res., v. 75, no. 13, p. 2569–2577.Google Scholar
  6. Anscombe, F. J., 1967, Topics in the investigation of linear relations fitted by the method of least squares: Jour. Roy. Stat. Soc., Ser. B, v. 29, p. 1–52.Google Scholar
  7. Björck, A., 1967, Solving linear least squares problems by Gram-Schmidt orthogonalization: BIT, Bd. 7, Heft 1, p. 1–21.Google Scholar
  8. Chapman, S., 1951, The earth's magnetism (2nd ed.): Methuen and Co., London, 127 p.Google Scholar
  9. Chapman, S., and Price, A. T., 1930, Electric and magnetic states of the interior of the earth: Phil. Trans. Roy. Soc., A., v. 229, p. 427–460.Google Scholar
  10. Golub, G. H., 1965, Numerical methods for solving linear least squares problems: Numer. Math., Bd. 7, Heft 3, p. 206–216.Google Scholar
  11. Hamming, R. W., 1962, Numerical methods for scientists and engineers, McGraw-Hill Book Co., New York, 411 p.Google Scholar
  12. Hannan, E. J., 1963, The estimation of seasonal variation in economic time series: Jour. Am. Stat. Assoc., v. 58, p. 31–44.Google Scholar
  13. Jenkins, G. M., and Watts, D. G., 1968, Spectral analysis and its application: Holden-Day, San Francisco, 525 p.Google Scholar
  14. Jordan, T. L., 1968, Experiments on error growth associated with some linear least squares procedures: Math. Comp., v. 22, no. 103, p. 579–588.Google Scholar
  15. Kendail, M. G., and Stuart, A., 1969, The advanced theory of statistics, Volumes 1 and 2 (3rd ed.), Charles Griffin & Co., Ltd., London, 1129 p.Google Scholar
  16. Matsushita, S., and Campbell, W. H., 1967, Physics of geomagnetic phenomena, Volumes I and II: Academic Press, New York, 1398 p.Google Scholar
  17. Mood, A. M., and Graybill, F. A., 1963, Introduction to the theory of statistics (2nd ed.), McGraw-Hill Book Co., New York, 443 p.Google Scholar
  18. Rikitake, T., and Sato, S., 1957, The geomagneticD st field of the magnetic storm on June 18–19, 1936: Earthquake Res. Inst. Bull., Tokyo Univ., v. 35, p. 7–21.Google Scholar
  19. Slutsky, E., 1937, The summation of random causes as the source of cyclic processes: Econometrica, v. 5, p. 105–146.Google Scholar
  20. Waugh, F. V., and Dwyer, P. S., 1945, Compact computation of the inverse of a matrix: Ann. Math. Stat., v. 16, no. 1, p. 259–271.Google Scholar

Copyright information

© Plenum Publishing Corporation 1971

Authors and Affiliations

  • R. S. Anderssen
    • 1
  • E. Seneta
    • 2
  1. 1.Computer CentreThe Australian National UniversityAustralia
  2. 2.Statistics DepartmentThe Australian National UniversityAustralia

Personalised recommendations