, Volume 24, Issue 1, pp 99–105 | Cite as

Statistical analysis of Fourier coefficients in a restricted harmonic dial



Due to the physical nature of certain periodic data, the harmonic dial points (the Fourier coefficients obtained from harmonic analysis of the data) are sometimes restricted to circular regions in the dial plane. It is proposed that a circular normal distribution (CND) truncated outside a circular region be used to describe the probabilistic behavior of the random phenomena. Recurrence relations for the population moments of a CND truncated outside a circular region are derived. These recurrence relations are used to obtain consistent asymptotically (jointly) normal estimators of the unknown parameters of the distribution. A numerical example based on the harmonic dial points representing the 27-day recurrence tendency of the daily international magnetic character-figureC i is given to illustrate the theory.


Stochastic Process Probability Theory Economic Theory Harmonic Analysis Dial 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bartels, J.: Statistical methods for research on diurnal variations, Terrestrial Magnetism and Atmospheric Electricity37, 1932, p. 291.Google Scholar
  2. —: Random fluctuations, persistence, and quasi-persistence in geophysical and cosmical periodicities, Terrestrial Magnetism and Atmospheric Electricity40, 1935, p. 1.Google Scholar
  3. Cramér, H.: Mathematical Methods of Statistics, Princeton, N.J., 1946.Google Scholar
  4. Dyer, D.D.: Estimation in a truncated circular normal distribution with ballistic applications, Operations Research22, 1974, p. 197.Google Scholar
  5. Forbush, S.E.: Time-variations of cosmic rays, in Encyclopedia of Physics49 (part 1), edited by S. Flügge and J. Bartels, New York 1966.Google Scholar
  6. Gröbner, W., andN. Hofreiter: Integraltafel, Zweiter Teil, Bestimmte Integrale, New York 1966.Google Scholar
  7. Haurwitz, B.: Tidal phenomena in the upper atmosphere, Technical Note No. 58, World Meteorological Organization, 1964.Google Scholar
  8. Mauchly, J.W.: A significance-test for ellipticity in the harmonic dial, Terrestrial Magnetism and Atmospheric Electricity45, 1940, p. 145.Google Scholar
  9. Owen, D.B.: Handbook of Statistical Tables, Reading, Mass., 1962.Google Scholar

Copyright information

© Physica-Verlag Rudolf Liebing KG 1977

Authors and Affiliations

  • D. Dyer
    • 1
  1. 1.Department of MathematicsUniversity of Texas at ArlingtonArlington

Personalised recommendations