Journal of Geodesy

, Volume 73, Issue 2, pp 67–78 | Cite as

Stochastic significance of peaks in the least-squares spectrum

  • S. D. Pagiatakis


The least-squares spectral analysis method is reviewed, with emphasis on its remarkable property to accept time series with an associated, fully populated covariance matrix. Two distinct cases for the input covariance matrix are examined: (a) it is known absolutely (a-priori variance factor known); and (b) it is known up to a scale factor (a-priori variance factor unknown), thus the estimated covariance matrix is used. For each case, the probability density function that underlines the least-squares spectrum is derived and criteria for the statistical significance of the least-squares spectral peaks are formulated. It is shown that for short series (up to about 150 values) with an estimated covariance matrix (case b), the spectral peaks must be stronger to be statistically significant than in the case of a known covariance matrix (case a): the shorter the series and the lower the significance level, the higher the difference becomes. For long series (more than about 150 values), case (b) converges to case (a) and the least-squares spectrum follows the beta distribution. The results of this investigation are formulated in two new theorems.

Key words. Least squares Spectrum Hindcast skill Regression Statistical test 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • S. D. Pagiatakis
    • 1
  1. 1.Natural Resources Canada, Earth Sciences, Geodetic Survey Division, 615 Booth Street, Ottawa, Ontario, Canada K1A 0E9 e-mail:; Tel.: 613 995 8720; Fax: 613 992 1468CA

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