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Stochastic significance of peaks in the least-squares spectrum

Authors
  • Pagiatakis, S. D.1
  • 1 Natural Resources Canada, Earth Sciences, Geodetic Survey Division, 615 Booth Street, Ottawa, Ontario, Canada K1A 0E9 e-mail: [email protected]; Tel.: 613 995 8720; Fax: 613 992 1468, CA
Type
Published Article
Journal
Journal of Geodesy
Publisher
Springer-Verlag
Publication Date
Mar 01, 1999
Volume
73
Issue
2
Pages
67–78
Identifiers
DOI: 10.1007/s001900050220
Source
Springer Nature
Keywords
License
Yellow

Abstract

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.

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