Open AccessAbout a new method of spectral analysis
Author(s) -
Anufriev A.
Publication year - 1987
Publication title -
geophysical journal of the royal astronomical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0016-8009
DOI - 10.1111/j.1365-246x.1987.tb00746.x
Subject(s) - amplitude , spectral analysis , maximum entropy spectral estimation , maximum entropy method , series (stratigraphy) , harmonic analysis , time series , mathematics , principle of maximum entropy , algorithm , mathematical analysis , statistical physics , geology , physics , statistics , optics , paleontology , quantum mechanics , spectroscopy
Summary A new method, non‐orthogonal spectral analysis (NSA), for the harmonic analysis of short time‐series is proposed. Using as an initial approximation the results of the maximum entropy method (MEM), the method allows an improvement in the precision of period estimates and also calculation of amplitude values. Pairs of ‘virtual frequencies’ are chosen close to each frequency of initial approximation, and corresponding ‘virtual amplitudes’ are found inverting the matrix which connects them with the Fourier transform of the time series. It is shown that when the amplitude of the left frequency of each pair equals zero, then the right frequency and amplitude are close to the true ones and exactly coincide with them when without noise. To calculate the true frequencies, an iterative algorithm is proposed based on a revealed connection between virtual amplitudes and deviations of the virtual frequencies from the true ones. Analytical estimates of the period and amplitude deviations arising from the presence of noise are developed. A method based on NSA is given for a graphical estimation of the precision of any harmonic analysis (not only NSA). Spectrum of residual amplitudes is introduced which equals zero when the frequencies to be estimated coincide with the true ones and the noise is absent. When with noise, these amplitudes are of order σ/√ N /2 where σ 2 is the variance of noise and N is the number of data describing the time series. The maximum of the residual amplitude which considerably exceeds this value points to a previously unknown harmonic at a frequency where this maximum occurs and with an amplitude approximately equal to the residual amplitude. The computer analysis of an artificial and natural time series carried out by the suggested method is discussed.