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Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit between data and trigonometric functions. Since the Fourier analysis, as the most used spectral method in science, generally boosts long-periodic noise in long gapped records, the LSSA is its superior alternative for analyzing long incomplete records such as most natural datasets.^[1]^[2]

LSSA is also known as Vaníček analysis after Canadian geodesist and geophysicist Petr Vaníček (sometimes Vaníček spectral analysis^[3] or Gauss-Vaníček spectral analysis^[2]); Vaníček published the method in 1969^[4] and 1971.^[5] It is sometimes also known as the Lomb method, the Lomb periodogram, or the Lomb-Scargle method, based on N.R. Lomb's^[6] and J.D. Scargle's^[7] independent simplifications of the method. Note however that even the Scargle's paper stated it meant no new method at all.^[7]

Historical background

The concept of least-squares fitting of data with trigonometric functions was first remarked briefly in a 1963 paper by Barning^[8]. It was first developed mathematically by Vaníček in 1969^[4] and in 1971^[5]. The method was then simplified in 1976 by Lomb^[6] and by Scargle^[7].

Main features

Processing any datasets, equidistant or incomplete, regardless of record length;
Rigorous testing of the statistical null hypothesis;
Straightforward significance level regime;
Weighing of the data on a per point basis rather than on a time interval basis;
Accurate simultaneous detection of field relative dynamics and eigenfrequencies;
Describes fields uniquely and relatively thanks to output’s linear background noise;
Removal of unwanted frequencies from a record during the processing;
Removal of periodic noise from a time series with minimal distortion of the spectrum of the remaining series;
Setting up of spectral resolution at will;
Outputs spectra in percentage variance (var%) or decibels (dB).

Applications

Vaníček analysis has many scientific applications — ranging from astronomy, geophysics, physics, microbiology, genetics and medicine, to mathematics and finance.^[9] This wide applicability stems from many useful properties of the least-squares fit. The most useful feature of the method is enabling for incomplete records to be spectrally analyzed, without the need to manipulate the record or to invent otherwise non-existent data.

Just like the Fourier transformation in signal processing can isolate individual components of a complex signal, concentrating them for easier detection and/or removal, the Vaníček analysis can do the same for scientific analyses by using the original i.e. "raw" dataset alone, without manipulating either the input data or output spectra. While to switch to such a sophistication as the signal fitting may not be justified in applications like electronics systems, the Vaníček Analysis is the method of choice in virtually all natural sciences, where results are normally more decision making than profit driven.

Since magnitudes in the Vaníček spectrum depict the contribution of a period to the variance of the time-series, of the order of (some) %^[4], in physical sciences this means that the method can be used for measuring field relative dynamics ^[9]^[2]. Generally, spectral magnitudes defined in the above manner enable the output's straightforward significance level regime.^[10] Alternatively, magnitudes in the Vaníček spectrum can also be expressed in dB.^[11] Note that magnitudes in the Vaníček spectrum follow β-distribution.^[12] Inverse transformation has been discussed in literature as well.^[13]

References

^ Press; et al. (2007). Numerical Recipes (3rd Edition ed.). Cambridge University Press. ISBN 0521880688. {{cite book}}: |edition= has extra text (help); Explicit use of et al. in: |author= (help)
^ ^a ^b ^c Omerbashich M., "Gauss–Vanicek spectral analysis of the Sepkoski compendium: no new life cycles", Pages 26-30, Computing in Science & Engineering, Volume 8, Number 4, (July-August, 2006) ISSN 1521-9615.
^ Taylor J., Hamilton S. Some tests of the Vanícek method of spectral analysis, Astrophysics and Space Science, International Journal of Cosmic Physics, D. Reidel Publishing Co., Dordrecht, Holland (1972).
^ ^a ^b ^c Vanícek P. Approximate Spectral Analysis by Least-squares Fit, Astrophysics and Space Science, Pages 387-391, Volume 4 (1969).
^ ^a ^b Vanícek P. Further development and properties of the spectral analysis by least-squares fit, Astrophysics and Space Science, Pages 10-33, Volume 12 (1971).
^ ^a ^b Lomb, N. R., "Least-squares frequency analysis of unequally spaced data," Astrophysics and Space Science 39, p.447–462 (1976).
^ ^a ^b ^c Scargle, J. D., "Studies in astronomical time series analysis II: Statistical aspects of spectral analysis of unevenly spaced data," Astrophysics and Space Science 302, p.757–763 (1976).
^ Barning, F.J.M. The numerical analysis of the light-curve of 12 lacertae, Bulletin of the Astronomical Institutes of the Netherlands, 17, p.22-28.
^ ^a ^b Omerbashich M. , Earth-Model Discrimination Method, pp.129, Ph.D. dissertation, University of New Brunswick, Canada (2003).
^ Beard, A.G., Williams, P.J.S., Mitchell, N.J. & Muller, H.G. A special climatology of planetary waves and tidal variability, J Atm. Solar-Ter. Phys. 63 (09), p.801-811 (2001).
^ Pagiatakis, S. Stochastic significance of peaks in the least-squares spectrum, J of Geodesy 73, p.67-78 (1999).
^ Steeves, R.R. A statistical test for significance of peaks in the least squares spectrum, Collected Papers of the Geodetic Survey, Department of Energy, Mines and Resources, Surveys and Mapping, Ottawa, Canada, p.149-166 (1981)
^ Craymer, M.R. The Least Squares Spectrum, Its Inverse Transform and Autocorrelation Function: Theory and Some Applications in Geodesy (compressed pdf), Ph.D. Dissertation, University of Toronto, Canada (1998).

External links

LSSA software freeware download (via ftp), from the Natural Resources Canada.

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 '''Least-squares spectral analysis''' (LSSA) is a method of estimating a [[frequency spectrum]], based on a [[least squares]] fit between data and [[trigonometric functions]]. Since the [[Fourier analysis]], as the most used spectral method in science, generally boosts long-periodic noise in long gapped records, the LSSA is its superior alternative for analyzing long incomplete records such as most [[natural]] [[dataset]]s.<ref name=pres> {{cite book | url = http://books.google.com/books?id=9GhDHTLzFDEC&pg=PA685&dq=%22spectral+analysis%22+%22vanicek%22+inauthor:press&as_brr=3&ei=10EKR6akEovqoQLOy9iqDQ&ie=ISO-8859-1&sig=Pt6HJ2hsLodcsrr2PUQDxnVSlPU | author = Press et al. | title = Numerical Recipes | edition = 3rd Edition | year = 2007 | publisher = Cambridge University Press | isbn = 0521880688}}</ref><ref name=sepk/>
-LSSA is also known as '''Vaníček analysis''' after Canadian [[geodesist]] and [[geophysicist]] [[Petr Vaníček]] (sometimes Vaníček spectral analysis<ref>Taylor J., Hamilton S. Some tests of the Vanícek method of spectral analysis, Astrophysics and Space Science, International Journal of Cosmic Physics, D. Reidel Publishing Co., Dordrecht, Holland (1972).</ref> or [[Carl Friedrich Gauss|Gauss]]-Vaníček spectral analysis<ref name=sepk>Omerbashich M., [http://arxiv.org/pdf/math-ph/0608014 "Gauss–Vanicek spectral analysis of the Sepkoski compendium: no new life cycles"], Pages 26-30, ''Computing in Science & Engineering'', Volume 8, Number 4, (July-August, 2006) ISSN 1521-9615.</ref>); Vaníček published the method in 1969<ref name=vana>Vanícek P. Approximate Spectral Analysis by Least-squares Fit, Astrophysics and Space Science, Pages 387-391, Volume 4 (1969).</ref> and 1971<ref name=vanb>Vanícek P. Further development and properties of the spectral analysis by least-squares fit, Astrophysics and Space Science, Pages 10-33, Volume 12 (1971).</ref>.  It is sometimes also known as the Lomb method, the Lomb–Scargle method, or the Lomb periodogram, based on the method's simplifications by N.R. Lomb<ref name=lomb/> and, independently, by J.D. Scargle<ref name=scar/>. Note however that even the Scargle's paper stated it meant no new method at all<ref name=scar/>.
+LSSA is also known as '''Vaníček analysis''' after Canadian [[geodesist]] and [[geophysicist]] [[Petr Vaníček]] (sometimes Vaníček spectral analysis<ref>Taylor J., Hamilton S. Some tests of the Vanícek method of spectral analysis, Astrophysics and Space Science, International Journal of Cosmic Physics, D. Reidel Publishing Co., Dordrecht, Holland (1972).</ref> or [[Carl Friedrich Gauss|Gauss]]-Vaníček spectral analysis<ref name=sepk>Omerbashich M., [http://arxiv.org/pdf/math-ph/0608014 "Gauss–Vanicek spectral analysis of the Sepkoski compendium: no new life cycles"], Pages 26-30, ''Computing in Science & Engineering'', Volume 8, Number 4, (July-August, 2006) ISSN 1521-9615.</ref>); Vaníček published the method in 1969<ref name=vana>Vanícek P. Approximate Spectral Analysis by Least-squares Fit, Astrophysics and Space Science, Pages 387-391, Volume 4 (1969).</ref> and 1971.<ref name=vanb>Vanícek P. Further development and properties of the spectral analysis by least-squares fit, Astrophysics and Space Science, Pages 10-33, Volume 12 (1971).</ref>   It is sometimes also known as the Lomb method, the Lomb [[periodogram]], or the Lomb-Scargle method, based on N.R. Lomb's<ref name=lomb/>  and J.D. Scargle's<ref name=scar/> independent simplifications of the method.  Note however that even the Scargle's paper stated it meant no new method at all.<ref name=scar/><!--QUESTION: is the "Note however ... no new method at all" line actually necessary (or factual?) here?-->
 ==Historical background==