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'''Least-squares spectral analysis''' (LSSA) is a method of estimating a [[frequency spectrum]], based on a [[least squares]] fit |
'''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/>. |
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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 (or the Lomb periodogram) and the Lomb–Scargle method, based on the contributions of N. R. Lomb<ref name=lomb/> and, independently, Jeffrey D. Scargle<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 (or the Lomb periodogram) and the Lomb–Scargle method, based on the contributions of N. R. Lomb<ref name=lomb/> and, independently, Jeffrey D. Scargle<ref name=scar/>. |
Revision as of 13:40, 12 October 2007
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 (or the Lomb periodogram) and the Lomb–Scargle method, based on the contributions of N. R. Lomb[6] and, independently, Jeffrey D. Scargle[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 Vanícek in 1969[4] and in 1971[5]. The method was then simplified in 1976 by Lomb, who pointed out its close connection to periodogram analysis[6]. It was subsequently further analyzed, modified, and applied by Scargle[7].
Scargle states that his paper "does not introduce a new detection technique, but instead studies the reliability and efficiency of detection with the most commonly used technique, the periodogram, in the case where the observation times are unevenly spaced," and further points out in reference to least-squares fitting of sinusoids compared to periodogram analysis, that his paper "establishes, apparently for the first time, that (with the proposed modifications) these two methods are exactly equivalent"[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 Vanícek 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 Vanícek spectrum can also be expressed in dB[11]. Note that magnitudes in the Vanícek spectrum follow ß-distribution[12]. Inverse transformation has been discussed in literature as well[13].
Implementation
The LSSA can be implemented in less than a page of MATLAB code.[14] For each frequency in a desired set of frequencies, sine and cosine functions are evaluated at the times corresponding to the data samples, and dot products of the data vector with the sinusoid vectors are taken and appropriately normalized; following the method known as Lomb/Scargle periodogram, a time shift is calculated for each frequency to orthogonalize the sine and cosine components before the dot product, as desribed by Craymer[13]; finally, a power is computed from those two amplitude components. This same process implements a discrete Fourier transform when the data are uniformly spaced in time and the frequencies chosen correspond to integer numbers of cycles over the finite data record.
As Craymer explains, this method treats each sinusoidal component independently, or out of context, even though they may not be orthogonal on the data points, whereas Vaníček's original method does a full simultaneous least-squares fit by solving a matrix equation, partitioning the total data variance between the specified sinusoid frequencies[13]. Such a matrix least-squares solution is natively available in MATLAB as the backslash operator[15].
References
- ^ Press; et al. (2007). Numerical Recipes (3rd Edition ed.). Cambridge University Press. ISBN 0521880688.
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(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 (1982).
- ^ 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)
- ^ a b c Craymer, M.R., The Least Squares Spectrum, Its Inverse Transform and Autocorrelation Function: Theory and Some Applications in Geodesy, Ph.D. Dissertation, University of Toronto, Canada (1998).
- ^ Richard A. Muller and Gordon J. MacDonald (2000). Ice Ages and Astronomical Causes: Data, Spectral Analysis, and Mechanisms. Springer. ISBN 3540437797.
- ^ Timothy A. Davis and Kermit Sigmon (2005). MATLAB Primer. CRC Press. ISBN 1584885238.
External links
- LSSA software freeware download (via ftp), from the Natural Resources Canada.