https://wikimd.org/index.php?action=history&feed=atom&title=Least-squares_spectral_analysisLeast-squares spectral analysis - Revision history2026-04-27T11:35:06ZRevision history for this page on the wikiMediaWiki 1.44.2https://wikimd.org/index.php?title=Least-squares_spectral_analysis&diff=5892829&oldid=prevPrab: CSV import2024-06-05T23:32:34Z<p>CSV import</p>
<p><b>New page</b></p><div>[[File:Linear_least_squares2.svg|thumb|Linear_least_squares2]] [[file:Linear_least_squares_example2.svg|right|thumb|Linear_least_squares_example2]] [[file:Periodogram.svg|right|thumb|Periodogram]] [[file:PDF_of_the_Beta_distribution.gif|thumb|PDF_of_the_Beta_distribution]] '''Least-squares spectral analysis''' (LSSA) is a method used in [[signal processing]] and [[time series analysis]] to estimate the [[power spectrum]] of a signal. Unlike traditional methods such as the [[Fourier transform]], LSSA is particularly useful for analyzing data with irregular sampling intervals or missing data points.<br />
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==Overview==<br />
LSSA is based on the principle of [[least squares]], which minimizes the sum of the squares of the differences between the observed and estimated values. This method is advantageous when dealing with non-uniformly sampled data, as it can provide more accurate spectral estimates compared to conventional techniques.<br />
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==History==<br />
The concept of least-squares spectral analysis was first introduced by [[Peter D. Welch]] in the 1960s. It has since been developed and refined by various researchers, becoming a valuable tool in fields such as [[astronomy]], [[geophysics]], and [[economics]].<br />
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==Mathematical Formulation==<br />
The LSSA method involves fitting a model of the form:<br />
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\[ y(t) = \sum_{k=1}^{N} A_k \cos(2\pi f_k t + \phi_k) \]<br />
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where \( y(t) \) is the observed data, \( A_k \) are the amplitudes, \( f_k \) are the frequencies, and \( \phi_k \) are the phases. The parameters \( A_k \), \( f_k \), and \( \phi_k \) are estimated by minimizing the sum of the squared differences between the observed data and the model.<br />
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==Applications==<br />
LSSA is widely used in various scientific and engineering disciplines. Some of the key applications include:<br />
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* [[Astronomy]]: Analyzing the light curves of variable stars.<br />
* [[Geophysics]]: Studying seismic data and earth tides.<br />
* [[Economics]]: Investigating economic cycles and trends.<br />
* [[Medicine]]: Analyzing physiological signals such as heart rate variability.<br />
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==Advantages==<br />
* Handles irregularly sampled data effectively.<br />
* Provides high-resolution spectral estimates.<br />
* Can be applied to data with missing points.<br />
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==Limitations==<br />
* Computationally intensive compared to traditional methods.<br />
* Requires careful selection of model parameters.<br />
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==Related Pages==<br />
* [[Fourier transform]]<br />
* [[Power spectrum]]<br />
* [[Signal processing]]<br />
* [[Time series analysis]]<br />
* [[Least squares]]<br />
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==See Also==<br />
* [[Lomb-Scargle periodogram]]<br />
* [[Autoregressive model]]<br />
* [[Wavelet transform]]<br />
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==References==<br />
{{Reflist}}<br />
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==External Links==<br />
{{Commons category|Least-squares spectral analysis}}<br />
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[[Category:Signal processing]]<br />
[[Category:Time series analysis]]<br />
[[Category:Mathematical methods in physics]]<br />
[[Category:Mathematical and quantitative methods (economics)]]<br />
[[Category:Spectral analysis]]<br />
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