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Journal ArticleDOI

Least - squares frequency analysis of unequally spaced data

N. R. Lomb1
01 Feb 1976-Astrophysics and Space Science (Kluwer Academic Publishers)-Vol. 39, Iss: 2, pp 447-462
TL;DR: In this article, the statistical properties of least-squares frequency analysis of unequally spaced data are examined and it is shown that the reduction in the sum of squares at a particular frequency is a X22 variable.
Abstract: The statistical properties of least-squares frequency analysis of unequally spaced data are examined. It is shown that, in the least-squares spectrum of gaussian noise, the reduction in the sum of squares at a particular frequency is aX22 variable. The reductions at different frequencies are not independent, as there is a correlation between the height of the spectrum at any two frequencies,f1 andf2, which is equal to the mean height of the spectrum due to a sinusoidal signal of frequencyf1, at the frequencyf2. These correlations reduce the distortion in the spectrum of a signal affected by noise. Some numerical illustrations of the properties of least-squares frequency spectra are also given.
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Journal ArticleDOI
Adrian M. Price-Whelan1, B. M. Sipőcz1, Hans Moritz Günther1, P. L. Lim1, Steven M. Crawford1, S. Conseil1, D. L. Shupe1, M. W. Craig1, N. Dencheva1, Adam Ginsburg1, Jacob T VanderPlas1, Larry Bradley1, David Pérez-Suárez1, M. de Val-Borro1, T. L. Aldcroft1, Kelle L. Cruz1, Thomas P. Robitaille1, E. J. Tollerud1, C. Ardelean1, Tomáš Babej1, Y. P. Bach1, Matteo Bachetti1, A. V. Bakanov1, Steven P. Bamford1, Geert Barentsen1, Pauline Barmby1, Andreas Baumbach1, Katherine Berry1, F. Biscani1, Médéric Boquien1, K. A. Bostroem1, L. G. Bouma1, G. B. Brammer1, E. M. Bray1, H. Breytenbach1, H. Buddelmeijer1, D. J. Burke1, G. Calderone1, J. L. Cano Rodríguez1, Mihai Cara1, José Vinícius de Miranda Cardoso1, S. Cheedella1, Y. Copin1, Lia Corrales1, Devin Crichton1, D. DÁvella1, Christoph Deil1, É. Depagne1, J. P. Dietrich1, Axel Donath1, M. Droettboom1, Nicholas Earl1, T. Erben1, Sebastien Fabbro1, Leonardo Ferreira1, T. Finethy1, R. T. Fox1, Lehman H. Garrison1, S. L. J. Gibbons1, Daniel A. Goldstein1, Ralf Gommers1, Johnny P. Greco1, P. Greenfield1, A. M. Groener1, Frédéric Grollier1, A. Hagen1, P. Hirst1, Derek Homeier1, Anthony Horton1, Griffin Hosseinzadeh1, L. Hu1, J. S. Hunkeler1, Ž. Ivezić1, A. Jain1, T. Jenness1, G. Kanarek1, Sarah Kendrew1, Nicholas S. Kern1, Wolfgang Kerzendorf1, A. Khvalko1, J. King1, D. Kirkby1, A. M. Kulkarni1, Ashok Kumar1, Antony Lee1, D. Lenz1, S. P. Littlefair1, Zhiyuan Ma1, D. M. Macleod1, M. Mastropietro1, C. McCully1, S. Montagnac1, Brett M. Morris1, M. Mueller1, Stuart Mumford1, D. Muna1, Nicholas A. Murphy1, Stefan Nelson1, G. H. Nguyen1, Joe Philip Ninan1, M. Nöthe1, S. Ogaz1, Seog Oh1, J. K. Parejko1, N. R. Parley1, Sergio Pascual1, R. Patil1, A. A. Patil1, A. L. Plunkett1, Jason X. Prochaska1, T. Rastogi1, V. Reddy Janga1, J. Sabater1, Parikshit Sakurikar1, Michael Seifert1, L. E. Sherbert1, H. Sherwood-Taylor1, A. Y. Shih1, J. Sick1, M. T. Silbiger1, Sudheesh Singanamalla1, Leo Singer1, P. H. Sladen1, K. A. Sooley1, S. Sornarajah1, Ole Streicher1, P. Teuben1, Scott Thomas1, Grant R. Tremblay1, J. Turner1, V. Terrón1, M. H. van Kerkwijk1, A. de la Vega1, Laura L. Watkins1, B. A. Weaver1, J. Whitmore1, Julien Woillez1, Victor Zabalza1, Astropy Contributors1 
TL;DR: The Astropy project as discussed by the authors is a Python project supporting the development of open-source and openly developed Python packages that provide commonly needed functionality to the astronomical community, including the core package astropy.
Abstract: The Astropy Project supports and fosters the development of open-source and openly developed Python packages that provide commonly needed functionality to the astronomical community. A key element of the Astropy Project is the core package astropy, which serves as the foundation for more specialized projects and packages. In this article, we provide an overview of the organization of the Astropy project and summarize key features in the core package, as of the recent major release, version 2.0. We then describe the project infrastructure designed to facilitate and support development for a broader ecosystem of interoperable packages. We conclude with a future outlook of planned new features and directions for the broader Astropy Project.

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TL;DR: It is found that motion-induced signal changes are often complex and variable waveforms, often shared across nearly all brain voxels, and often persist more than 10s after motion ceases, which increase observed RSFC correlations in a distance-dependent manner.

2,713 citations


Cites methods from "Least - squares frequency analysis ..."

  • ...(2004), using a method based on the Lomb-Scargle periodogram (Lomb, 1976)....

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  • ...To compute the frequency content of uncensored data, we applied a least squares spectral analysis adapted for nonuniformly sampled data, as described in Mathias et al. (2004), using a method based on the Lomb-Scargle periodogram (Lomb, 1976)....

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Journal ArticleDOI
TL;DR: The results show that for most drivers studied, skin conductivity and heart rate metrics are most closely correlated with driver stress level, indicating that physiological signals can provide a metric of driver stress in future cars capable of physiological monitoring.
Abstract: This paper presents methods for collecting and analyzing physiological data during real-world driving tasks to determine a driver's relative stress level. Electrocardiogram, electromyogram, skin conductance, and respiration were recorded continuously while drivers followed a set route through open roads in the greater Boston area. Data from 24 drives of at least 50-min duration were collected for analysis. The data were analyzed in two ways. Analysis I used features from 5-min intervals of data during the rest, highway, and city driving conditions to distinguish three levels of driver stress with an accuracy of over 97% across multiple drivers and driving days. Analysis II compared continuous features, calculated at 1-s intervals throughout the entire drive, with a metric of observable stressors created by independent coders from videotapes. The results show that for most drivers studied, skin conductivity and heart rate metrics are most closely correlated with driver stress level. These findings indicate that physiological signals can provide a metric of driver stress in future cars capable of physiological monitoring. Such a metric could be used to help manage noncritical in-vehicle information systems and could also provide a continuous measure of how different road and traffic conditions affect drivers.

1,777 citations


Cites methods from "Least - squares frequency analysis ..."

  • ...A Lomb periodogram [15] was used to calculate the power spectrum [33], [34] of the heart rate time series because it can directly use unevenly sampled interbeat interval data and because it is robust to missed beats [35]....

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  • ...The EKG electrodes were placed in a modified lead II configuration to minimize motion artifacts and to maximize the amplitude of the R-waves, since both the heart rate [13] and heart rate variability (HRV) [14], [15] algorithms used in this analysis depend on R-wave peak detection....

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Journal ArticleDOI
TL;DR: The Lomb-Scargle periodogram is a common tool in the frequency analysis of unequally spaced data equivalent to least-squares fitting of sine waves as discussed by the authors, and it can be used to detect eccentric orbits of exoplanets.
Abstract: The Lomb-Scargle periodogram is a common tool in the frequency analysis of unequally spaced data equivalent to least-squares fitting of sine waves. We give an analytic solution for the generalisation to a full sine wave fit, including an offset and weights (χ 2 fitting). Compared to the Lomb-Scargle periodogram, the generalisation is superior as it provides more accurate frequencies, is less susceptible to aliasing, and gives a much better determination of the spectral intensity. Only a few modifications are required for the computation and the computational effort is similar. Our approach brings together several related methods that can be found in the literature, viz. the date-compensated discrete Fourier transform, the floating-mean periodogram, and the “spectral significance” estimator used in the SigSpec program, for which we point out some equivalences. Furthermore, we present an algorithm that implements this generalisation for the evaluation of the Keplerian periodogram that searches for the period of the best-fitting Keplerian orbit to radial velocity data. The systematic and non-random algorithm is capable of detecting eccentric orbits, which is demonstrated by two examples and can be a useful tool in searches for the orbital periods of exoplanets.

1,367 citations


Cites background or methods from "Least - squares frequency analysis ..."

  • ...Lomb (1976) showed that if data are Gaussian noise, the termsŶC 2 /ĈC and ŶS 2 /Ŝ S in Eq....

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  • ...The analytic solution for the generalised Lomb-Scargle periodogram can be obtained in a straightforward manner in the same way as outlined in Lomb (1976)....

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  • ...The equation for the periodogram was given by Barning (1963), and also Lomb (1976) and Scargle (1982), who furthermore investigated its statistical behaviour, especially the statistical significance of the detection ofa signal....

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  • ...As already mentioned by Lomb (1976) Eq....

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  • ...For a time series (ti , yi) with zero mean (y = 0), the LombScargle periodogram is defined as (normalisation from Lomb 1976): p̂(ω) = 1 ŶY ŶC 2 τ̂ ĈCτ̂ + ŶS 2 τ̂ Ŝ Ŝτ (1) = 1 ∑ i y 2 i [∑ i yi cosω(ti − τ̂) ]2 ∑ i cos2ω(ti − τ̂) + [∑ i yi sinω(ti − τ̂) ]2 ∑ i sin 2ω(ti − τ̂) …...

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Journal ArticleDOI
16 Mar 2012-Cell
TL;DR: This study demonstrates that longitudinal iPOP can be used to interpret healthy and diseased states by connecting genomic information with additional dynamic omics activity and reveals extensive heteroallelic changes during healthy and disease states and an unexpected RNA editing mechanism.

1,142 citations


Cites methods from "Least - squares frequency analysis ..."

  • ...For the integrated analysis, per omics set, for each time-series curve the Lomb-Scargle transformation (Hocke and Kämpfer, 2009; Lomb, 1976; Scargle, 1982, 1989) for unevenly sampled gapped time-series data was implemented (Ahdesmaki et al....

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  • ...For the integrated analysis, per omics set, for each time-series curve the Lomb-Scargle transformation (Hocke and Kämpfer, 2009; Lomb, 1976; Scargle, 1982, 1989) for unevenly sampled gapped time-series data was implemented (Ahdesmäki et al., 2007; Glynn et al., 2006; Van Dongen et al., 1999; Yang…...

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References
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46,339 citations


"Least - squares frequency analysis ..." refers background in this paper

  • ...For simplicity let us put is the bivariate normal probability function for variables x and y, with zero mean, unit variance and correlation e. Equation 26.3.2 of the Handbook of Mathematical Functions (Abramowitz and Stegun, 1964) gives where Z(x) is the normal probability function for a variable with zero mean and unit variance....

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  • ...…bivariate normal probability function for variables x and y, with zero mean, unit variance and correlation e. Equation 26.3.2 of the Handbook of Mathematical Functions (Abramowitz and Stegun, 1964) gives where Z(x) is the normal probability function for a variable with zero mean and unit variance....

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Journal ArticleDOI
TL;DR: In this paper, the spectral analysis using least squares (LS) is further developed to remove any undesired influence on the spectrum, which can be used for irregularly spaced as well as equidistantly spaced data.
Abstract: The concept of spectral analysis using least-squares is further developed to remove any undesired influence on the spectrum. The influence of such a ‘systematic noise’ can be eliminated without the necessity of knowing the magnitudes of the noise constituents. The technique can be used for irregularly spaced as well as equidistantly spaced data. The response to random noise is found to be constant in the frequency domain and its expected level is derived. Presence of random noise in the analyzed time series is shown to transform the spectrum merely linearly. Examples of applications of the technique are presented.

298 citations

Journal ArticleDOI
TL;DR: In this paper, a method for analyzing quasi-periodic light and radial velocity variations is presented, which gives a clear interpetation to the periodogram, reduces the alias problem, allows detection of very weak components, has potential for greater precision in frequency measurement, and is a shorter computation than the usual method.
Abstract: We present a method fof Fourier analyzing quasi-periodic light and radial velocity variations. The scheme gives a particularly clear interpetation to the periodogram, reduces the alias problem, allows detection of very weak components, has potential for greater precision in frequency measurement, and is a shorter computation than the usual method Our developement points out the possibility of error in evaluating the zero-frequency component by the usual method. We derive, under appropriate assumptions, a rigorous evaluation of the phase of each component. (auth)

40 citations


"Least - squares frequency analysis ..." refers methods in this paper

  • ...A slightly modified form of periodogram analysis has been devised by Gray and Desikachary (1973), in which prewhitening is carried out in the frequency domain instead of the time domain....

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Journal ArticleDOI

24 citations


"Least - squares frequency analysis ..." refers methods in this paper

  • ...A model of the observed short period variation in the ~ Vir 1934 velocities (Shobbrook et al., 1972) was set up....

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