SigSpec

SigSpec (acronym of SIGnificance SPECtrum) is a statistical technique to provide the reliability of periodicities in a measured (noisy and not necessarily equidistant) time series.^[1] It relies on the amplitude spectrum obtained by the Discrete Fourier transform (DFT) and assigns a quantity called the spectral significance (frequently abbreviated by “sig”) to each amplitude. This quantity is a logarithmic measure of the probability that the given amplitude level would be seen in white noise, in the sense of a type I error. It represents the answer to the question, “What would be the chance to obtain an amplitude like the measured one or higher, if the analysed time series were random?”

SigSpec may be considered a formal extension to the Lomb-Scargle periodogram,^[2][3] appropriately incorporating a time series to be averaged to zero before applying the DFT, which is done in many practical applications. When a zero-mean corrected dataset has to be statistically compared to a random sample, the sample mean (rather than the population mean only) has to be zero.

Probability density function (pdf) of white noise in Fourier space

Considering a time series to be represented by a set of ${\displaystyle K}$  pairs ${\displaystyle (t_{k},x_{k})}$ , the amplitude pdf of white noise in Fourier space, depending on frequency and phase angle may be described in terms of three parameters, ${\displaystyle \alpha _{0}}$ , ${\displaystyle \beta _{0}}$ , ${\displaystyle \theta _{0}}$ , defining the “sampling profile”, according to

${\displaystyle \tan 2\theta _{0}={\frac {K\sum _{k=0}^{K-1}\sin 2\omega t_{k}-2\left(\sum _{k=0}^{K-1}\cos \omega t_{k}\right)\left(\sum _{k=0}^{K-1}\sin \omega t_{k}\right)}{K\sum _{k=0}^{K-1}\cos 2\omega t_{k}-{\big (}\sum _{k=0}^{K-1}\cos \omega t_{k}{\big )}^{2}+{\big (}\sum _{k=0}^{K-1}\sin \omega t_{k}{\big )}^{2}}},}$ 

${\displaystyle \alpha _{0}={\sqrt {{\frac {2}{K^{2}}}\left(K\sum _{k=0}^{K-1}\cos ^{2}\left(\omega t_{k}-\theta _{0}\right)-\left[\sum _{l=0}^{K-1}\cos \left(\omega t_{k}-\theta _{0}\right)\right]^{2}\right)}},}$ 

${\displaystyle \beta _{0}={\sqrt {{\frac {2}{K^{2}}}\left(K\sum _{k=0}^{K-1}\sin ^{2}\left(\omega t_{k}-\theta _{0}\right)-\left[\sum _{l=0}^{K-1}\sin \left(\omega t_{k}-\theta _{0}\right)\right]^{2}\right)}}.}$ 

In terms of the phase angle in Fourier space, ${\displaystyle \theta }$ , with

${\displaystyle \tan \theta ={\frac {\sum _{k=0}^{K-1}\sin \omega t_{k}}{\sum _{k=0}^{K-1}\cos \omega t_{k}}},}$ 

the probability density of amplitudes is given by

${\displaystyle \phi (A)={\frac {KA\cdot \operatorname {sock} }{2<x^{2}>}}\exp \left(-{\frac {KA^{2}}{4<x^{2}>}}\cdot \operatorname {sock} \right),}$ 

where the sock function is defined by

${\displaystyle \operatorname {sock} (\omega ,\theta )=\left[{\frac {\cos ^{2}\left(\theta -\theta _{0}\right)}{\alpha _{0}^{2}}}+{\frac {\sin ^{2}\left(\theta -\theta _{0}\right)}{\beta _{0}^{2}}}\right]}$ 

and ${\displaystyle <x^{2}>}$  denotes the variance of the dependent variable ${\displaystyle x_{k}}$ .

False-alarm probability and spectral significance

Integration of the pdf yields the false-alarm probability that white noise in the time domain produces an amplitude of at least ${\displaystyle A}$ ,

${\displaystyle \Phi _{\operatorname {FA} }(A)=\exp \left(-{\frac {KA^{2}}{4<x^{2}>}}\cdot \operatorname {sock} \right).}$ 

The sig is defined as the negative logarithm of the false-alarm probability and evaluates to

${\displaystyle \operatorname {sig} (A)={\frac {KA^{2}\log e}{4<x^{2}>}}\cdot \operatorname {sock} .}$ 

It returns the number of random time series one would have to examine to obtain one amplitude exceeding ${\displaystyle A}$  at the given frequency and phase.

Applications

SigSpec is primarily used in asteroseismology to identify variable stars and to classify stellar pulsation (see references below). The fact that this method incorporates the properties of the time-domain sampling appropriately makes it a valuable tool for typical astronomical measurements containing data gaps.

See also

• Spectral density estimation

References

1. P. Reegen (2007). "SigSpec - I. Frequency- and phase-resolved significance in Fourier space". Astronomy and Astrophysics. 467 (3): 1353–1371. arXiv:physics/0703160. Bibcode:2007A&A...467.1353R. doi:10.1051/0004-6361:20066597. S2CID 15076973.
2. N. R. Lomb (1976). "Least-squares frequency analysis of unequally spaced data". Astrophysics and Space Science. 39 (2): 447–462. Bibcode:1976Ap&SS..39..447L. doi:10.1007/BF00648343. S2CID 2671466.
3. J. D. Scargle (1982). "Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data". The Astrophysical Journal. 263: 835–853. Bibcode:1982ApJ...263..835S. doi:10.1086/160554.

• M. Breger; S. M. Rucinski; P. Reegen (2007). "The Pulsation of EE Camelopardalis". The Astronomical Journal. 134 (5): 1994–1998. arXiv:0709.3393. Bibcode:2007AJ....134.1994B. doi:10.1086/522795. S2CID 120843648.
• M. Gruberbauer; K. Kolenberg; J. F. Rowe; D. Huber; J. M. Matthews; P. Reegen; R. Kuschnig; C. Cameron; T. Kallinger; W. W. Weiss; D. B. Guenther; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2007). "MOST photometry of the RRdLyrae variable AQLeo: two radial modes, 32 combination frequencies and beyond". Monthly Notices of the Royal Astronomical Society. 379 (4): 1498–1506. arXiv:0705.4603. Bibcode:2007MNRAS.379.1498G. doi:10.1111/j.1365-2966.2007.12042.x. S2CID 55678660.
• M. Gruberbauer; H. Saio; D. Huber; T. Kallinger; W. W. Weiss; D. B. Guenther; R. Kuschnig; J. M. Matthews; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2008). "MOST photometry and modeling of the rapidly oscillating (roAp) star γ Equulei". Astronomy and Astrophysics. 480 (1): 223–232. arXiv:0801.0863. Bibcode:2008A&A...480..223G. doi:10.1051/0004-6361:20078830. S2CID 54726017.
• D. B. Guenther; T. Kallinger; P. Reegen; W. W. Weiss; J. M. Matthews; R. Kuschnig; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2007). "Searching for p-modes in η Bootis & Procyon using MOST satellite data". Communications in Asteroseismology. 151: 5–25. Bibcode:2007CoAst.151....5G. doi:10.1553/cia151s5.
• D. B. Guenther; T. Kallinger; K. Zwintz; W. W. Weiss; J. Tanner (2007). "Seismology of Pre-Main-Sequence Stars in NGC 6530" (PDF). The Astrophysical Journal. 671 (1): 581–591. Bibcode:2007ApJ...671..581G. doi:10.1086/522880. S2CID 54866017.
• D. Huber; H. Saio; M. Gruberbauer; W. W. Weiss; J. F. Rowe; M. Hareter; T. Kallinger; P. Reegen; J. M. Matthews; R. Kuschnig; D. B. Guenther; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2008). "MOST photometry of the roAp star 10 Aquilae". Astronomy and Astrophysics. 483 (1): 239–248. arXiv:0803.1721. Bibcode:2008A&A...483..239H. doi:10.1051/0004-6361:20079220. S2CID 3032930.
• T. Kallinger; D. B. Guenther; J. M. Matthews; W. W. Weiss; D. Huber; R. Kuschnig; A. F. J. Moffat; S. M. Rucinski; D. Sasselov (2008). "Nonradial p-modes in the G9.5 giant ε Ophiuchi? Pulsation model fits to MOST photometry". Astronomy and Astrophysics. 478 (2): 497–505. arXiv:0711.0837. Bibcode:2008A&A...478..497K. doi:10.1051/0004-6361:20078171. S2CID 18201762.
• T. Kallinger; P. Reegen; W. W. Weiss (2008). "A heuristic derivation of the uncertainty for frequency determination in time series data". Astronomy and Astrophysics. 481 (2): 571–574. arXiv:0801.0683. Bibcode:2008A&A...481..571K. doi:10.1051/0004-6361:20077559. S2CID 18481860.
• P. Reegen (2005). ""SigSpec - reliable computation of significance in Fourier space", in The A-Star Puzzle, Proceedings IAU Symp. 224, eds. J. Zverko, J. Ziznovsky, S.J. Adelman, W.W. Weiss". The A-Star Puzzle, Proceedings of IAU Symposium 224. Cambridge, UK: Cambridge University Press. pp. 791–798. ISBN 0-521-85018-5.
• P. Reegen; M. Gruberbauer; L. Schneider; W. W. Weiss (2008). "Cinderella - Comparison of INDEpendent RELative Least-squares Amplitudes". Astronomy and Astrophysics. 484 (2): 601–608. arXiv:0710.2963. Bibcode:2008A&A...484..601R. doi:10.1051/0004-6361:20078855. S2CID 11390524.
• C. Schoenaers; A. E. Lynas-Gray (2007). "A new slowly pulsating subdwarf-B star: HD 4539". Communications in Asteroseismology. 151: 67–76. Bibcode:2007CoAst.151...67S. doi:10.1553/cia151s67.
• M. Zechmeister; M. Kuerster (2009). "The gemeralised Lomb-Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms". Astronomy and Astrophysics. 496 (2): 577–584. arXiv:0901.2573. Bibcode:2009A&A...496..577Z. doi:10.1051/0004-6361:200811296. S2CID 10408194.
• K. Zwintz; T. Kallinger; D. B. Guenther; M. Gruberbauer; D. Huber; J. Rowe; R. Kuschnig; W. W. Weiss; J. M. Matthews; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker; M. P. Casey (2009). "MOST photometry of the enigmatic PMS pulsator HD 142666". Astronomy and Astrophysics. 494 (3): 1031–1040. arXiv:0812.1960. Bibcode:2009A&A...494.1031Z. doi:10.1051/0004-6361:200811116. S2CID 54503935.
• K. Zwintz; M. Hareter; R. Kuschnig; P. J. Amado; N. Nesvacil; E. Rodriguez; D. Diaz-Fraile; W. W. Weiss; T. Pribulla; D. B. Guenther; J. M. Matthews; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2009). "MOST observations of the young open cluster NGC 2264". Astronomy and Astrophysics. 502: 1239–252. doi:10.1051/0004-6361/200911863. S2CID 123505620.

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