Astronomy & Astrophysics manuscript no. GLSperiodogram c ESO 2009 January 20, 2009 The generalised Lomb-Scargle periodogram A new formalism for the floating-mean and Keplerian periodograms M. Zechmeister1 and M. K¨urster1 Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, 69117 Heidelberg, Germany e-mail: [email protected] Received / Accepted 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 (χ2 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. Key words. methods: data analysis – methods: analytical – methods: statistical – techniques: radial velocities 1. Introduction the mean of the data was subtracted, which assumes that the mean of the data and the mean of the fitted sine function are the The Lomb-Scargle periodogram (Scargle 1982) is a widely used same. One can overcome this assumption with the introduction tool in period searches and frequency analysis of time series. It is of an offset c, resulting in a further generalisation of this peri- = + equivalent to fitting sine waves of the form y a cos ωt b sin ωt. odogram to the equivalent of weighted full sine wave fitting; i.e., While standard fitting procedures require the solution of a set of y = a cos ωt + b sin ωt + c. Cumming et al. (1999), who called linear equations for each sampled frequency, the Lomb-Scargle this generalisation “floating-mean periodogram”, argue that this method provides an analytic solution and is therefore both con- approach is superior: “... the Lomb-Scargle periodogram fails to ffi venient to use and e cient. The equation for the periodogram account for statistical fluctuations in the mean of a sampled si- was given by Barning (1963), and also Lomb (1976) and Scargle nusoid, making it non-robust when the number of observations (1982), who furthermore investigated its statistical behaviour, is small, the sampling is uneven, or for periods comparable to especially the statistical significance of the detection of a signal. or greater than the duration of the observations.” These authors = For a time series (ti, yi) with zero mean (y 0), the Lomb- provided a formal definition and also a sophisticated statistical Scargle periodogram is defined as (normalisation from Lomb treatment, but do not use an analytical solution for the computa- 1976): tion of this periodogram. arXiv:0901.2573v1 [astro-ph.IM] 16 Jan 2009 2 2 1 YCˆ YSˆ Basically, analytical formulae for a full sine, least-squares pˆ(ω) = τˆ + τˆ (1) spectrum have already been given by Ferraz-Mello (1981), call- ˆ ˆ ˆ YY CCτˆ SS τˆ ing this date-compensated discrete Fourier transform (DCDFT). 2 2 We prefer to adopt a notation closely related to the Lomb- 1 i yi cos ω(ti τˆ) i yi sin ω(ti τˆ) = − + − Scargle periodogram calling it the generalised Lomb-Scarglepe- 2 cos2 ω(t τˆ) 2 i yi i i i sin ω(ti τˆ) riodogram (GLS). Shrager (2001) tries for such an approach but P − P − (2) did not generalise the parameterτ ˆ in Eq. (3). Moreover, our gen- P P P eralised equations, which are derived in the following (Sect. 2), where the hats are used in this paper to symbolise the classical have a comparable symmetry to the classical ones and also allow expressions. The parameterτ ˆ is calculated via us to pointout equivalencesto the “spectral significance” estima- sin 2ωt tor used in the SigSpec program by Reegen (2007) (Sect. 4). tan2ωτˆ = i i . (3) cos2ωt Pi i However, there are two shortcomings.P First, the Lomb- 2. The generalised Lomb-Scargle periodogram Scargle periodogram does not take the measurement errors into (GLS) account. This was solved by introducing weighted sums by Gilliland & Baliunas (1987) and Irwin et al. (1989) (equiva- The analytic solution for the generalised Lomb-Scargle peri- lent to the generalisation to a χ2 fit). Second, for the analysis odogram can be obtained in a straightforward manner in the 2 M. Zechmeister and M. K¨urster: The generalised Lomb-Scargle periodogram same way as outlined in Lomb (1976). Let yi be the N mea- So the sums YC and YS use the weighted mean subtracted data surements of a time series at time ti and with errors σi. Fitting a and are calculated in the same way as for the Lomb-Scargle pe- full sine function (i.e. including an offset c): riodogram (but with weights). The generalised Lomb-Scargle periodogram p(ω) in Eq. (4) y(t) = a cos ωt + b sin ωt + c is normalised to unity and therefore in the range of 0 p 1, 2π = ≤ =≤ at given frequency ω (or period P = ω ) means to minimise the with p 0 indicating no improvement of the fit and p 1 a 2 2 squared difference between the data yi and the model function “perfect” fit (100% reduction of χ or χ = 0). y(t): As the full sine fit is time-translation invariant, there is N [y y(t )]2 also the possibility to introduce an arbitrary time reference 2 i i 2 2 χ = − = W wi[yi y(ti)] point τ (t t τ; now, e.g. CC = w cos ω(t τ) 2 − i i i i i=1 σi → −2 2 − − X X ( wi cos ω(ti τ)) ), which will not affect the χ of the fit. If where this parameter−τ is chosen as P P 1 1 1 wi = W = wi = 1 2CS W σ2 σ2 tan2ωτ = (19) i i CC SS X X − 1 wi sin 2ωti 2 wi cos ωti wi sin ωti are the normalised weights . Minimisation leads to a system of = − 2 2 (three) linear equations whose solution is derived in detail in wi cos2ωti ( wi cos ωti) ( wi sin ωti) Appendix A.1. Furthermore, it is shown in A.1 that the relative P − P −P χ2-reduction p(ω) as a function of frequency ω and normalised P h P P i the interaction term in Eq. (5) disappears, CS τ = wi cos ω(ti to unity by χ2 (the χ2 for the weighted mean) can be written as: − 0 τ) sin ω(ti τ) wi cos ω(ti τ) wi sin ω(ti τ) = 0 (proof − − − −P χ2 χ2(ω) in Appendix A.2) and in this case we append the index τ to the ω = 0 − time dependent sums.P The parameterP τ(ω) is determined by the p( ) 2 (4) χ0 times ti and the measurement errors σi for each frequency ω. So 1 when using τ as defined in Eq. (19) the periodogram in Eq. (5) p(ω) = SS YC2 + CC YS 2 2CS YC YS (5) YY D · · − · · becomes · 1 YC2 YS 2 with: h i p(ω) = τ + τ . (20) YY CCτ SS τ D(ω) = CC SS CS 2 (6) " # · − Note that Eq. (20) has the same form as the Lomb-Scargle pe- and the following abbreviations for the sums: riodogram in Eq. (1) with the difference that the errors can be weighted (weights wi in all sums) and that there is an additional Y = wiyi (7) second term in CCτ, SS τ, CS τ and tan2ωτ (Eqs. (13)–(15) and Eq. (19), respectively) which accounts for the floating mean. C = X wi cos ωti (8) The computational effort is similar as for the Lomb-Scargle S = X wi sin ωti (9) periodogram. The incorporation of the offset c requires only two additional sums for each frequency ω (namely S = w sin ωt X i i and C = w cos ωt or S and C respectively). The effort is YY = YYˆ Y Y YYˆ = w y2 (10) i i τ τ − · i i even weaker when using Eq. (5) with keeping CS insteadP of us- ing Eq. (20)P with the parameter τ introduced via Eq. (19) which YC(ω) = YCˆ Y C YCˆ = X wiyi cos ωti (11) − · needs an extra preceding loop in the algorithm. If the errors are YS (ω) = YSˆ Y S YSˆ = X wiyi sin ωti (12) taken into account as weights, also the multiplication with wi − · must be done. CC(ω) = CCˆ C C CCˆ = X w cos2 ωt (13) − · i i For fast computation of the trigonometricsums the algorithm of Press & Rybicki (1989) can be applied, which has advan- SS (ω) = SSˆ S S SSˆ = X w sin2 ωt (14) − · i i tages in the case of large data sets and/or many frequency steps. Another possibility are trigonometric recurrences2 as described CS (ω) = CSˆ C S CSˆ = X wi cos ωti sin ωti (15) − · in Press et al. (1992). Note also that the first sum in SS can be Note that sums with hats correspondX to the classical sums.