A&A 496, 577–584 (2009) Astronomy DOI: 10.1051/0004-6361:200811296 & c ESO 2009 Astrophysics The generalised Lomb-Scargle periodogram A new formalism for the floating-mean and Keplerian periodograms M. Zechmeister and M. Kürster Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany e-mail: [email protected] Received 5 November 2008 / Accepted 19 December 2008 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 ff y = ω + ω of an o set c, resulting in a further generalisation of this peri- equivalent to fitting sine waves of the form a cos t b sin t. 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- ffi approach is superior: “... the Lomb-Scargle periodogram fails to 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)andScargle 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. y y = or greater than the duration of the observations”. These authors For a time series (ti, i) with zero mean ( 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. ⎡ ⎤ ⎢ 2 2 ⎥ Basically, analytical formulae for a full sine, least-squares 1 ⎢ YCˆ τ YSˆ τ ⎥ pˆ(ω) = ⎣⎢ ˆ + ˆ ⎦⎥ (1) spectrum have already been given by Ferraz-Mello (1981), call- YYˆ CCˆ τˆ SSˆ 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) ⎬ = ⎪ + ⎪(2) Scargle periodogram calling it the generalised Lomb-Scargle pe- y2 ⎩ 2 ω − τ 2 ω − τ ⎭ i i i cos (ti ˆ) i sin (ti ˆ) riodogram (GLS). Shrager (2001) tries for such an approach but did not generalise the parameterτ ˆ in Eq. (3). Moreover, our gen- where the hats are used in this paper to symbolise the classical eralised equations, which are derived in the following (Sect. 2), expressions. The parameterτ ˆ is calculated via have a comparable symmetry to the classical ones and also allow us to point out equivalences to the “spectral significance” estima- sin 2ωt tan 2ωτˆ = i i · (3) tor used in the SigSpec program by Reegen (2007) (Sect. 4). i cos 2ωti However, there are two shortcomings. 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)andIrwin 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 Article published by EDP Sciences 578 M. Zechmeister and M. Kürster: 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). ω y(t) = a cos ωt + b sin ωt + c The generalised Lomb-Scargle periodogram p( )inEq.(4) 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 with p = 0 indicating no improvement of the fit and p = 1a 2 2 the squared difference between the data yi and the model func- “perfect” fit (100% reduction of χ or χ = 0). tion y(t): As the full sine fit is time-translation invariant, there is also the possibility to introduce an arbitrary time reference N y − y 2 2 [ i (ti)] 2 τ t → t − τ CC = w 2 ω t − τ − χ = = W wi[yi − y(ti)] point ( i i ;now,e.g. i cos ( i ) σ2 w ω − τ 2 ff χ2 i=1 i ( i cos (ti )) ), which will not a ect the of the fit. If this parameter τ is chosen as where 1 1 w = = 1 w = ωτ = 2CS i W σ2 i 1 tan 2 σ2 i − W i CC SS w sin 2ωt −2 w cos ωt w sin ωt are the normalised weights1. Minimisation leads to a system of = i i i i i i (19) 2 2 (three) linear equations whose solution is derived in detail in wi cos 2ωti − ( wi cos ωti) −( wi sin ωti) Appendix A.1. Furthermore, it is shown in Appendix A.1 that 2 = w ω − the relative χ -reduction p(ω) as a function of frequency ω and the interaction term in Eq. (5) disappears, CS τ i cos (ti χ2 χ2 τ)sinω(ti − τ) − wi cos ω(ti − τ) wi sin ω(ti − τ) = 0 (proof normalised to unity by 0 (the for the weighted mean) can be written as: in Appendix A.2) and in this case we append the index τ to the τ ω χ2 − χ2 ω time dependent sums. The parameter ( ) is determined by the 0 ( ) σ ω p(ω) = (4) times ti and the measurement errors i for each frequency .So χ2 τ 0 when using as defined in Eq. (19) the periodogram in Eq. (5) 1 becomes p(ω) = SS · YC2 + CC · YS 2 − 2CS · YC · YS (5) YY · D YC2 YS 2 ω = 1 τ + τ · with: p( ) (20) YY CCτ SSτ 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: ff riodogram in Eq. (1) with the di erence that the errors can be weighted (weights wi in all sums) and that there is an additional Y = wiyi (7) second term in CCτ, SSτ, CS τ and tan 2ωτ (Eqs. (13)−(15) C = wi cos ωti (8) and (19), respectively) which accounts for the floating mean. The computational effort is similar as for the Lomb-Scargle = w ω S i sin ti (9) periodogram. The incorporation of the offset c requires only two ω = w ω additional sums for each frequency (namely S i sin ti = w ω ff YY = YYˆ − Y · Y YYˆ = w y2 (10) and C i cos ti or S τ and Cτ respectively). The e ort is i i even weaker when using Eq. (5) with keeping CS instead of us- τ YC(ω) = YCˆ − Y · C YCˆ = wiyi cos ωti (11) ing Eq. (20) with the parameter introduced via Eq. (19)which needs an extra preceding loop in the algorithm. If the errors are YS(ω) = YSˆ − Y · S YSˆ = wiyi sin ωti (12) taken into account as weights, also the multiplication with w i must be done. CC(ω) = CCˆ − C · C CCˆ = w cos2 ωt (13) i i For fast computation of the trigonometric sums the algorithm SS(ω) = SSˆ − S · S SSˆ = w sin2 ωt (14) of Press & Rybicki (1989) can be applied, which has advan- i i tages in the case of large data sets and/or many frequency steps. 2 CS(ω) = CSˆ − C · S CSˆ = wi cos ωti sin ωti. (15) Another possibility are trigonometric recurrences as described in Press et al. (1992). Note also that the first sum in SS can be Note that sums with hats correspond to the classical sums.