[Paper Review] The generalised Lomb-Scargle periodogram. A new formalism for the floating-mean and Keplerian periodograms

M. Zechmeister, M. Kürster|ArXiv.org|Jan 16, 2009

Stellar, planetary, and galactic studiesPhysics and Astronomy22 references689 citations

TL;DR

This paper introduces the Generalised Lomb-Scargle Periodogram (GLS), a unified analytical framework for fitting full sine waves with offset and weights to unequally spaced data. It improves upon the classical Lomb-Scargle method by enabling accurate frequency estimation, reducing aliasing, and providing better spectral intensity determination through weighted least-squares fitting, with computational cost comparable to the original method.

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.

Motivation & Objective

• To develop a unified analytical formalism that generalizes the Lomb-Scargle periodogram to include a floating mean and weighted data fitting.
• To overcome the limitations of the classical Lomb-Scargle method, which assumes zero mean and ignores measurement errors.
• To unify and clarify equivalences between existing methods such as the date-compensated discrete Fourier transform, floating-mean periodogram, and the spectral significance estimator in SigSpec.
• To extend the formalism to the Keplerian periodogram for detecting orbital periods in radial velocity data, including eccentric orbits.
• To provide a computationally efficient, systematic, and non-random algorithm for detecting exoplanet orbital periods from radial velocity time series.

Proposed method

• Derives an analytic solution for the least-squares fitting of a full sine wave model: $ y(t) = a\cos\omega t + b\sin\omega t + c $, including an offset $ c $ and weighted data.
• Uses weighted sums with normalised weights $ w_i = \frac{1}{\sigma_i^2} / \sum \frac{1}{\sigma_i^2} $ to incorporate measurement errors.
• Applies the method of Lagrange multipliers to derive the minimum $ \chi^2 $ condition, leading to a system of three linear equations for parameters $ a $, $ b $, and $ c $.
• Solves the system analytically to express the periodogram power $ p(\omega) $ in terms of weighted sums: $ Y, C, S, CC, SS, CS $, and determinant $ D = CC\cdot SS - CS^2 $.
• Introduces a frequency-dependent phase shift $ \tau $ via $ \tan 2\omega\tau = \frac{2CS}{CC - SS} $, generalising the classical $ \hat{\tau} $ from Lomb-Scargle.
• Applies the formalism to the Keplerian periodogram by fitting a Keplerian orbit model to radial velocity data, enabling detection of eccentric orbits.

Experimental results

Research questions

• RQ1 How can the Lomb-Scargle periodogram be generalised to include a floating mean and measurement weights while retaining analytical efficiency?
• RQ2 What is the relationship between the generalised Lomb-Scargle periodogram and existing methods such as the date-compensated discrete Fourier transform and the SigSpec spectral significance estimator?
• RQ3 Can the generalised formalism improve frequency detection accuracy and reduce aliasing in unevenly sampled time series?
• RQ4 How can the GLS be adapted to detect orbital periods in radial velocity data, particularly for eccentric exoplanet orbits?
• RQ5 What is the computational cost and performance gain of the GLS compared to the classical Lomb-Scargle and floating-mean periodograms?

Key findings

• The generalised Lomb-Scargle periodogram (GLS) provides a superior alternative to the classical Lomb-Scargle method by allowing full sine wave fitting with offset and weights, resulting in more accurate frequency estimates.
• The GLS formulation reduces susceptibility to aliasing and improves the determination of spectral intensity compared to the standard method.
• The GLS achieves this with only minor modifications to the classical algorithm, maintaining computational efficiency comparable to the original Lomb-Scargle periodogram.
• The GLS formalism unifies and clarifies equivalences between the date-compensated discrete Fourier transform, the floating-mean periodogram, and the spectral significance estimator used in SigSpec.
• The GLS enables the construction of a systematic and non-random Keplerian periodogram capable of detecting eccentric orbits in radial velocity data, as demonstrated in two example cases.
• The analytical solution for the GLS periodogram is derived using weighted least-squares fitting, with the periodogram power expressed as $ p(\omega) = \frac{1}{YY \cdot D} \left[ SS \cdot YC^2 + CC \cdot YS^2 - 2CS \cdot YC \cdot YS \right] $, where $ D = CC\cdot SS - CS^2 $.