SparSpec: a new method for fitting multiple sinusoids with irregularly sampled data

Abstract

Context. The location of pure frequencies in the spectrum of an irregularly sampled time series is an important topic in astrophysical data analysis. Especially in the domain of asteroseismology, a highly precise and unambiguous study of frequencies in photometric light or radial velocity curves is required. Aims. Due to sampling irregularities and large observational gaps, the classic methods for frequency estimation (prewhitening techniques, clean, cleanest, etc.) sometimes suffer false detections. We propose a new framework for this problem that allows a more precise and unambiguous frequency location. Methods. Multisine fitting is addressed as the sparse representation of the data in an overcomplete dictionary of frequencies, hence the name SparSpec for the method. We model the data as the sum of an arbitrarily large number of pure frequencies, discretised on a fixed grid. Among all the many representations fitting the data, we seek the one with the fewest non-zero amplitudes. This solution can be computed by minimising a convex criterion with no local minima. A computationally efficient and convergent optimisation strategy is derived and a user-friendly software implementing SparSpec is provided online at http://www.ast.obs-mip.fr/Softwares. Results. The method is first illustrated on a simple test example where SparSpec correctly locates the frequencies while classic methods fail. Then, simulations on more realistic artificial time series reveal the interest of this new methodology in terms of robustness toward sampling aliases. An application to the radial velocity curve of the pre-main sequence Herbig Ae star HD 104237 is finally presented, where the method is able to determine oscillation frequencies even in the presence of strong low-frequency perturbations such as orbital movements. While SparSpec mainly confirms previously published studies for the four more important frequencies, it suggests some ambiguity about the position of a fifth frequency. Additional simulations show that the SparSpec results are more plausible.

The JET Alfvén Eigenmode Local Manager for the real-time detection and tracking of a frequency-degenerate spectrum of MHD instabilities

“…This has prompted the developments and the applications of various methods 4 to the analysis of MHD data in thermonuclear fusion plasmas, such as the Singular Value (SVD) [18,19] decomposition of unevenly sampled data using a very small number of measurement points. As some of the mathematical background of this method has been presented elsewhere [11,12,25,26], here we only briefly review its theoretical foundations, with a more compete overview given in Appendix-A to facilitate the reading of this contribution.In the standard tokamak coordinate system (toroidal angle , poloidal angle θ), and taking explicitly into account the usual 2D boundary conditions along the longitudinal (toroidal) axis and on the plane perpendicular to it (the poloidal direction), magnetic perturbations can be represented by functions involving toroidal (n) and poloidal (m) harmonics. Considering now the usual case of a perturbation with a specific toroidal mode number n, this can be written as ( , ) i t in im mn m n e e A e…”

“…This has prompted the developments and the applications of various methods 4 to the analysis of MHD data in thermonuclear fusion plasmas, such as the Singular Value (SVD) [18,19] decomposition of unevenly sampled data using a very small number of measurement points. As some of the mathematical background of this method has been presented elsewhere [11,12,25,26], here we only briefly review its theoretical foundations, with a more compete overview given in Appendix-A to facilitate the reading of this contribution.…”

“…Many computationally efficient algorithms have been developed to solve eq. (3), and the SparSpec code uses one based on an iterative Block Coordinate Descent [11,12] algorithm. A version of this algorithm has recently been adapted to perform the required mode decomposition analysis using the rather modest computational resources available to process real-time JET data [26].…”

“…This has prompted the development and implementation within the AELM of a new real-time algorithm for the detection, discrimination and tracking of the individual components in such a frequency-degenerate spectrum, which is based on the method of Sparse Representation of Signals [11,12]. This paper reports on the real-time development of the Sparse Representation method within the AELM hardware and software infrastructure, and on its specific application to the sub-millisecond detection, discrimination and tracking of the individual components in the multi-harmonic, frequencydegenerate spectrum of stable AEs which are excited in the JET tokamak by an array of external antennae used for MHD diagnostic purposes.…”

“…As some of the mathematical background of this method has been presented elsewhere [11,12,25,26], here we only briefly review its theoretical foundations, with a more compete overview given in Appendix-A to facilitate the reading of this contribution.…”