The Multivariate Extension of the Lomb-Scargle Method

TL;DR

This paper extends the Lomb-Scargle method to multivariate time series, enabling more accurate spectral analysis of non-uniformly sampled data while maintaining statistical advantages like noise rejection.

Contribution

The authors develop an n-dimensional Lomb-Scargle method by redefining the shifting parameter, preserving orthogonality and extending the 1D approach to multivariate data.

Findings

The n-D Lomb-Scargle method maintains statistical benefits of the 1D version.

It provides improved parameter estimation for non-uniformly sampled multivariate data.

Application results demonstrate the method's effectiveness with real and simulated data.

Abstract

The common methods of spectral analysis for multivariate ($n$-dimensional) time series, like discrete Frourier transform (FT) or Wavelet transform, are based on Fourier series to decompose discrete data into a set of trigonometric model components, e. g. amplitude and phase. Applied to discrete data with a finite range several limitations of (time discrete) FT can be observed which are caused by the orthogonality mismatch of the trigonometric basis functions on a finite interval. However, in the general situation of non-equidistant or fragmented sampling FT based methods will cause significant errors in the parameter estimation. Therefore, the classical Lomb-Scargle method (LSM), which is not based on Fourier series, was developed as a statistical tool for one dimensional data to circumvent the inconsistent and erroneous parameter estimation of FT. The present work deduces LSM for $n$-dimensional data sets by a redefinition of the shifting parameter $\tau$, to maintain orthogonality of the trigonometric basis. An analytical derivation shows, that $n$-D LSM extents the traditional 1D case preserving all the statistical benefits, such as the improved noise rejection. Here, we derive the parameter confidence intervals for LSM and compare it with FT. Applications with ideal test data and experimental data will illustrate and support the proposed method.