Multivariate Frequency and Amplitude Estimation for Unevenly Sampled Data Using and Extending the Lomb-Scargle Method

The Lomb-Scargle method (LSM) constitutes a robust method for frequency and amplitude estimation in cases where data exhibit irregular or sparse sampling. Conventional spectral analysis techniques, such as the discrete Fourier transform (FT) and wavelet transform, rely on orthogonal mode decomposition and are inherently constrained when applied to non-equidistant or fragmented datasets, leading to significant estimation biases. The classical LSM, originally formulated for univariate time series, provides a statistical estimator that does not assume a Fourier series representation. In this work, we extend the LSM to multivariate datasets by redefining the shifting parameter τ to preserve the orthogonality of trigonometric basis functions in Rn. This generalization enables simultaneous estimation of the frequency, phase, and amplitude vectors while maintaining the statistical advantages of the LSM, including consistency and noise robustness. We demonstrate its application to solar activity data, where sunspots serve as intrinsic markers of the solar dynamo process. These observations constitute a randomly sampled two-dimensional binary dataset, whose characteristic frequencies are identified and compared with the results of solar research. Additionally, the proposed method is applied to an ultrasound velocity profile measurement setup, yielding a three-dimensional velocity dataset with correlated missing values and significant temporal jitter. We derive confidence intervals for parameter estimation and conduct a comparative analysis with FT-based approaches.