Least Absolute Deviations Estimation for Sinusoidal Models

2026-06-11

AI Summary

This study addresses the challenge of robust parameter estimation in sinusoidal regression under heavy-tailed noise and outliers. The authors propose an efficient estimation method based on least absolute deviation (LAD), employing a coordinate descent strategy to jointly optimize amplitude and frequency parameters. The amplitude is updated via a weighted median, while the frequency is refined through a combination of periodogram-based grid search and local optimization. The algorithm’s modular design circumvents the high computational complexity of traditional simplex methods. Theoretical guarantees are established, including strong consistency and asymptotic normality of the resulting estimators. Experimental results on synthetic data and real-world time series—such as Mauna Loa CO₂ concentrations and airline passenger counts—demonstrate the method’s superior robustness and performance in the presence of non-Gaussian noise.

Abstract

We study robust parameter estimation in sinusoidal regression models within a least absolute deviations (LAD) framework. While classical approaches rely predominantly on least-squares formulations, they are known to be sensitive to heavy-tailed noise and outliers. We formulate the estimation problem as direct minimization of the LAD objective and propose a simple, modular coordinate descent algorithm that exploits the partial convexity of the objective: amplitude parameters are updated via weighted median computations, leading to substantial computational improvements over traditional simplex-based optimization methods, while frequency parameters are estimated via a periodogram-inspired grid search with local refinement. We establish strong consistency and asymptotic normality of the proposed estimator under mild regularity conditions. Empirically, we demonstrate the method's effectiveness on both synthetic datasets and real-world time series, including the Mauna Loa atmospheric CO2 data, air passenger data, and UK drivers' deaths data, where robustness to non-Gaussian noise is essential. The proposed approach provides a simple, interpretable, and robust alternative to least-squares-based methods for sinusoidal signal estimation.

Problem

• sinusoidal regression
• robust estimation
• outliers
• heavy-tailed noise
• parameter estimation

Innovation

• least absolute deviations
• sinusoidal regression
• coordinate descent
• robust estimation
• weighted median