Advancements in Robust Least Squares Approximation Techniques: A Comparative Analysis

This study presents a comprehensive comparative analysis of recent advancements in robust least squares approximation techniques, with emphasis on their theoretical foundations, computational methods, and practical applications. While traditional least squares methods remain widely used for data fitting, signal processing, and estimation tasks, they are highly sensitive to noise and outliers, reducing effectiveness in real-world settings. To address these limitations, robust approaches have emerged, including penalised least squares, polynomial discrete penalised least squares, meshfree moving least squares (MLS), fuzzy least squares, and augmented MLS methods. The purpose of this research is to evaluate these techniques in terms of robustness, accuracy, and computational efficiency across various domains, including finance, engineering, geospatial analysis, and stochastic modelling. Methodologically, the study integrates theoretical insights, case studies, and a controlled numerical illustration with added noise, outliers, and irregular sampling, evaluated using Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and a robustness index (RI). Results show that each method offers distinct advantages. MLS techniques perform well with irregular data, fuzzy least squares are suited for uncertain environments, and penalised approaches balance complexity with stability. In quantitative evaluation, augmented MLS achieved the lowest error and highest robustness index, while runtime analysis revealed trade-offs between accuracy and computational cost. Contributions include a unified framework for comparing methods, explicit clarification of modelling assumptions, and discussion of orthogonal and generalised polynomial systems for enhanced stability. Limitations involve the restricted scope of benchmarks and case studies, which may limit generalisability. Nonetheless, the findings provide actionable guidance for practitioners facing noisy, uncertain, or irregular data, and highlight opportunities for hybrid and adaptive robust least squares methods in emerging fields.