Regularization of Divergent Series via Classical Summability Methods with Applications to Reliability and Engineering Systems

Abstract

Accurate estimation of remaining useful life (RUL) from noisy and oscillatory degradation data remains a fundamental challenge in prognostics and health management (PHM), especially in safety-critical and resource-constrained settings. While modern data-driven methods achieve high predictive accuracy, they often require extensive training, lack interpretability, and provide limited guarantees on reliability. This study addresses these challenges by proposing a Cesàro-based regularization framework for constructing stable, interpretable, and conservative health indicators from degradation signals. The main objective is to develop a parameter-free and computationally efficient smoothing method that ensures safety-aware degradation tracking. The novelty of this work lies in adapting classical summability theory to PHM and establishing an explicit finite-sample bound that guarantees the smoothed estimate remains conservatively biased relative to the most recent observation, even under mild non-monotonic conditions. The method applies Cesàro averaging to a univariate health index derived from sensor fusion and is evaluated on NASA C-MAPSS FD001 and FD004 datasets using a consistent experimental framework. It is compared with standard smoothing techniques, including SMA, EWMA, Kalman filtering, and isotonic regression. Results show that the proposed approach achieves competitive or improved performance across RMSE, MAE, PHM score, and prognostic horizon, while significantly reducing variance and computational complexity. The Cesàro regularization provides a theoretically grounded, lightweight, and interpretable solution for degradation modeling. Its conservative nature enhances reliability, making it particularly suitable for real-time predictive maintenance applications in aerospace, energy, and industrial systems.