# Robust HMM-Based Remaining Useful Life Estimation Using a Ridge-Regularized EM Algorithm

## TL;DR

This paper introduces a robust method for estimating the remaining useful life of mechanical systems using a hidden Markov model with ridge regularization and a Huber-based estimator.

## Contribution

The novel contribution is a ridge-regularized EM algorithm for HMM-based RUL estimation that improves accuracy and reduces sensitivity to outliers.

## Key findings

- The ridge-regularized EM algorithm significantly reduces parameter variance and improves predictive accuracy compared to WLS-EM.
- The proposed method provides smoother and more reliable RUL prediction trajectories in real-world data.
- The framework maintains the probabilistic structure of the HMM while enhancing robustness to limited data and outliers.

## Abstract

Estimating the remaining useful life (RUL) of engineering systems is crucial for maintenance planning and the reliability of complex mechanical units. Accurate RUL predictions support timely interventions and help to prevent unexpected failures. This study proposes a statistically robust framework that models degradation signals up to the end of life using a hidden Markov model (HMM) with a simple-failure structure and an absorbing terminal state. The proposed method estimates state-dependent linear emission parameters and transition probabilities using a ridge-regularized expectation–maximization (EM) algorithm. The ridge penalty stabilizes slope estimates under limited data, while a robust Huber-based scale estimator reduces sensitivity to outliers in the sensor-derived health indicator. RUL is computed as a weighted expected time to absorption, combining transient-state survival characteristics with smoothed posterior-state probabilities obtained via the forward–backward algorithm. This yields a low-variance state-aware estimator that preserves the probabilistic structure of the HMM. Simulation studies show that the proposed ridge-regularized EM significantly reduces parameter variance and improves predictive accuracy compared with the baseline weighted least squares EM (WLS-EM). A real-data case analysis demonstrates further improvements in RUL estimation accuracy and smoother, more reliable prediction trajectories. Overall, the framework provides a robust and interpretable approach for practical prognostics applications.