[Paper Review] Remaining Useful Life Estimation Using Functional Data Analysis

Qiyao Wang, Shuai Zheng+3 more|arXiv (Cornell University)|Apr 12, 2019

Fault Detection and Control SystemsEngineering30 references19 citations

TL;DR

This paper proposes a functional Multilayer Perceptron (functional MLP) using Functional Data Analysis (FDA) to improve Remaining Useful Life (RUL) estimation by modeling sensor time series as continuous stochastic processes. The method captures within-unit correlations and across-unit variability, achieving state-of-the-art performance on NASA C-MAPSS benchmark data with a 26.89% average RMSE improvement and 70.54% average score improvement over LSTM.

ABSTRACT

Remaining Useful Life (RUL) of an equipment or one of its components is defined as the time left until the equipment or component reaches its end of useful life. Accurate RUL estimation is exceptionally beneficial to Predictive Maintenance, and Prognostics and Health Management (PHM). Data driven approaches which leverage the power of algorithms for RUL estimation using sensor and operational time series data are gaining popularity. Existing algorithms, such as linear regression, Convolutional Neural Network (CNN), Hidden Markov Models (HMMs), and Long Short-Term Memory (LSTM), have their own limitations for the RUL estimation task. In this work, we propose a novel Functional Data Analysis (FDA) method called functional Multilayer Perceptron (functional MLP) for RUL estimation. Functional MLP treats time series data from multiple equipment as a sample of random continuous processes over time. FDA explicitly incorporates both the correlations within the same equipment and the random variations across different equipment's sensor time series into the model. FDA also has the benefit of allowing the relationship between RUL and sensor variables to vary over time. We implement functional MLP on the benchmark NASA C-MAPSS data and evaluate the performance using two popularly-used metrics. Results show the superiority of our algorithm over all the other state-of-the-art methods.

Motivation & Objective

• To address limitations in existing data-driven RUL estimation methods that fail to model temporal correlations within units and random variations across units.
• To develop a novel deep learning framework that treats sensor time series as continuous functional data to better capture dynamic degradation patterns.
• To improve RUL prediction accuracy by explicitly modeling time-varying relationships between sensor variables and remaining life.
• To evaluate the proposed method on the standard NASA C-MAPSS benchmark dataset using RMSE and a score-based metric.
• To demonstrate superior performance compared to state-of-the-art models like LSTM, CNN, and HMMs in RUL estimation.

Proposed method

• The functional MLP treats each equipment's sensor time series as a sample of a continuous stochastic process over time, enabling FDA-based modeling of functional data.
• It incorporates within-unit temporal correlations and across-unit random variations by modeling the entire time series as a functional observation.
• The model uses functional principal component analysis (FPCA) to extract low-dimensional functional features from raw time series data.
• A multilayer perceptron is trained on the functional principal component scores to predict RUL, allowing flexible, time-varying relationships between features and RUL.
• The method leverages the functional representation to avoid strong assumptions about fixed mappings across time, unlike RNNs or LSTMs.
• The model is trained end-to-end using backpropagation with loss functions based on RMSE and a non-symmetric score metric penalizing overestimation more heavily.

Experimental results

Research questions

• RQ1 Can modeling sensor time series as continuous functional processes improve RUL estimation accuracy compared to standard deep learning models?
• RQ2 How does the functional MLP handle within-unit temporal dependencies and across-unit variability in degradation patterns?
• RQ3 Does the functional approach reduce bias from fixed mapping assumptions inherent in RNNs and LSTMs?
• RQ4 To what extent does the functional MLP outperform state-of-the-art methods on the NASA C-MAPSS benchmark dataset?
• RQ5 How does the model perform across different engine types and failure modes in the C-MAPSS dataset?

Key findings

• The functional MLP achieved a 26.89% average improvement in RMSE over LSTM across all four C-MAPSS data subsets.
• On the score metric, the functional MLP showed a 70.54% average improvement over LSTM, indicating superior handling of overestimation.
• For FD001, the functional MLP achieved an RMSE of 13.36, outperforming LSTM’s 16.14, with a 17.22% improvement.
• In the score metric, the functional MLP achieved 2.0×10² for FD001, compared to LSTM’s 3.4×10², representing a 41.18% improvement.
• Visual comparisons showed that the estimated RUL closely tracked the true RUL, especially near end-of-life, indicating strong predictive reliability.
• The model demonstrated consistent superiority across all four C-MAPSS data subsets (FD001–FD004), confirming robustness to varying failure modes.