# Remaining Useful Life Estimation of Bearings Using Data-Driven Ridge Regression
## 2 Related Works  
During the last decade, numerous methods have been proposed to estimate RUL of the mechanical system using measurement data and suitable models [20; 21]. RUL estimation methods are generally categorized into model-based and data-driven approaches [22]. To estimate the current health status and forecast future failure, model-based methods use mathematical models of the degradation trend of machines such as the partial differential equations [23] and state-space model [24; 25]. However, they require comprehensive domain knowledge to establish the degradation trend of the failure process of the system. Moreover, an adequate physical model is infeasible to derive, especially for real complex systems, since the essential data are hard to obtain. Since the degradation process is a functional relationship between measured data and health status, the data-driven methods apply statistical tools and machine learning algorithms to identify the degradation trend of bearings based on the observed data [22; 24]. Time series prediction methods Kalman filter [26] and autoregressive and integrated moving average [27] and machine learning techniques such as support vector machine (SVM) [28] and its variation [29] are used to characterize the degradation process. [PERSON] et al. [30] extract the main feature by using the principal component analysis (PCA) from the vibration signals of ball bearings. The least-square SVM constructs a regression model to predict RUL. Since most monitoring signals of bearings remain stable for over 90% of the time and only suddenly rise when it is close to failure, a single regression model built upon training samples is not effective in predicting RUL. [PERSON] et al. [31] separate the fault detection module and the RUL prediction module. The fault detection module uses PCA to estimate the fault type based on the frequency feature of the envelope analysis. A regression model predicts the RUL value if the fault is detected. [PERSON] et al. [32] divided the entire degradation process of bearings into multiple health states where a regression model is locally applied. After selecting the main features out of the extracted fault features, they condensed the degradation trend into three representative states by using unsupervised learning and supervised learning. SVM is then used as the primary tool to predict the RUL value of the regression problem. Bayesian approaches are also used to estimate the RUL value. [PERSON] et al. [33] proposed a data-driven approach consisting of offline and online phases to predict the RUL value. In the offline phase, the proposed method builds different health indicators representing degradation as a function of time using unsupervised variable selection, PCA, and trend extraction. In the online phase, the K-nearest neighbors algorithm finds the most similar offline health indicator based on the time series data. Then, the Bayesian filter estimates the degradation state using the linear model of the selected health indicator. The method is evaluated using battery and turbofan engine degradation simulation data from the National Aeronautics and Space Administration data repository. [PERSON] et al. [34] developed a collaboration algorithm combining the Bayesian method and expectation conditional maximization algorithm to estimate RUL of the newly made system using real-time sensing data. Residual life distributions and posterior distributions are first calculated through the Bayesian updating method based on random initial priori distributions. Then, the prior distributions are revised and improved for future predictions by the expectation conditional maximization algorithm. A set of simulation results is used to verify the applicability of the proposed fusion algorithm. While previous Bayesian approaches [33; 34] consider the specific degradation model and the error fluctuation for the given degradation path, it is hard to guarantee the optimal static model by only evaluating the estimation performance based on the partial information.