# Remaining Useful Life Estimation of Bearings Using Data-Driven Ridge Regression
## 1 Introduction  
Modern industrial companies must continuously maintain their production resources by using appropriate maintenance strategies to improve availability, reliability, and safety while reducing their maintenance costs [1; 2; 3]. In this domain, predictive maintenance is a promising research direction because it enables manufacturers to monitor the working condition of machinery and diagnose faults or predict the next failure [4; 5]. While the interventions are conducted after the failure occurs in conventional corrective maintenance, it is executed according to the predicted health condition of the equipment in predictive maintenance [6]. The bearing prevents direct metal to metal contact between multiple elements of most motor-driven machines for various industrial applications [7; 8]. Besides, it reduces energy consumption as sliding motion is replaced with low friction rolling. The faults of the rolling element bearings result in significant machinery failures because it incurs friction, heat, and, ultimately, the wear and tear of parts. Predicting the remaining useful life (RUL) of bearings is one of the major goals of predictive maintenance for mechanical systems [1]. The RUL of a system is the length of remaining useful time before it is repaired or replaced. Engineers can schedule maintenance, optimize operating efficiency, and avoid unplanned downtime based on the RUL estimation. However, the RUL prediction is challenging since the failure is typically the result of the long and slow degradation of multiple components of the complex system [9; 10]. The traditional model-based prognostic approach provides reliable results by using an analytical model to reflect thedegradation behavior of a simple system [4; 11]. However, the analytical model is generally difficult to obtain because of the nonlinear complex relationships and stochastic degradation mechanisms of practical systems. On the other hand, the data-driven prognostic approach converts the monitoring and operational data into relevant information and degradation models [4; 5; 12]. This approach utilizes various tools, including statistical method and machine learning, to build the degradation model of the mechanical systems and equipment. While deep learning (DL) has achieved considerable success in computer vision and natural language processing, it faces two fundamental problems, namely safety and insufficient data, to predict the RUL value of mechanical bearings. DL models are mostly used as black-box models without formal guarantees due to their nonlinear and large-scale nature [13]. As a consequence, DL models are vulnerable to input uncertainties or adversarial attacks. Such disruptions can be either of an adversarial nature [14] or merely compression and cropping [15]. These fundamental drawbacks limit the adoption of DL models for safety-critical applications [16; 17; 18]. Another major challenge is the insufficient data to apply the DL technique for the RUL prediction of bearings. In most cases, the failure data from machines are not available, or only a few failure cases exist due to regular maintenance and the infrequent incidents [5; 11]. However, the DL technique still requires a large amount of data due to its inefficient learning procedures regarding the amount of data [19]. We propose a hybrid prognostic method combining model-based and data-driven approaches. Compared to previous works of the analytical model-based approaches [7], the adaptive sliding window method optimizes the local model parameters of the degradation model based on the ridge regression of the time series health indicator. Furthermore, the mathematical degradation models considerably reduce the amount of required data to implement data-driven approaches. We evaluate the performance of the proposed scheme compared to existing Bayesian approaches for predicting the RUL value of the experimental data of mechanical bearings.