## 2 Problem Description  
The open-access PRONOSTIA bearing datasets were produced by the FEMTO-ST Institute [15, 16]. In the bearing experimental platform, namely PRONOSTIA, two accelerometers were mounted on the bearing housing to measure vibration in directions parallel and perpendicular to the gravity axis. Data were collected at 10 s intervals at a 25.6 kHz sampling rate for a 0.1 s duration. Hence, an observation contained 2560 sample points in each time cycle. The bearing datasets were collected under three different operating conditions in variation of load and speed. For each operating condition (e.g., speed: 1800 rpm and load: 4000 N), only two sets of data from the entire run-to-failure experiments were provided. These are used as learning instances in this study to build the RUL estimation model. Partial data from run-to-failure experiments obtained by truncating the time series after a certain number of time cycles are also provided and used as test instances. In the 2012 PHM Prognostic Challenge [15, 16], RUL was defined as the time span until the vibration amplitude exceeds 20 g.  
Figure 1 shows vibration signals in the horizontal direction, i.e., perpendicular to gravity, under the operating condition I, i.e., speed of 1800 rpm and load of 4000 N. A total of seven datasets are provided including two learning instances labelled as Bearing1-1 and Bearing1-2, and five test instances labelled as Bearing1-3, 1-4, 1-5, 1-6, and 1-7. The two run-to-failure datasets show very different lifespans. A large variation in data length can be seen among the seven bearing instances (from 1 h to 7 h). The test bearing has 13 rolling elements with 3.5 mm diameter. Its inner race diameter is 22.1 mm and outer race diameter is 29.1 mm. The bearing has a contact angle of zero degrees. For the 2012 PHM challenge, no information about the type of failure, e.g., fatigue spalling and its location or poor lubrication, was given. The data provider from the FEMTO-ST Institute has advised that theoretical models based on bearing characteristic  
Figure 1: Raw signal, Root-Mean-Square, Crest-Factor, and Kurtosis of bearing instances under the operating condition of 1800 rpm and 4000 N (Note: red lines depict a moving average trend in each chart): (**a**) Bearing1-1; (**b**) Bearing1-2; (**c**) Bearing1-3; (**d**) Bearing1-4; (**e**) Bearing1-5; (**f**) Bearing1-6; (**g**) Bearing1-7. (Note: the all subfigures have the same range of time cycle in x-axis [0–3000] to illustrate very different lifespans of bearings in the PRONOSTIA datasets). frequencies were highly unlikely to be useful in identifying bearing fault types, because degradation of all bearing components may take place at the same time. In addition, theoretical reliability framework, e.g., L10 law, ball-pass-frequency-inner (BPFI), ball-pass-frequency-outer (BPFO) does not match the experimental observations very well [15]. There have been a variety of statistical measures that can be used to extract prognostic features from vibration signals, including Peak, Peak-to-Peak, Energy, Variance, Standard Deviation (SD), Root-Mean-Square, Entropy, Kurtosis, Skewness, Crest Factor, K-Factor, Impulse Factor, Margin Factor (i.e., Clearance Factor), Shape Factor (i.e., Form Factor, Wave Factor), and the trigonometric function based features SD-IHC and SD-IHS (i.e., SD of inverse hyperbolic cosine (IHC), SD of inverse hyperbolic sine (IHS)) [3,7]. For each time cycle containing a set of data points that are within the signal sampling duration, these statistical measures can be computed. Some statistical parameters are reported to have the ability to indicate the severity of bearing faults [15]. For example, signal energy is generally proportional to the severity of bearing damage and hence parameters such as Root-Mean-Square and Peak-to-Peak should increase as faults progress.