# Remaining Useful Life Estimation of Bearings Using Data-Driven Ridge Regression
## 6 Evaluation Setup
### PHM Challenge Problem  
PRONOSTIA is an experimental platform developed by the FEMTO-ST institute to evaluate and validate various fault detection and diagnostic algorithms of ball bearings [45]. The platform consists of three parts, namely rotating part, a measurement part, and a degradation generation part. It provides experimental measurements of accelerated degradation of mechanical bearings with constant or variable rotating speed and load force. Dedicated sensors measure the vibration and temperature of the platform. Two miniature accelerometers are installed orthogonally on the external race of the bearing to monitor the vertical and horizontal vibrations. The temperature sensor is located near the external ring of the bearing. These measurements are used to extract the health indicator of ball bearings. PRONOSTIA platform provides various run-to failure experimental data since it enables the experimental degradation within a few hours. Seventeen experimental data cases are generated under three different operating conditions of rotating speed and load force, as summarized in Table 2. The experimental data includes 6 run-to-failure training sets and 11 remaining test sets to build RUL estimation models and evaluate the accuracy. Note that we use the notation i_j to denote \(i\)th dataset of operating condition \(i\) where \(i\in\{\texttt{A},\texttt{B},\texttt{C}\}\). The vibration and temperature signals are recorded with specific sampling frequencies of 25.6 kHz and 10 Hz, respectively. Each experiment is terminated if the vibration signal reaches 20 g to prevent testbed damages. Hence, RUL is defined as time to accelerometer exceeding 20 g. Note that the RUL prediction method is only able to use the measurements of ball bearings since the detailed information of the degradation process is not provided. Figure 2 depicts a horizontal vibration raw signal of various cases Bearing A_1, Bearing A_2, Bearing B_1, Bearing B_2, Bearing C_1, Bearing C_2 obtained during a whole experiment. The degradation processes of bearings have considerably different behaviors with various experimental duration (1-7 h). Efficient estimator design is challenging due to a small amount of training data with a high variation of the life duration of all bearings. As the machine progressively approaches to failure, the vibration signal impulsiveness increases. While the acceleration of operating condition A is slowly growing over time, it is suddenly raised under condition C. Furthermore, we observe significantly different behaviors even under the same operating condition, such as cases Bearing A_1 and Bearing A_2. The raw vibration signals do not have a clear correlation with various operating conditions. In a typical framework, the machine learning trains a regression model based on the data of some bearings and uses it to predict the RUL value of the other bearings. However, even if we obtain the measurements of the same type of bearing under the identical operating condition, their probabilistic density functions are considerably different due to various uncertainties. Hence, most existing machine learning algorithms result in a substantial performance loss because of the violation of the i.i.d. condition even with the same operating condition. While many mathematical models using frequency domain features efficiently detect the faults of the inner and outer races of ball bearings, they are hard to apply in practice due to the extraction difficulty of the frequency features under complex