# An Ensemble Learning and RUL Prediction Method Based on Bearings Degradation Indicator Construction
## 2 Basic Theory and Algorithm
### Feature Selection
#### 2.2.1 FCBF Feature Selection Method and Markov Blanket  
The fast correlation-based filter (FCBF) solution feature selection method is based on the idea of significance and adopts the backward sequential search strategy to find the feature subset quickly and effectively. Symmetrical uncertainty (SU) was used as a correlation metric to select symmetrical features and remove redundant features [36].  
Calculate the symmetric uncertainty of each feature:  
\[SU(f_{i},R)=2\left[\frac{IG(R\left|f_{i}\right.)}{H\left(R\right)+H\left(f_{i }\right)}\right],\]  
where \(H(R)\) and \(H(f_{i})\) represent the information entropy of the real RUL value \(R\) and feature \(f_{i}\), respectively [37]; \(IG(R|f_{i})\) represents the information gain \((IG)\) and measures the reduction in uncertainty about the real RUL value \(R\) given the value of feature \(f_{i}\).  
Given a threshold value \(\lambda\), if \(SU(f_{i},R)\geq\lambda\), \(f_{i}\) is a strongly correlated feature for the real RUL value \(R\), it should be retained or deleted otherwise.  
In this paper, symmetric uncertainty SU in the FCBF feature selection method is adopted as the metric standard to approximate the Markov Blanket. We applied approximate Markov blanket [25,36] to identify and delete redundant features. Feature redundancy can be determined using the Markov blankets [38] concept. Formally, it is defined as:  
**Definition 1** (Markov Blankets).: _In the feature set \(F\), for a given feature \(f_{i}\in F\), if there is \(f_{i}\perp\{F-M_{i}-\{f_{i}\},R\}\mid M_{i}\), the feature subset \(M_{i}\subset F\left(f_{i}\
otin M_{i}\right)\) is the Markov Blanket of \(f_{i}\)._In the above definition, \(\bot\) denotes independent and \(|M_{i}\) denotes conditional on \(M_{i}\). In other words, the Markov Blanket condition in the definition states, where the feature set \(F\) is divided into three mutually exclusive parts: feature \(f_{i}\), feature subset \(M_{i}\), and feature subset \(F-M_{i}-\{f_{i}\}\). These three subsets have no intersection, and the union is the feature set \(F\). If feature subset \(M_{i}\) is given, the feature \(f_{i}\) is independent of the feature subset \(F-M_{i}-\{f_{i}\}\) and the real RUL value \(R\).  
**Definition 2** (Approximate Markov Blankets).: _For the two features \(f_{i}\) and \(f_{j}(i\
eq j)\), the condition of \(f_{i}\) being the approximate Markov blanket of \(f_{j}\) is: \(SU(f_{i},R)>SU(f_{j},R)\) and \(SU(f_{i},R)<SU(f_{i},f_{j})\)._  
The approximate Markov Blanket (AMB) is computed by comparing the correlation between feature \(f_{i}\) and feature \(f_{j}\), and the \(SU\) value of \(f_{i}\) and the real RUL value \(R\). If the correlation \(SU\) between different features is large, then \(f_{j}\) is an AMB.  
**Definition 3** (Predominant feature).: _A feature \(f_{i}\) is a predominant feature of \(F\) if it does not have any approximate Markov blanket in \(F\). Predominant features are not removed at any stage._  
The process of using the AMB feature selection method to find and delete redundant features is as follows:  
FCBF consists of two stages: obtaining the subset of relevant features and selecting the predominant features from the subset. A relevant feature \(f_{i}\) is predominant if no other relevant feature \(f_{j}\) exists, such that \(f_{j}\) is an AMB for \(f_{i}\). The feature subset composed of all predominant features is the initial bearing degradation indicator subset \(F^{*}\), which represents the degradation state of bearings.