# Data-Driven Health Assessment in Flight Control System
## 4 Health Assessment Method
### Weight Assignment Based on Rough Sets Reduction Algorithm  
For the two-level weight calculation in Figure 2, there are three commonly used assignment methods [17]: subjective, objective, and subjective/objective weighting methods. During the weight assignment process, it is necessary to minimize the subjective factors' impact, and an objective weighting method based on rough sets reduction is used in this section, which removes an attribute from the set firstly and evaluates its importance to determine its weight.  
The main idea of rough set is to ensure that the classification ability of the information itself does not change. A new classification method is formed by the relative simplicity of information knowledge, under the condition that the simplicity of knowledge does not change the original classification, and then the expression of new knowledge is formed. The brief process of knowledge in each message can be described using specific mathematical formulas, which makes it capable of processing most rough sets of data. As the knowledge structure of the information is preserved, rough sets processing method is widely used in machine learning, pattern recognition, and data mining.  
Rough set algorithm does not need priori data, it only needs to mine the hidden rules from the knowledge itself and extract the importance of attribute components. So, we can obtain the importance of component attributes on information classification, which can be integrated with a weighted comprehensive model to establish objective feature parameter weight distribution methods.  
Based on this, the idea of rough sets reduction is to continuously remove certain attributes from the original complete attribute set, and then observe whether the postclassification state has changed greatly; if it does, the importance of this attribute is higher, otherwise the importance is lower. When using the rough set reduction algorithm to calculate the weight of each channel, the membership function of the relative degradation degree of each channel is first constructed, and then the weight value is derived from the attribute importance of each channel, which can reduce the inaccurate weight setting caused by the bias of human subjective judgment, thereby improving the robustness of the evaluation results.  
The specific assignment steps are shown as follows:  
(1) Constructing decision table: Constructing a decision table with different attributes and importance, the lower evaluation factor in Figure 2 is used as the condition attribute in decision table \(C\ =\ \left\{\ c_{1}\ \ \ c_{2}\ \ \cdots\ \ c_{n}\ \right\}\), and the upper factor is used as decision attribute \(D\ =\ \left\{\ d_{1}\ \ \ d_{2}\ \ \ d_{3}\ \ \ d_{4}\ \right\}\), \(n\ =\ n_{1}+n_{2}+n_{3}+n_{4}\).  
(2) Calculating the attribute conditional information entropy: Supposing \(X\) is a subset of attributes in the flight control system evaluation factors, and the \(x\) is a specific attribute, the conditional information entropy of \(x\) for \(X\) is:  
\[I(X)\ =\ 1-\frac{\sum\limits_{i\ =\ 1}^{n}|X_{i}|^{2}}{|U|}, \tag{8}\]  
where \(U\) is a finite nonempty set of flight control system.  
(3) Calculating the importance of a single attribute: Excluding an attribute \(c\), the importance of \(c\) in \(C\) based on conditional information entropy is computed as:  
\[Sig(c)\ =\ I(D/C)-I(D/C-\{c\}), \tag{9}\]  
(4) Calculating the weights: Based on the importance calculation of a single attribute, the weight of the attribute can be obtained as:  
\[w(c_{i})\ =\ \frac{sig(c_{i})}{\sum\limits_{i\ =\ 1}^{n}sig(c_{i})}, \tag{10}\]