# A Diagnosability-Integrated Design Approach Based on Graph Theory
## 3 Diagnosability Integrated Design Method Details
### Diagnosis Method Determination  
#### 3.2.1 Design of Maximal Clique Classifier  
The set of maximal cliques determined through Section 3.1.3 is denoted as  
\[C^{*}=\{\mathcal{C}_{1},\mathcal{C}_{2},\cdots,\mathcal{C}_{mi}\} \tag{3}\]  
where \(\mathcal{C}_{j}\) represents the \(j\)-th maximal clique in the set. According to the previous section, the system faults contained in the maximal clique \(\mathcal{C}_{j}\) are easily distinguishable under the measurement signal of the measurement point, and so, a classifier can be set for any maximal clique \(\mathcal{C}_{j}\) in the set \(C^{*}\), and the classifier can identify the system fault states by the signal of the current measurement point. In classification problems, the difficulty of the classification problem is closely related to the kind of classification. In general, the more kinds of classification, the more detailed features are required and the more resources are consumed. In order to reduce the difficulty of classification, researchers try to use a cascade approach to solve the classification problem in the current classification problem [22], that is, the problems are first roughly classified by using features, and then the characteristics are finely classified for each rough classification, so as to achieve the purpose of correctly classifying the target samples. Studies have shown that the cascade classification approach has good classification results [23]. In this section, cascade classifiers of each maximal clique in maximal clique set \(C^{*}\) were designed to reduce classification categories as far as possible, so as to select fewer features in feature extraction and identify the state of the system. Based on the structure shown in Equation (3), construct the matrix  
\[ \begin{array}{l}W=\left[w_{kj}\right]_{(m+1)\times ni}\\ k=0,1,2,\cdots,m\\ j=1,2,\cdots,ni\end{array}  \tag{4}\]  
where \(w_{kj}=0,1\), if \(w_{kj}=1\) denotes the \(j\)-th maximal clique \(\mathcal{C}_{j}\) in the maximal cluster clique set \(\mathcal{C}^{\star}\) can recognize the fault state \(f_{k}\), and vice versa, \(w_{kj}=0\). In this paper, the method of setting up classifiers for each maximal clique can be followed as follows. The following is a concrete example to illustrate the calculation process of Algorithm 3. ```
Input:\(C^{\star}=\{\mathcal{C}_{1},\mathcal{C}_{2},\cdots,\mathcal{C}_{mi}\}\) \(\mathcal{F}=\{f_{0},f_{1},\cdots,f_{m}\}\) Output: classifier design results
1. \(W_{i}\) is determined by \(\mathcal{C}_{i}\) according to Formula (4)
2. while\(\mathcal{F}\
eq\varnothing\)do
3. \(W_{i}\) adds up the rows to get the vector \(w=[w_{0},w_{1},\cdots,w_{m}]^{T}\)
4. if\(w_{0}=w_{1}=\cdots=w_{m}\)then
5. Set the classifier on \(\mathcal{C}_{j}^{i}\in\mathcal{C}^{\star}\), the number of categories is \(w_{0}\). 6. \(\mathcal{F}\leftarrow\varnothing\)
7. else
8. \(n=\min\{w_{0},w_{1},\ldots,w_{m}\). 9. \(i=\operatorname*{argmin}\left\{w_{j}\right\}\)
10. if\(n=1\) and \(|i|=1\)then
11. \(\exists\mathcal{C}_{j}^{i}\in\mathcal{C}^{\star}\), s.t \(w_{ji}=1\)
12. the binary classifier is set on \(\mathcal{C}_{j}^{i}\), the positive class is \(f_{i}\), the negative class is all the remaining faults
13. \(\mathcal{F}\leftarrow\mathcal{F}-\{f_{i}\}\)
14. \(\mathcal{C}^{\star}\leftarrow\mathcal{C}^{\star}-\left\{\mathcal{C}_{j}^{i}\right\}\)
15. endif
16. if\(n=1\) and \(|i|>1\)then
17. Select \(\mathcal{C}_{j}^{i}\) so that it can cover the most faults in \(f_{i}\),
18. \(\mathcal{F}=\mathcal{F}-\{f_{i}\}\)
19. \(\mathcal{C}_{i}=\mathcal{C}_{i}-\left\{\mathcal{C}_{j}^{i}\right\}\)
20. endif
21. if\(n>1\)then
22. the binary classifier is set on \(\mathcal{C}_{j}^{i}\), the positive class is \(f_{i}\), the negative class is all the remaining faults
23. \(\mathcal{F}\leftarrow\mathcal{F}-\{f_{i}\}\)
24.