# Interclass Interference Suppression in Multi-Class Problems
## 2 Interclass Interference of Multi-Class Problems  
Interclass interference is an inherent problem in feature extraction of multi-class problems. The removal of key features of some classes as redundant features is the direct cause of interclass interference in multi-class problems. In this section, this paper will use an example of multi-class vibration fault diagnosis of steam turbine shown in Table 1 to introduce the concept of interclass interference and reveal its essence. According to expert knowledge, the vibration signal of 0.4f-0.6f, 1f and 2f, namely the features \(a_{1}\), \(a_{2}\) and \(a_{3}\), are the essential features that can significantly discriminate oil film whirl, unbalance and misalignment from other faults, respectively [47; 48; 49]. Note that f denotes the rotation frequency in Table 1.  
In machine learning, feature selection is an essential step in classification, which can reduce dimensionality. In Table 1, \(a_{1}\) is the feature that can significantly distinguish the oil film whirl fault from other two fault classes including unbalance and misalignment. Meanwhile, the fact that attribute \(a_{1}\) is the intrinsic characteristics of oil film whirl also meets the physical laws [50; 51]. However, feature \(a_{1}\) cannot distinguish unbalance and misalignment faults. In the feature selection process, features that can distinguish all classes have the highest priorities to be preserved. Feature \(a_{1}\) may be deleted because it cannot distinguish between unbalance and misalignment fault. If feature \(a_{1}\) is deleted, the generalization ability of this multi-classifier will be bad because the essential feature of the oil film whirl is deleted. This phenomenon is called interclass inference in this paper. In the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Fault Instances**} & \multicolumn{2}{c}{**Conditional Attributes (C)**} & \multicolumn{2}{c}{**Decision Attributes (D)**} \\  & **0.4f\(\_\)0.6f (\(a_{1}\))** & **1f (\(a_{2}\))** & **2f (\(a_{3}\))** & **Fault (d)** \\ \hline \(x_{1}\) & Low & High & Low & Unbalance \\ \(x_{2}\) & Low & High & Medium & Unbalance \\ \(x_{3}\) & Low & High & High & Unbalance \\ \(x_{4}\) & Low & Medium & Low & Unbalance \\ \(x_{5}\) & Low & Medium & Medium & Unbalance \\ \(x_{6}\) & Low & Low & High & Misalignment \\ \(x_{7}\) & Low & Medium & Medium & Misalignment \\ \(x_{8}\) & Low & Medium & High & Misalignment \\ \(x_{9}\) & High & Low & Low & Oil Film Whirl \\ \(x_{10}\) & High & Low & Medium & Oil Film Whirl \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Decision table for vibration fault diagnosis of a steam turbine. following part of this section, this paper will use a rough set for feature selection, elaborate the interclass inference and analyze the bad influence of interclass inference in detail. Rough set is a popular feature selection method. This paper uses rough sets for feature selection because the attribute reduction of rough sets removes unnecessary and unimportant features and has great advantages in feature selection. In rough set theory, data are usually stored in the form of a decision table \(<U,A=C\cup D,V,f>\), where \(U\) is the universe, \(A\) is the set of attributes, \(C\) is the set of conditional attributes, \(D\) is the set of decision attributes, \(V\) is the set of all attributive values and \(f\) is the information function. For any conditional attribute subset \(B\subseteq C\), there exits an equivalence relation \(IND(B)\), and it is defined by \(IND(B)=\{(x,y)\in U\times U|f(x,a)=f(y,a),\forall a\in B\}\). The set of all equivalence classes is denoted as \(U/B\). For every \(x\in U\), the equivalence classes of \(x\), denoted by \([x]_{B}\), are defined by \([x]_{B}=\{y\in U|(x,y)\in IND(B)\}\). Let \(X\) be a subset of \(U\); then, the B-lower approximation \(\underline{B}(X)\) and B-upper approximation \(\overline{B}(X)\) of \(X\) are defined by \(\underline{B}(X)=\{x\in U|[x]_{B}\subseteq X\}\) and \(\overline{B}(X)=\{x\in U|[x]_{B}\cap X\
eq\varnothing\}\), respectively.