# Interclass Interference Suppression in Multi-Class Problems
## 2 Interclass Interference of Multi-Class Problems
The B-positive region \(POS_{B}(D)\) in the relation \(IND(D)\) is defined by \(POS_{B}(D)=\bigcup\limits_{X\in U/D}\underline{B}(X)\). The dependency degree \(\gamma_{B}(D)\) of \(U/D\) on \(B\) is defined by \(\gamma_{B}(D)=|POS_{B}(D)|/|U|\), where \(|F|\) denotes the cardinality of set \(F\). If \(B\) is a subset of \(C\) such that \(\gamma_{B}(D)=\gamma_{C}(D)\), then \(B\) is a reduct of \(C\)[50]. Currently, there are many attribute reduction algorithms, among which one of the most common-used ones is the dependency degree-based algorithm proposed in [51]. Therefore, this paper used it for feature selection. According to the feature selection algorithm proposed in the literature [51], a feature subset \(\{a_{2},a_{3}\}\) is obtained after attribute reduction. In this feature subset \(\{a_{2},a_{3}\}\), two rules namely \({}^{\prime}(a_{2}=low)\{a_{3}=medium\}\Rightarrow(d=oil~{}film~{}whirl)^{\prime}\) and \({}^{\prime}(a_{2}=low)\{a_{3}=low\}\Rightarrow(d=oil~{}film~{}whirl)^{\prime}\), are needed to diagnose the fault of oil film whirl. The support coefficients of two rules are both \(1/10\). If the essential feature of oil film whirl, namely feature \(a_{1}\) is used, only one rule \({}^{\prime}(a_{1}=high)\Rightarrow(d=oil~{}film~{}whirl)^{\prime}\) is needed to diagnose the fault of oil film whirl, and the support coefficient of this rule increases to \(2/10\). The rule obtained by using essential features and non-essential features can both realize the perfect classification of the existing instances of oil film whirl in Table 1. However, we can obtain shorter rules and larger support coefficients based on the essential feature \(a_{1}\). In general, it can reflect the characteristics of data better and contribute to better generalization ability if there are fewer rules and larger support coefficients. Thus, the rule \({}^{\prime}(a_{1}=high)\Rightarrow(d=oil~{}film~{}whirl)^{\prime}\) is a more direct reflection of the oil film whirl compared with the previous two rules and accords with well expert knowledge. However, in the process of feature selection, the essential feature \(a_{1}\) of oil film whirl is deleted. Obviously, the deletion of essential features deviates from the intrinsic characteristics and thus seriously decreases the generalization ability of multi-classifiers. In this paper, the deletion of essential features that can significantly discriminate a certain class from the other classes in the feature selection of multi-classifiers is called interclass interference. Next, we will analyze the reason for interclass interference, namely the reason why essential features are deleted. We performed feature selection through the algorithm in [50]. First, we calculate the significance of conditional attribute \(a_{1}\), \(a_{2}\) and \(a_{3}\) in Table 1: we can obtain that \(\gamma_{a_{1}}(d)=\frac{2}{10}\), \(\gamma_{a_{2}}(d)=\frac{3}{10}\) and \(\gamma_{a_{3}}(d)=0\). The feature \(a_{2}\) has the greatest significance, so it is selected first. Then, on the basis of attribute \(a_{2}\), we calculate the significance of \(a_{1}\) and \(a_{3}\). We obtain \(\gamma_{\{a_{1},a_{2}\}}(d)=\frac{6}{10}\) and \(\gamma_{\{a_{2},a_{3}\}}(d)=\frac{8}{10}\). The feature \(a_{3}\) has a greater significance, so it is selected. Because of \(\gamma_{\{a_{2},a_{3}\}}(d)=\gamma_{\{a_{1},a_{2},a_{3}\}}(d)\), the feature selection ends and the selected feature subset is \(\{a_{2},a_{3}\}\). In this process, the reason for the deletion of attribute \(a_{1}\) is not because it has no contribution to classification. The essential features of the imbalance \(a_{2}\) and misalignment \(a_{3}\) are capable of classifying not only the instances of imbalance and misalignment but also the existing instances of oil film whirl correctly. After selecting feature \(a_{2}\) and \(a_{3}\), the essential feature \(a_{1}\) becomes redundant in terms of the existing instances of oil film whirl and thus is deleted. From the above analysis, the essence of interclass inference can be revealed.