# Dynamic Online Label Distribution Feature Selection Based on Label Importance and Label Correlation
## 4 Proposed Method
### Feature Selection Based on NRSs  
The processing method of the labeled distribution neighborhood decision system is similar to that of the multi-label decision system. By extending the rough set importance theory of labeled distribution data (Equation (6)), combining the positive domain theory of labeled distribution decision (Equation (12)) and label correlation (Equation (14)), we obtain the importance of the decision attribute \(D=L=\{l_{1},l_{2},\ldots,l_{m}\}\) on the set feature subset \(B(B\subseteq C)\):  
\[\gamma_{B}\Big{(}D^{L}\Big{)}=\sum_{L_{j}\in L}^{m}\frac{\text{Card}\Big{(} \text{PoS}_{B}\Big{(}D^{L_{j}}\Big{)}\Big{)}LW\big{(}l_{j}\big{)}}{\text{Card} (U)} \tag{15}\]  
The above equation reflects the importance of the decision attributes corresponding to the decision positive domain and feature subset. It solves the problem of granularity selection and feature association of label distribution NRS. In the neighborhood decision system \(NDS=<U,C\cup D,\delta>\), for feature subset \(B\subseteq C,a\in C-B\), the importance of \(a\) to \(B\) is defined as follows:\[\text{SIG}(a,B,D)=\gamma_{B\cup u}(D)-\gamma_{B}(D) \tag{16}\]  
In the new importance model, we add label importance and label relevance to the NRS model. The new NRS model reflects the fusion of feature information and label relevance. For the above NRS model, we construct a method to compute the most reducible feature sets based on a greedy forward search. In this method, steps 1 and 2 perform the preprocessing when the labeled data arrive. Our reduced set starts from the empty set and calculates the label weights \(LW(L)\) for the entire label space. This step requires traversing the entire label space and constructing an undirected graph. Assuming that the number of labels in the label space is \(L\), the time complexity of calculating the correlation between each pair of labels is \(O\big{(}|L|^{2}\big{)}\), and the time complexity of calculating the weight of each label is \(O(1)\). Therefore, the time complexity of steps 1 and 2 is \(\text{O}\big{(}|L|^{2}+1\big{)}=\text{O}\big{(}|L|^{2}\big{)}\). Steps 3 \(-\) 6 are divided into two parts: calculating the neighborhood of the instance, and analyzing whether the instance and the neighborhood are important. First, the nearest hit or miss is determined for each instance by selecting the average approximate neighborhood as the domain granularity criterion. Assuming the instance space is \(U\), the time complexity of this step is \(\text{O}\big{(}|U|^{2}\big{)}\). Next, the neighborhood corresponding to the instance and the labeling granularity of the labeling space are determined, and then the decision positive domain and attribute importance are calculated. The time complexity of determining the neighborhood of each instance is \(\text{O}(n\log n)\); the time complexity of determining the marking granularity of the marking space is \(\text{O}\big{(}|U|^{2}\big{)}\); and the time complexity of determining the decision positive domain and importance calculation corresponding to each sample is \(\text{O}(1)\). Therefore, the overall time complexity of instance domain calculation is \(O\big{(}|U|^{2}+|U|^{2}+|U|\log|U|+1+1\big{)}=O\big{(}|U|^{2}\big{)}\). Therefore, the time complexity of Algorithm 1 is \(\text{O}\big{(}|L|^{2}+|U|^{2}\big{)}\). ```
0: neighborhood decision system \(\langle U,C\cup D\rangle\).