# Dynamic Online Label Distribution Feature Selection Based on Label Importance and Label Correlation
## 4 Proposed Method
### Improvement of NRS
Here, \(\Theta_{D^{L_{j}}}(x_{i})\) indicates whether the neighborhood is centered on \(x_{i}\), and \(\theta_{D^{L_{j}}}^{L_{j}}\) represents the radius. A larger \(\theta_{D^{L_{j}}}\) value indicates the sample is closer to \(x_{i}\). When \(d_{x_{i}}^{L_{j}}=0\), the neighborhood granularity is equivalent to the equivalence class._  
**Example 1**. : _Continue to Table 1. Assume 2-norm is applied to \(\Delta_{D^{L_{1}}}(x_{i},x_{j})\), we take \(D^{L_{1}}\) for example, and then we can compute \(\Theta_{D^{L_{1}}}(x_{i})\) as: \(\Theta_{D^{L_{1}}}(x_{1})=\{x_{1},x_{2},x_{3}\}\), \(\Theta_{D^{L_{1}}}(x_{2})=\{x_{1},x_{2},x_{3},x_{5},x_{9}\}\), \(\Theta_{D^{L_{1}}}(x_{3})=\{x_{2},x_{3},x_{4},x_{5},x_{7}\}\), \(\Theta_{D^{L_{1}}}(x_{4})=\{x_{3},x_{4},x_{5},x_{7}\}\), \(\Theta_{D^{L_{1}}}(x_{5})=\{x_{2},x_{3},x_{4},x_{5},x_{7}\}\), \(\Theta_{D^{L_{1}}}(x_{6})=\{x_{6},x_{8},x_{10}\}\), \(\Theta_{D^{L_{1}}}(x_{7})=\{x_{3},x_{4},x_{5},x_{7}\}\), \(\Theta_{D^{L_{1}}}(x_{8})=\{x_{6},x_{8},x_{10}\}\), \(\Theta_{D^{L_{1}}}(x_{9})=\{x_{1},x_{2},x_{9}\}\), \(\Theta_{D^{L_{1}}}(x_{10})=\{x_{6},x_{8},x_{10}\}\)._  
_Based on the above definition, we can find that the neighborhood granularity of instance in the label space forms the granularity system that can cover the instance set. Then, we can summarize the following properties: (1) \(\Theta_{D^{L_{j}}}(x_{i})\
eq\emptyset\), for \(x_{i}\
otin\Theta_{D^{L_{j}}}(x_{i})\); (2) \(\bigcup_{i=1}^{i}\Theta_{D^{L_{j}}}(x_{i})=U\)._  
**Definition 10**. : _In the neighborhood decision system \(NDS=<U,C\cup D>\), \(U=\{X_{1},X_{2},\ldots,X_{n}\}\) represents the sample space, and \(D^{L}=\{D^{L_{1}},D^{L_{2}},\ldots,D^{L_{m}}\}\) represents the label distribution's feature space. By defining the multi-label decision space, we can expand the decision positive domain of the single-label decision using Equation (4). For a certain feature subset \(B\subseteq C\), the lower approximation of the decision attribute \(D^{i}\) with respect to \(B\) is as follows:_  
\[\text{POS}\Big{(}D^{i}\Big{)}^{\prime}=\underline{N_{B}}D^{L_{j}}=\bigcup_{i= 1}^{n}\Big{\{}x_{i}\mid\delta_{B}(x_{i})\subseteq\Theta_{D^{L_{j}}}(x_{i}),x_ {i}\in U\Big{\}} \tag{11}\]  
In this equation, the neighborhood particle \(\delta_{B}(x_{i})\) is obtained from Definition 9, and the label particle \(\Theta_{D^{L_{j}}}(x_{i})\) is obtained from Definition 10. Through this method, we extend the NRS and solve the problem of selecting the neighborhood granularity of rough sets in labeled distribution learning. **Example 2**. : _Continue to Table 1. We take \(L_{1}\) for example, for feature set \(B=(f_{1},f_{2})\), and then we can compute \(\delta_{B}(x_{i})\) according to Definition 9: \(\delta_{B}(x_{1})=(x_{1},x_{4},x_{8})\), \(\delta_{B}(x_{1})=(x_{2},x_{3},x_{5})\); \(\delta_{B}(x_{3})=(x_{3},x_{5})\), \(\delta_{B}(x_{4})=(x_{4},x_{5},x_{8})\),\(\delta_{B}(x_{5})=(x_{1},x_{5},x_{3})\),\(\delta_{B}(x_{6})=(x_{6},x_{9})\); \(\delta_{B}(x_{7})=(x_{7},x_{3})\), \(\delta_{B}(x_{8})=(x_{8},x_{1})\), \(\delta_{B}(x_{9})=(x_{9},x_{6})\),\(\delta_{B}(x_{10})=(x_{10},x_{9})\)._  
\begin{tableAccording to Definition 11, using Examples 1 and 2, we can calculate \(\text{POS}\big{(}D^{i}\big{)}^{\prime}=(x_{2},x_{3},x_{7})\). In LDL, because the labels of each instance are always related in some way, it is necessary to consider the importance of labels and the correlation between them. **Definition 11**. : _For a sample \(x_{i}\) and the corresponding feature vector \(Y_{i}\), that is, \(D=\{(x_{i},Y_{i})\mid 1\leq i\leq N,x_{i}\in U,Y_{i}\in L\}\), \(N\) is the number of instances in the training set, and \(l_{i},l_{j}\in\) and \(L(1\leq i,j\leq k)\) are any two labels in label space \(L\).