# Dynamic Online Label Distribution Feature Selection Based on Label Importance and Label Correlation
## 4 Proposed Method  
### Improvement of NRS  
Given a decision system \(NDT=\langle U,C,D\rangle\), \(U=\{x_{1},x_{2},\ldots,X_{n}\}\) represents a non-empty set of instances, \(C\) represents the feature set corresponding to the instance set, and \(D\) represents the decision attribute set \(D^{L}=\left\{D^{L_{1}},D^{L_{2}},\ldots,D^{L_{m}}.\right\}\) Traditional single-label and multi-label neighborhood information particle partitioning methods are not suitable for labeled distribution data. For general data, a group of instances with the same attribute value or label value is called an equivalence class. Similarly, for mixed data, a group of instances with similar attribute values or label values is called a neighborhood class. In the method proposed in this paper, the margin of particles in the sample is used for granulating the neighborhood size. **Definition 7**. : _Given a sample \(x\), the margin of \(x\) relative to a set of samples \(U\) is defined as follows [12]:_  
\[m(x)=\Delta(x,NS(x))-\Delta(x,NT(x)) \tag{7}\]  
_where \(NS(x)\) denotes the instance from \(U\) that has the shortest distance from \(x\) and whose label class is different from that of \(x\). In addition, \(NT(x)\) denotes the instance from \(U\) that has the shortest distance from \(x\) and has the same label class as \(x\). We call these instances the nearest miss and the nearest hit, respectively. Moreover, \(\Delta(x,NS(x))\) denotes the distance between \(x\) and \(NS(x)\), and \(\Delta(x,NT(x))\) denotes the distance between \(x\) and \(NT(x)\). We call \(\delta(x)=\left\{y\mid\Delta(x,y)\leq m(x)\right\}\) the neighborhood particle about \(x\). To facilitate the setting of neighborhood information particles, we set \(m(x)=0\) when \(m(x)<0\)._  
_A sample may have a positive or negative effect on different labels. Thus, for a given sample, the degree of granularity depends on the label used._  
**Definition 8**. : _For a sample \(x\) and label \(l_{k}\in L\), the margin of \(x\) with respect to \(l_{k}\) is \(m_{l_{k}}(x)=\Delta_{l_{k}}\big{(}x,NS_{l_{k}}(x)\big{)}-\Delta_{l_{k}}\big{(} x,NT_{l_{k}}(x)\big{)},l_{k}\in L\)._  
_As noted above, each sample has a different label and, correspondingly, a different granularity. Depending on the different decision views, we need to combine all the single-label granularities of a given sample to form a label distribution granularity [44]. Therefore, in this paper, we choose the average granularity (i.e., the average nearest neighborhood, also known as the neutral view) to represent the label distribution granularity of a sample [12]:_  
\[m^{\text{neut}}\left(x\right)=\frac{1}{L}\sum_{i=1}^{L}m_{l_{k}}(x) \tag{8}\]  
_To solve the problem of the granularity selection of \(\delta\), combining Equations (1) and (9), the new neighborhood of the sample is defined as_  
\[\delta_{B}(x_{i})=\left\{x_{j}\mid x_{j}\in U,\Delta_{B}\big{(}x_{i},x_{j} \big{)}\leq m^{\text{neut}}\left(x_{i}\right)\right\} \tag{9}\]  
We define a new neighborhood information particle to solve the problem of selecting the neighborhood granularity, which is caused by label distribution data. In addition, the average nearest neighbor reflects the relationship between features in an instance. This new neighborhood model considers the relationships between features and is based on improved neighborhood information. **Definition 9**. : _Given \(\forall x_{i}\in U\), a label space \(D^{L}=\left\{D^{L_{1}},D^{L_{2}},\ldots,D^{L_{m}}.\right\},D^{L_{j}}\in D^{L}\), the label space granularity on label \(D^{L_{j}}\) is defined as follows:_\[\Theta_{D^{L_{j}}}(x_{i})=\left\{ \begin{array}{l}\left\{x_{j}\mid\Delta_{D^{L_{j}} }(x_{i},x_{j})\leq\theta_{D^{L_{j}}}\wedge\Delta_{D^{L_{j}}}\big{(}x_{i},x_{j} \big{)}\
eq 0,x_{j}\in U\right\}\right\}\text{, if }d_{x_{i}}^{L_{j}}\
eq 0\\ \left\{x_{j}\mid\Delta_{D^{L_{j}}}\big{(}x_{i},x_{j}\big{)}=0\right\}\text{, if }d_{x_{i}}^{L_{j}}=0\end{array} \right. \tag{10}\]  
_where \(\theta_{D^{L_{j}}}=\frac{\sum_{j=1}^{j=\pi}\sum_{i=1}^{i=\pi}d_{x_{i}}^{L_{j}}} {m\times n}\).