# Transmission Line Fault-Cause Identification Based on Hierarchical Multiview Feature Selection
## 2 Hierarchical Multiview Feature Selection (HMVFS)
### The Objective Function  
Given the notation defined and a fault dataset (\(\mathbf{X}\), \(\mathbf{Y}\)), the problem of HMVFS is transformed into determining weight matrix \(\mathbf{W}\) and then ranking features for selection. We formulate the optimization problem as  
\[\min_{\mathbf{W},\mathbf{M}}\mathbf{\Psi}(\mathbf{W},\mathbf{M})\ +\ \alpha\Phi(\mathbf{W})\ +\ \beta\Omega(\mathbf{W})\ =\ \min_{\mathbf{W},\mathbf{M}}\lVert\mathbf{X}\mathbf{W}-\mathbf{Y}-\mathbf{B} \otimes\mathbf{M}\rVert_{F}^{2}\ +\ \alpha\lVert\mathbf{W}\rVert_{2,1}\ +\ \beta\sum\limits_{v=1}^{m}\lVert\mathbf{W}_{v}\rVert_{F}, \tag{1}\]  
where \(m\) is the view number; \(m=2\) in this paper.  
In this formulation, \(\mathbf{\Psi}(\mathbf{W},\mathbf{M})\) is the loss function that measures the calculation distance to achieve minimum regression error, which is derived from the least square loss function. Furthermore, the \(\varepsilon\)-dragging is introduced to drag binary outputs in \(\mathbf{Y}\) away along two opposite directions. The outputs for positive digits will become \(1+\varepsilon_{\text{i}}\) and the outputs for negative digits will be \(-\varepsilon_{\text{i}}\), in which all of the \(\varepsilon\)s are nonnegative. The treatment that enlarges the distance between data points from different classes helps to develop a compact optimization model for classification [30]. \(\mathbf{B}\in\{-1,1\}^{n\times c}\) in the formulation is a constant matrix, and its element \(B_{ij}\) is defined as  
\[B_{ij}\ =\ \left\{\begin{array}{ll}+1,&Y_{ij}=1\\ -1,&Y_{ij}=0.\end{array}\right. \tag{2}\]  
\(B_{ij}\) denotes the dragging direction for elements in label matrix \(\mathbf{Y}\). \(\mathbf{M}\in\mathbb{R}^{n\times c}\) is a nonnegative matrix that records all \(\varepsilon\)s. The operator \(\otimes\) is the Hadamard product operator of matrices. Thus, \(\mathbf{B}\otimes\mathbf{M}\) represents the dragging distance, and we have a new label matrix after the \(\varepsilon\)-dragging:  
\[\mathbf{Y}\prime\ =\ \mathbf{Y}\ +\ \mathbf{B}\otimes\mathbf{M}. \tag{3}\]  
With the least square loss function defined as  
\[\Psi(\mathbf{W})\ =\ \lVert\mathbf{X}\mathbf{W}\ -\ \mathbf{Y}\rVert_{F}^{2}, \tag{4}\]  
we can attain our loss function \(\mathbf{\Psi}(\mathbf{W}\),\(\mathbf{B}\),\(\mathbf{M}\)).  
\[\Psi(\mathbf{W},\mathbf{B},\mathbf{M})\ =\ \lVert\mathbf{X}\mathbf{W}\ -\ \mathbf{Y}\ -\ \mathbf{B}\otimes\mathbf{M}\rVert_{F}^{2}. \tag{5}\]  
Next, regularization items used in the formulation are \(l_{2,1}\)-norm and \(F\)-norm, and we take row-wise feature selection and view-wise feature selection into account.  
\[\Phi(\mathbf{W})\ =\ \lVert\mathbf{W}\rVert_{2,1}=\sum_{i=1}^{d}\sqrt{\sum_{j=1 }^{c}w_{ij}^{2}}. \tag{6}\]  
\[\Omega(\mathbf{W})\ =\ \sum_{v=1}^{m}\lVert\mathbf{W}_{v}\rVert_{F}=\sum_{v=1}^{ m}\sqrt{\sum_{i=1}^{d}\sum_{j=1}^{c}w_{ij}^{2}}. \tag{7}\]  
\(l_{2,1}\)-norm measures the distance of features as a whole and forces the weights of unimportant features to be assigned small values so that it can perform feature selection among all features. Similarly, \(F\)-norm measuring the distance between views forces the weights of unimportant views to be assigned small values [31]. The weight matrix \(\mathbf{W}\) is regulated by these penalty terms, and hierarchical feature selection is completed with row-wise and view-wise selection. \(l_{2,1}\)-norm penalty corresponds to the low-level feature selection, and \(F\)-norm penalty corresponds to the high-level feature selection.  
Therefore, the objective function of the HMVFS model is obtained and represented as (1). \(\alpha\) and \(\beta\) are nonnegative constants that tune hierarchical feature selection. This model is also available with more than two views.