## 3 The Proposed Pattern Recognition Method
### Feature Ranking and Selection Based on Improved Laplacian Score Algorithm  
Although the extracted features can be used to identify the working status of equipment from a variety of aspects, they have different sensitivities to different working conditions. In addition, too many inputs for the FCM will consume more clustering time and reduce the clustering accuracy. Therefore, it is necessary to remove the insensitive features and refine the feature vectors. In this paper, an improved Laplacian score (LS) algorithm is introduced, in order to rank the extracted features. Following this, some optimal features are selected, according to their importance and locality preserving. Suppose that the experiment is comprised of \(M\) samples, and that each sample has \(T\) features. In this study, \(L_{r}\) denotes the Laplacian score of the \(r\)th feature, and \(f_{ir}\) denotes the \(r\)th feature of the \(i\)th sample, where \(i=1,2,\ldots,M\), \(r=1,2,\ldots,T\). The feature vector of \(M\) samples can be used to construct a nearest neighbor graph \(G\), and the weighting matrix \(S\) can then be calculated as the similarity matrix of graph \(G\). The Laplacian score of the \(r\)th feature can be computed as follows [33]:  
\[L_{r}=\frac{\sum\limits_{i,j=1}^{M}\left(\widetilde{f}_{ir}-\widetilde{f}_{ir} \right)^{2}S_{ij}}{\sum\limits_{i=1}^{M}\left(\widetilde{f}_{ir}-\frac{1}{M} \sum\limits_{j=1}^{M}\widetilde{f}_{ir}\right)^{2}D_{ii}}=\frac{\widetilde{ \mathbf{f}}_{r}^{\mathrm{T}}\mathbf{L}\widetilde{\mathbf{f}}_{r}}{\widetilde {\mathbf{f}}_{r}^{\mathrm{T}}\mathbf{D}\widetilde{\mathbf{f}}_{r}^{\mathrm{ T}}}, \tag{22}\]  
where \(\mathbf{f}_{r}=\left(f_{1r},f_{2r},\cdots,f_{Mr}\right)^{T}\), \(\mathbf{D}=\mathbf{S}\), \(\mathbf{I}=\left(1,1,\cdots,1\right)^{T}\), and \(\mathbf{L}\) is the Laplacian matrix of graph \(G\), and can be calculated as \(\mathbf{L}=\mathbf{D}-\mathbf{S}\), \(\widetilde{\mathbf{f}}_{r}=\mathbf{f}_{r}-(\mathbf{f}_{r}^{\mathrm{T}}\mathbf{ D}\mathbf{I})\mathbf{I}/(\mathbf{I}^{T}\mathbf{D}\mathbf{I})\). After calculating the LS value of each feature, the features can be ranked from a low score to a high score. However, the Laplacian score algorithm depends too much on the local structure information of adjacent samples. The selected features, based on LS, possess a stronger representational capacity for the similarity of adjacent samples, and a weaker distinguishing capability for global samples. To the best of our knowledge, the Fisher score (FS) algorithm is usually employed to measure the separation degree between samples. In this work, FS is coupled with the LS algorithm, named the Laplacian-Fisher score (LFS), to select the optimal features from the entire 28 features. The Fisher score of the \(r\)th feature can be written as follows:  
\[F_{r}=\left[\sum_{ti=1}^{T}M_{ti}(\mu_{rti}-\mu)^{2}\right]\left/\left[\sum_{ti= 1}^{T}M_{ti}\sigma_{ri}^{2}\right],\right. \tag{23}\]  
where \(ti\) denotes the category of the \(i\)th sample, \(M_{ti}\) denotes the number of samples of the \(t\)th category, and \(\mu_{rti}\) and \(\sigma_{rti}\) represent the mean and standard deviation of the \(r\)th features, respectively. The feature with the largest Fisher score has a stronger distinguishing capability for global samples. In order to guarantee the selected features with a smaller LS and larger FS, a weighting factor \(\alpha\) is introduced into the improved LS algorithm, and the LFS can be calculated as:  
\[LF_{r}=|\alpha L_{r}-(1-\alpha)F_{r}| \tag{24}\]  
where \(\alpha\in[0,1]\). The improved LS method is degenerated to the basic LS method for \(\alpha=1\), while the improved LS becomes the basic FS method for \(\alpha=0\). In order to illustrate the superiority of the improved LS method, the comparisons of LFS, LS, and FS are conducted. Two data sets: Glass and Wall-Following Robot Navigation (WFRN) [34], are selected from the UCI database for the simulations, to estimate the ranking performance of the three methods.