# Research on Fault Recognition of Roadheader Based on Multi-Sensor and Multi-Layer Local Projection
## 4 Multi-Sensor and Multi-Layer Fault Recognition Based on LPP  
### Locality-Preserving Projection  
Locality-preserving projection (LPP) is a manifold learning algorithm proposed by [PERSON] based on the Laplacian eigenmaps (LE) algorithm [23]. This algorithm retains the advantages of non-linear manifold learning and linear learning, and its idea is to approximate the linear learning of LE. LPP can extract the most discriminative features for dimensionality reduction and is a dimensionality reduction method that preserves local information and reduces the interference of multiple external factors in signals, thereby preserving the local non-linear relationships when analyzing high-dimensional original data [24; 25; 26]. Compared with other non-linear dimensionality reduction methods, LPP can incorporate new analysis samples into the original manifold and find the corresponding mapping in the reduced subspace by calculation, thus enabling the LPP algorithm to evaluate and analyze new sample data. The process of the LPP algorithm can be mainly divided into three steps [23; 27; 28; 29; 30]: (1) finding nearest neighbors; (2) computing the affinity matrix; and (3) constructing the objective function, as shown in Figure 8.  
The deduction process of LPP is as follows:  
The original sample set is \(X=\{x_{1},x_{2},x_{3}\ldots x_{n}\}\), where the dimension of \(X\) is \(D\), \(x_{i}\in R^{D},i=0,1,2,\ldots,n\). Then, \(Y=\{y_{1},y_{2},y_{3}\ldots y_{n}\}\) is the low-dimensional mapping of the original sample set in the manifold space, where the dimension of \(Y\) is \(d\), \(d<<n\). Figure 8: Flowchart of LPP. The objective function needs to be minimized when the LPP algorithm is used for low-dimensional mapping. After minimization, the original neighbor structure remains unchanged. \[\text{argmin}\sum\limits_{i,j=0}^{n}\left\|y_{i}-y_{j}\right\|^{2}W_{ij} \tag{8}\]  
where \(W_{ij}\) is the edge weight between \(x_{i}\) and \(x_{j}\) on the nearest neighbor graph \(G\). For a neighbor graph \(G\), if two vertices \(j\) and \(i\) are neighbors, \(G_{i,j}=1\); otherwise, \(G_{i,j}=0\). The formula for calculating the edge weights of the nearest neighbor graph is shown as follows:  
\[W_{ij}= \begin{cases}e^{-\frac{|x_{i}-x_{j}|^{2}}{t}},G_{i,j}=1\\ 0,G_{i,j}=0\end{cases}  \tag{9}\]  
where \(t\) is constant. By introducing the Heat-Kemal theory for calculation, the edge weight formula can be simplified as:  
\[W_{ij}= \begin{cases}1,G_{i,j}=1\\ 0,G_{i,j}=0\end{cases}  \tag{10}\]  
The Laplace change is used to transform Formula (8). \[ \begin{array}{l}\sum\limits_{i,j=0}^{n}\left\|y_{i}-y_{j}\right\|^{2}W_{ij} \\ =\sum\limits_{i,j=0}^{n}\left({y_{i}}^{2}+{y_{j}}^{2}-2{y_{i}}{y_{j}}\right)W _{ij}\\ =\sum\limits_{i,j=0}^{n}{y_{i}}^{2}W_{ij}+\sum\limits_{i,j=0}^{n}{y_{j}}^{2}W _{ij}-2\sum\limits_{i,j=0}^{n}{y_{i}}{y_{j}}W_{ij}\\ =\sum\limits_{i,j=0}^{n}{y_{i}}^{2}D_{ii}+\sum\limits_{i,j=0}^{n}{y_{j}}^{2}D_ {jj}-2\sum\limits_{i,j=0}^{n}{y_{i}}{y_{j}}W_{ij}\\ =2Y^{T}LY\end{array}  \tag{11}\]  
where \(D_{ii}=\sum_{j}W_{ij}\), \(L\) is a Laplacian matrix, and \(L=D-W\). Therefore, the objective function can be converted to the following formula:  
\[\text{argmin}_{Y^{T}DY}Y^{T}LY \tag{12}\]  
Using the Lagrange algorithm, Formula (12) can be converted to solve the eigenvalue problem of the following formula:  
\[LY=\lambda DY \tag{13}\]  
The low-dimensional space features of the original data can be obtained by solving Formula (13).