# Modified Local Linear Embedding Algorithm for Rolling Element Bearing Fault Diagnosis
## 5 Fault Diagnosis Model Based on M-LLE Algorithm  
Rolling element bearing is an indispensable component of industrial infrastructure equipment, its health state directly determines the health status of the mechanical system. The timely detection of the mechanical system and taking effective measures for bearing failure are indispensable works in real production process, and also a worthy of the further study of the direction [26]. Because the working condition is complex and the bearing is susceptible to noise interference, it often causes vibration signals show the characteristics of high-dimensional nonlinearity. Obviously, it is difficult to directly carry out effective feature extraction and make fault diagnosis. In this paper, M-LLE algorithm is used to get an effective dimensionality reduction from the obtained fault signals. Therefore, it is convenient to carry out effective and accurate fault diagnosis. Based on above method of neighborhood construction and weight construction, M-LLE algorithm is proposed to improve the robustness and accuracy of the original LLE algorithm. The details of M-LLE algorithm are shown in Algorithm 1.  
```
0: Initialize high-dimensional sample points \(x_{i}\), neighborhood size \(k\), and intrinsic dimension \(d\)
0: Low-dimensional embedded coordinates \(\{y_{1},y_{2},\ y_{3},\dots,\ y_{n}\}\)
1:for\(i:=1\) to \(n\)do
2: Obtain neighborhood \(U_{i}\) by improved neighborhood construction algorithm
3: Calculate distance weight matrix \(W_{i}^{S}\) by Equation (10)
4: Get structural weight matrix \(W_{i}^{D}\) by combining Equations (12) and (13)
5: Acquire local weight matrix \(W_{i}=W_{i}^{S}\times W_{i}^{D}\)
6: According to the Equation (3) to get local embedded coordinates \(y_{i}\)
7:end
8:return\(\{y_{1},y_{2},\ y_{3},\dots,\ y_{n}\}\)
```  
**Algorithm 1** M-LLE Algorithm  
In the process of rolling element bearing fault diagnosis, we obtain the original vibration signals by using the same sampling frequency to carry on the data acquisition. The rolling element bearing is working under a certain load and speed. The collected data is divided into training sample set and testing sample set, and adopting the frequency domain analysis to get signal feature extraction.  
However, the characteristics obtained only by signal feature extraction method may tend to characteristic redundancy, or result in partial feature conflict. In order to reduce the inaccuracy of feature extraction, M-LLE algorithm is used to ensure that the dimensionality reduction is under the same geometric relationship premise. This method can effectively extract key characteristics of fault data, and avoid the error caused by artificial selection. Subsequently, fault classifier is trained by training samples which have been reduced to intrinsic dimension, and put low-dimensional testing samples into the trained classifier. Finally, according to the output of classifier, we can determine the working condition and fault type of rolling element bearing. The detailed implementation process of fault diagnosis model based on M-LLE algorithm is shown in Figure 2.