# Modified Local Linear Embedding Algorithm for Rolling Element Bearing Fault Diagnosis
## 6 Experiments and Analysis
### Rolling Element Bearing Fault Data Set
Therefore, we use M-LLE algorithm to reduce the dimension of 20-dimensional fault data, and obtain the intrinsic characteristic information, which is the fundamental basis for judging the fault type. In order to verify the effectiveness of M-LLE algorithm, this paper uses four kinds of state bearing data: normal state, inner race fault, outer race fault and ball fault to make the experiment. Divide each status of rolling element bearing data into 600 groups, and add labels to them. Detailed data are shown in Table 3.  
We reduce the dimensions of the obtained 20-dimensional data by using LLE algorithm and M-LLE algorithm. Figure 7 shows the different state of the fault data distribution in three-dimensional space. What's more, the \(X\) axis, \(Y\) axis and \(Z\) axis represent the three features (\(K_{1}\), \(K_{2}\) and \(K_{3}\)) respectively, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number** & **Characteristic Equation** & **Number** & **Characteristic Equation** \\ \hline
1 & \(c_{1}=\sum\limits_{n=1}^{N}x(n)/N\) & 11 & \(c_{11}=\sum\limits_{k=1}^{K}s(k)/K\) \\
2 & \(c_{2}=\sqrt{\sum\limits_{n=1}^{N}(x(n)-c_{1})^{2}/(N-1)}\) & 12 & \(c_{12}=\frac{\sum_{k=1}^{K}(s(k)-c_{11})^{2}}{(K-1)}\) \\
3 & \(c_{3}=\left(\sum\limits_{n=1}^{N}\sqrt{|x(n)|}/N\right)^{2}\) & 13 & \(c_{13}=\frac{\sum_{k=1}^{K}(s(k)-c_{2})^{4}}{K_{12}^{2}}\) \\
4 & \(c_{4}=\sqrt{\sum\limits_{n=1}^{N}(x(n))^{2}/N}\) & 14 & \(c_{14}=\frac{\sum_{k=1}^{K}f_{ks}(k)}{\sum_{k=1}^{K}s(k)}\) \\
5 & \(c_{5}=max|x(n)|\) & 15 & \(c_{15}=\sqrt{\sum_{k=1}^{K}(f_{k}-c_{44})^{s}(k)}\) \\
6 & \(c_{6}=\sum\limits_{n=1}^{n}(x(n)-c_{1})^{4}/(N-1)c_{2}\)\({}^{4}\) & 16 & \(c_{16}\)\(c_{16}=\sqrt{\frac{\sum_{k=1}^{K}f_{ks}^{2}(k)}{\sum_{k=1}^{K}f_{ks}^{2}(k)}}\) \\
7 & \(c_{7}=c_{5}/c_{3}\) & 17 & \(c_{17}=\frac{\sum_{k=1}^{K}f_{ks}^{2}(k)}{\sum_{k=1}^{K}s(k)\sum_{k=1}^{K}f_{ks }(k)}\) \\
8 & \(c_{8}=c_{5}/c_{4}\) & 18 & \(c_{18}=c_{15}/c_{14}\) \\
9 & \(c_{9}=c_{4}/\left(\frac{1}{N}\sum\limits_{n=1}^{n}|x(n)|\right)\) & 19 & \(c_{19}=\frac{\sum_{k=1}^{K}(f_{ks}-c_{44})^{s}(k)}{K_{15}^{2}}\) \\
10 & \(c_{10}=c_{5}/\left(\frac{1}{N}\sum\limits_{n=1}^{n}|x(n)|\right)\) & 20 & \(c_{20}=\frac{\sum_{k=1}^{K}(f_{ks}-c_{44})^{1/2}s(k)}{K_{15}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Feature extraction in time domain and frequency domain. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Operating Condition** & **Training Samples** & **Testing Samples** & **Label** \\ \hline Normal & 400 & 200 & 1 \\ Outer race & 400 & 200 & 2 \\ Inner race & 400 & 200 & 3 \\ Ball & 400 & 200 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The details of each state of rolling element bearing data. which are obtained by dimensionality reduction. The \"**\" indicates the normal state data of the rolling bearing, \"\(\Delta\)\" indicates the inner ring fault status data of the rolling bearing, \"\(\bigcirc\)\" indicates the outer ring fault state data of the rolling bearing, and \"\(\square\)\" indicates the ball fault state data of the rolling bearing. Figure 7a is the result of reducing dimension by using the traditional LLE algorithm. As is shown in the figure, although four different states of the fault data have clustering effect, there are some data points aliasing among the bearing inner race failure, outer race failure and ball failure. Figure 7b is the result of reducing dimension by using M-LLE algorithm. There is obvious clustering phenomenon in different color bearing state data and there is no aliasing of data comparing with the traditional LLE algorithm. It will lay a solid foundation for the data classification and fault identification in the next step. According to the fault diagnosis model based on M-LLE algorithm is proposed in the fourth part of this paper, we can obtain final bearing fault diagnosis results by using the KNN classification algorithm.