# Modified Local Linear Embedding Algorithm for Rolling Element Bearing Fault Diagnosis
## 6 Experiments and Analysis  
### Artificial Data Set  
In this paper, we select three kinds of artificial data sets, which are Swiss-roll, S-curve and Punctured-sphere, in order to verify the effectiveness of the improved LLE algorithm. In Figure 3, the graphs a(1), b(1) and c(1) are the three-dimensional manifolds when the artificial data sets are Swiss-roll, S-curve and Punctured-sphere without noise. In order to verify the effectiveness of manifold learning from the intuitive and quantitative aspects, the evaluation criteria of residual error is introduced in this paper to depict the effect of dimensionality reduction. \[r=\frac{\sum_{i=1}^{n}[D_{X}(i)-D_{Y}(i)]^{2}}{n-2} \tag{15}\]  
We measure the embedded quality by using the residual error \(r\) between the coordinates \(D_{X}(i)\) in original higher-dimensional space and the embedded coordinates \(D_{Y}(i)\) in low-dimensional space [7]. The smaller residual error, the better effectiveness of dimensionality reduction. It is also means the result of dimensionality reduction can preserve the topology of original manifold better. For each kind of artificial data set, the number of sample points is 4000, the manifold is reduced to two dimensions, and the neighborhood size is 25. Figure 3: \(Cont.\)  
Figure 2: The fault diagnosis model based on M-LLE algorithm. For original artificial data sets, they are shown in Figure 3a(1),b(1),c(1). As is shown in Figure 3a(1), the Swiss-roll data set is scroll-like in three-dimensional space, where the color on the manifold is from deep to shallow to represent the data points of the different coordinates in the high dimensional space. As is shown in Figure 3b(1), the S-curve data set is S-shaped in three-dimensional space. The effective expansion of Swiss-roll and S-curve manifolds in two-dimensional space is a rectangular and the sample points of different colors can be separated well. As is shown in Figure 3c(1), the Punctured-sphere data set is a non-closed sphere in a three-dimensional space. When the dimension reduction algorithm is valid, its manifold in two-dimensional space is a circle with separated colors. For these three kinds of artificial data sets with no noise pollution, Figure 3a(2),b(2),c(2) are shows the embedded results of LLE algorithm. It can be clearly seen that distortion has occurred on embedded manifold, so they are can't recover the rectangular state of Swiss-roll and S-curve in two dimensions. What's more, at the edge of Figure 3c(2), some data points are stacked. However, the embedded manifolds which obtained by M-LLE algorithm can get good recoveries of the rectangular in Figure 3a(3),b(3), and the circle in Figure 3c(3). It is clear that data points of different colors can be completely separated and expanded. In addition to the intuitive graphical display, we can also see from the quantitative indicators. The residual errors in Figure 3a(2),b(2) are 0.3292 and 0.2129, but the residual errors of using M-LLE algorithm in Figure 3a(3),b(3) are 0.2612 and 0.1886, which are smaller than using LLE algorithm. It can also be judged by the residual errors in Figure 3c(2),c(3). When we adopt LLE algorithm in Punctured-sphere data set, the residual error of dimensionality reduction is 0.1167. There is no doubt that 0.1167 is higher than 0.1092 in Figure 3 c(3). Therefore, using M-LLE algorithm can effectively preserve the topology of original manifolds, and it is helpful for dimensionality reduction. The above simulation results have verified the effectiveness of M-LLE algorithm in the noiseless data. Next, in order to prove the robustness of M-LLE algorithm, we introduce the noisy artificial data sets in this study. The settings of the parameters are shown in Table 1.  
Figure 3: The dimensionality reduction effect of M-LLE algorithm in noiseless artificial data set. As is shown in Figure 4(a1),b(1),c(1), we add 5% noise to Swiss-roll data set, 10% noise to S-curve data set and 15% noise to Punctured-sphere data set.