# Fault Detection of Bearing Systems through EEMD and Optimization Algorithm
## 3 Simulation of Bearing Response and Fault Detection Results
Obviously, maximizing the Q through PSO provides an effective measure for quantifying the classification performance in terms of bearing defect category. Figure 5 illustrates convergence curves of the PSO searching process for a set of weights having maximum Q. We tried several independent PSO runs to seek a best convergence result, and used the weights as a trained data. Figure 5 shows the best training run and test run. The figure shows that the value Q increased from 0.56 to 9.01 in the  
Figure 4: The first four EEMD-decomposed intrinsic mode functions (IMFs) of simulated time-series data: (**a**) healthy bearing model; (**b**) inner-race damaged bearing model; (**c**) outer-race damaged bearing model; and (**d**) rolling-element damaged bearing model. training case. Similarly, the test case results in a relatively small increase of Q (from 3.53 to 5.86). This trend of variation contributes to the fact that group-to-group or \(d(i,j)\) remains the same, while the point-to-point distance or \(d_{k}^{\prime}\) increases more significantly. Note that half of the total data sets were used for optimization _training_, while the other half were used for _testing_ the PCA classification. Figure 6 shows the PCA-based localization results for comparison; before and after performance of the weight-optimization of feature vectors using the training time-series segments towards different damage locations, i.e., inner, outer-race, and rolling elements. Apparently, the first three principal components of PCA show that feature vectors of inner-race and rolling element damage are mostly inseparable in three-dimensional space, as shown in Figure 6a. On the other hand, weight-optimized feature vectors show clear distinction and separation between the two groups of defects (see Figure 6b). Having finished training, we applied the weight-optimized parameter vectors to the test data set. Figure 7 shows the comparison of the PCA results of the test data using the trained parameter vectors. Figure 7a shows that the inner ring defect and the rolling element defect are partially mixed in a reduced three-dimensional space. The figure was obtained by performing PCA that had not been subjected to the optimization process, the elements of each group of defects were widespread, and there was a risk that the type of defects may have been misleading. However, Figure 7b shows that the feature vectors optimized by PSO were tightly grouped together by the same defect, so that it was easier to visually distinguish the inner ring defects and rolling element defects. Figure 5: Particle Swarm Optimization (PSO)-based optimization result of the weightings of parameter vectors extracted from time-segments of various damaged bearing models. Figure 6: Principal component analysis (PCA) representation of training characteristic vectors from simulated bearing model: conditions of intact (\(\blacksquare\)), inner-race defect (\(\blacktriangle\)), outer-race defect (\(\blacklozenge\)), and rolling-element defect (\(\blacklozenge\)), (\(\blacklozenge\)) before, and (\(\blacklozenge\)) after PSO optimization. Likewise, Figures 8 and 9 shows the Isomap classification results of training and testing data sets, respectively. Figure 8a shows that the boundaries of the inner-race and rolling element defects partially overlapped. Obviously, this training process enhanced the performance of Isomap representation in different types of bearing defect, as shown in Figure 8b. Figure 9 shows the visualization result of Isomap classification in three-dimensional space using testing data sets. Similar to the PCA results without the weight optimization process, the inner ring defect and the rolling element defect were positioned within the same group adjacent to each other (see Figure 9a). Figure 9b shows the result of optimization through PSO, wherein the inner-race defect and the rolling element defect were clearly distinguishable from each other.