## 5 Fault Diagnosis Using the Proposed Method  
### Case Study 1: Fault Diagnosis of Motor Bearing  
The bearing dataset provided by Case Western Reserve University [25] is analyzed in this section. The dataset is detailed in Section 3.2.1. The dataset included ten health classes under four loads, and we treated the same health state under different loads as one class. Additionally, 15 trials were carried out for each experiment to reduce the randomness effect.  
First, we investigated the selection of the input dimension \(N_{in}\) of modified sparse filtering. 10% of the samples were randomly selected to train the proposed model. The remaining samples were equally divided into testing (to adjust the parameters) and validation dataset. The weight decay term of softmax regression was equal to \(1\times 10^{-5}\) and \(\lambda=1\). The output dimension \(N_{out}\) was half of the input dimension. The diagnosis results, in comparison to the original method with different \(N_{in}\)  
Figure 10: Illustration of the proposed two-stage learning method.  
are displayed in Figure 11. As seen in the figure, all testing accuracies of the proposed method were over 98.1%. Although the non-monotonicity phenomenon was still present, the testing accuracies of the proposed method were higher than the original method. This suggests that the deficiency of parameter selection was improved.  
The overfitting phenomenon always arises when the training dataset is small. Therefore, we investigated the selection of proper percentages in training samples. The diagnosis accuracies are shown in Figure 12. The testing accuracies of the proposed method are higher than the original method in each condition. Furthermore, the proposed method obtained 95.3% accuracy, with a small standard deviation of 1.03% using only 1% of samples for training. Such results indicate that the proposed method performs well and can overcome the overfitting problem.  
We selected the parameters of \(N_{in}\) = 100, \(N_{out}\) = 100, and \(\lambda\) = 1; softmax regression was equal to 1 \(\times\) 10\({}^{-5}\). The average accuracy of validation dataset was 99.85%, higher than the 99.66% obtained by the original method. To compare features, t-SNE [35] is used. This technique enabled us to embed these 100-D vectors in a 3D image in such a way that the vectors which were in close proximity to each other in the 100-D space were also in close proximity in the 3D plot [36]. The results of validation dataset processed by t-SNE are shown in Figure 13. The mapped features of the different types are  
Figure 11: Diagnosis results using various input dimensions of modified and original sparse filtering.  
Figure 12: Diagnosis results obtained by different percentages of samples using modified and original sparse filtering.  
demonstrably separated, and the features of the same type are gathered together; the distance between each type is large enough to distinguish different health conditions.  
As shown in Section 3, the nature of overfitting produces unclear and similar patterns for different vectors of the weight matrix. Figure 14 shows the inner product result of the weight matrix of the proposed method when the input dimension is equal to 200. 15 row vectors of \(\mathbf{W}\) were randomly selected and plotted in Figure 15a. The corresponding frequency spectra of them are displayed in Figure 15b; notably, most of the row vectors are approximately orthogonal. Additionally, the bandwidths of the row vectors are narrow in the frequency domain, as shown in Figure 15b. This suggests that the modified sparse filtering can extract more discriminative features with less redundancy. As a result, the accuracies obtained by the proposed method are higher because the learned features are constrained to be more meaningful and dissimilar.  
Figure 14: Results of \(\mathbf{W}\mathbf{W}^{\text{T}}\) of \(N_{in}=200\) using the proposed method.  
Figure 13: Visualization of features of validation dataset processed by t-SNE.