# Fisher Discriminative Sparse Representation Based on DBN for Fault Diagnosis of Complex System
## 4 Case Study
### Visualization of Learned Representation  
To demonstrate the efficacy of the proposed FDSR based on DBN for feature extraction and diagnosis in industrial systems, we introduced the visualization technique called t-distributed stochastic neighbor embedding (t-SNE) [50; 51]. T-SNE is a tool for dimensionality reduction that is well-suited for the visualization of high-dimension datasets. We chose faults 1, 2, 6, 8, 12, 13, 14, and 18 (Bold fault types in Table 3) and the normal data to complete the experiment. Step, random, slow drift, sticking, and unknown fault types were included to fully reflect the efficacy of the FDSR algorithm. Here, the DBN was used to extract the features with 16 dimensions from raw process data, hasten the dictionary learning, and eliminate redundant information for diagnosis. Then, t-SNE, the dimension reduction technique, was used to convert the 16-demensional features to a three-dimensional (3D) map. The resulting scatterplot maps are shown in Figure 8. To better demonstrate the visualization of the clustering performance, Figure 8b is the 3D view at an azimuth of 90 degrees and a top view of 90 degrees, and Figure 8a is the view at an azimuth of \(-\)37.5 degrees and a top view of 30 degrees. Furthermore, the features were used to learn the dictionary with 176 dimensions using the iterative projection method. The discriminative sparse coefficients were updated using the quadratic programming algorithm. Then, we applied t-SNE to convert the discriminative coefficients to a 3D map and the resulting scatterplot maps are shown in Figure 8. We used the same technique to visualize the raw process data shown in Figure 9. Only three conditions were completely separated, and the remainder are indistinguishable in Figure 9. The features extracted by DBN were also poor, with four conditions separated completely, as shown in Figure 8. The coefficients acquired from FDSR clustered best where the data points of nine conditions were well-separated in Figure 10. This verifies the good classification ability of the FDSR algorithm investigated in this study. The result also indicates that \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Classifier** & **Normal** & **Fault 1** & **Fault 2** & **Fault 6** & **Fault 8** & **Fault 12** & **Fault 13** & **Fault 14** & **Fault 18** \\  & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** \\ \hline DBN & 63.2 & 71.0 & 82.7 & 83.0 & 76.5 & 71.4 & 62.0 & 63.4 & 62.0 \\ BP & 81.2 & 83.2 & 76.5 & 75.2 & 77.2 & 64.5 & 65.8 & 66.6 & 63.2 \\ DBN+SVM & 88.1 & 80.1 & 86.5 & 84.2 & 88.1 & 78.2 & 69.2 & 72.0 \\ DBN+SRC & 79.2 & 82.1 & 75.8 & 83.2 & 72.5 & 76.5 & 71.2 & 73.9 & 65.3 \\ DBN+FDSR & 99.2 & 98.2 & 96.5 & 97.8 & 96.8 & 98.9 & 99.3 & 92.3 & 94.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comprehensive comparison of classifier performance. faults 12, 13, 14, an 18 are hard to classify without an effective method, whereas the FDSR algorithm investigated here solved this problem well. This also verifies in the classification accuracy of the different conditions displayed in Table 3.  
Figure 8: Visualization of features extracted by DBN ((**a**) is the view at an azimuth of \(-37.5\) degrees and a top view of 30 degrees, and (**b**) is the 3D view at an azimuth of 90 degrees and a top view of 90 degrees). Figure 10: Visualization of FDSR algorithm result ((**a**) is the view at an azimuth of \(-37.5\) degrees and a top view of 30 degrees, and (**b**) is the 3D view at an azimuth of 90 degrees and a top view of 90 degrees). Figure 9: Visualization of raw data ((**a**) is the view at an azimuth of \(-37.5\) degrees and a top view of 30 degrees, and (**b**) is the 3D view at an azimuth of 90 degrees and a top view of 90 degrees). From the trends shown by the three maps (Figures 8-10), the DBN itself is an effective feature-extraction method for industry process fault diagnosis.