# Deep Transfer Learning-Based Fault Diagnosis Using Wavelet Transform for Limited Data
## 5 Performance Evaluation
### Fault Data Analysis  
Figure 4 depicts the output flow rate and the power spectrum of the healthy state \(f_{0}\) and three different types of a single fault, \(f_{1}\), \(f_{2}\), and \(f_{4}\), as shown in Table 1. The faults \(f_{1}\), \(f_{2}\), and \(f_{4}\) correspond to the single fault type due to cylinder leak, blocked inlet, and increased bearing friction, respectively. Note that the unit of the volumetric flow rate is the liters per minute (lpm). We set the severity level \(l=4\) for all faults. Figure 3: Designed CNN architecture trained from scratch: \((c,a\times b)\) means \(c\) filters with the size of \(a\times b\). In Figure 3(a), the overall shapes of the output flow rates are similar between \(f_{0},f_{1}\), and \(f_{2}\), while the bearing friction fault \(f_{4}\) heavily decreases the output flow rate. Thus, the time domain analysis of the industrial signals is not enough to characterize each fault type. Some of the effects of different fault types are more noticeable in the spectrum analysis of the frequency domain, as shown in Figure 3(b). The power spectrum includes several resonant peaks. In particular, since the triplex pump model has three cylinders, it inherently has peaks at \(3\times 15.8\) Hz, or \(47.4\) Hz, as well as harmonics at multiples of \(47.4\) Hz where \(15.8\) Hz corresponds to the inverse of the pump motor speed, \(950\) rpm. The faults of \(f_{0},f_{1}\), and \(f_{2}\) have three peaks at \(15.8\) Hz, \(47.4\) Hz, and \(2\times 47.4\) Hz while \(f_{4}\) has three slightly shifted peaks. The power spectrum value of \(f_{1}\) at \(15.8\) Hz is higher than those of \(f_{0}\) and \(f_{2}\). Thus, it shows that the characteristics of both time and frequency domains are critical for classifying the fault types. However, it is still not trivial to distinguish \(f_{0}\) and \(f_{2}\) even in the power spectrum. Next, we analyze how the severity level of the fault condition affects the signal in both the time and frequency domains. Figure 5 presents the output flow rate and the power spectrum of the healthy state \(f_{0}\) and the blocked inlet fault \(f_{2}\) with different severity levels \(l=2,4,6\). We also report two output flow rates of the healthy state due to different noise realizations. In Figure 4(a), two outputs of \(f_{0}\) and three outputs of \(f_{2}\) with different severity levels are very similar in the time domain. On the other hand, Figure 4(b) shows that the high severity level affects the characteristics of the power spectrum. The blocked inlet fault \(f_{2}\) with the severe condition \(l=6\) results in several peaks at a high frequency of greater than \(70\) Hz. Furthermore, it also increases the power by around \(30\) Hz. However, the minor fault conditions with \(l=2,4\) still have a power spectrum similar to that of the healthy state  
Figure 4: Output flow rate and power spectrum of \(f_{0},f_{1},f_{2},f_{4}\). Figure 5: Output flow rate and power spectrum of \(f_{0}\) and \(f_{1}\) with different severity levels, \(l=2,4,6\). \(f_{0}\), making the fault diagnosis difficult. Note that the simulation parameter related to the blocked inlet \(f_{2}\) with \(l=2\) is 0.74, closer to the threshold of the healthy state, which is 0.8. Thus, we observe that the severity level could affect the fault classification performance even with the same fault type. To further illustrate the time and frequency characteristics of different faults, we show the wavelet images of various fault types \(f_{0},\ldots,f_{7}\) with a fixed severity level of \(l=6\) in Figure 6. Remember that the CWT technique converts the time-series signals into time-frequency images. We note that the wavelet of the same fault type could be different due to the different realization of the noise.