# Transfer Learning-Based Fault Diagnosis under Data Deficiency
## 3 Application
### Training Neural Networks of Target Domain by Transfer Learning  
Once the NN is trained in the source domain, it can be exploited by TL in the target domain. There are two ways for this: One is named TL1, which uses the same architecture in the target domain as that of the source domain. The other is TL2, which, in addition to the same architecture, initial values for the training are given by the pre-trained NN parameters of the source domain. Whichever it is, the training is carried out using the architecture of the source domain by 300 times for the training datasets in the target domain as given in Table 1. In the training, the initial values of the parameters are assumed at random in TL1 whereas they are given by the pre-trained NN parameters in TL2. The trained NNs are applied to the test datasets of Table 1, from which the diagnosis accuracies are obtained as defined previously in the source domain. Since the 300 number of accuracy is obtained as a result, the accuracy is represented by the distribution, which occurs due to the randomness of NN solutions. Furthermore, the number of fault datasets vary by IR. So this is repeated for each IR from 0.1 to 1.0 by the increment of 0.1. Note that the smaller the IR, the more the data deficiency becomes greater. To make a comparison of the TL-based approach, the ordinary machine learning (ML) approach is also performed. In this case, the training and validation are carried out solely by the training datasets in the target domain to determine the optimum architecture. The procedure is the same \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feature** & **Formula** & **Feature** & **Formula** \\ \hline Mean & \(\frac{1}{N}\sum\limits_{i}^{N}x_{i}\) & Crest factor & \(\frac{\max\left(|x|\right)}{\sqrt{\frac{1}{N}\sum\limits_{i}^{N}x_{i}^{2}}}\) \\ Root amplitude & \(\left(\frac{1}{N}\sum\limits_{i}^{N}\sqrt{|x_{i}|}\right)^{2}\) & Clearance factor & \(\frac{\max\left(|x|\right)}{\left(\frac{1}{N}\sum\limits_{i}^{N}\sqrt{|x_{i}|} \right)^{2}}\) \\ RMS & \(\sqrt{1}\sum\limits_{i}^{N}x_{i}^{2}\) & Shape factor & \(\frac{\sqrt{\frac{1}{N}\sum\limits_{i}^{N}x_{i}^{2}}}{\frac{1}{N}\sum\limits_ {i}^{N}|x_{i}|}\) \\ Standard deviation & \(\sqrt{\frac{1}{N-1}\sum\limits_{i}^{N}(x_{i}-\overline{x})^{2}}\) & Impulse factor & \(\frac{\max\left(|x|\right)}{\frac{1}{N}\sum\limits_{i}^{N}|x_{i}|}\) \\ Peak & \(\max(|x_{i}|)\) & Peak-to-peak & \(\max(x_{i})-\min(x_{i})\) \\ Skewness & \(\frac{\sum_{i}^{N}(x_{i}-\overline{x})^{3}}{(N-1)\sigma^{3}}\) & Root sum of squares & \(\sqrt{\sum\limits_{i}^{N}x_{i}^{2}}\) \\ Kurtosis & \(\frac{\sum_{i}^{N}(x_{i}-\overline{x})^{4}}{(N-1)\sigma^{4}}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Equations for statistical features. Figure 6: Diagnosis accuracy as a function of the number of hidden nodes: (**a**) the source domain; (**b**) the target domain with IR of 0.5.  
as that addressed in the source domain. The process is repeated for each IR from 0.1 to 1.0 by the increment of 0.1. Figure 6b gives the result of validation when IR is 0.5, where the optimum number of hidden nodes is found at 16. The results for other IRs are found in Table 3. Once this is done, the NN has trained again by 300 times to obtain the 300 sets of parameters under the architecture using the training datasets with the initial values assumed at random. Finally, the trained NNs are applied to the test datasets of Table 1, from which the diagnosis accuracies are obtained, representing the distribution due to the randomness of NN solutions. The results of accuracy distribution in TL1, TL2, and ML are given as a function of IR as shown in Figure 7a-c, respectively. In the figures, the bottom, top, and red lines of the box indicate the 25 th, 75 th percentile, and median of the distribution, respectively. The top and bottom of the extended dotted lines are the 5 th and 95 th percentiles.