## 4 Result and Analysis
### TL Approach: TCA and JDA  
Aiming to reduce the effect of a domain shift, a TL approach for solving the fault diagnosis problem is developed based on TCA and JDA, using k-NN as the classifier. Two important parameters to be determined for the TCA and JDA algorithms are the value of \(\lambda\), which controls the level of optimisation and the dimensions of the features. The value of \(\lambda\) is to be determined first while holding the feature dimension at 6. The predictive accuracy for the target domain cases varies with the value of \(\lambda\), as plotted in Figure 9. For TCA, the predictive accuracy shows a general increasing trend as \(\lambda\) decreases from 0.1 to 0.001 and remains constant at 97.99% for \(\lambda\leq 0.0007\). For JDA, the predictive accuracy stays around 40% for \(\lambda\) between 0.1 and 0.003 and jumps to around 95% for \(\lambda\) smaller than 0.0002. Hence, the optimal value of \(\lambda\) for both TCA and JDA was taken as 0.0001. Figure 8: An example of misclassification following the non-TL approach. Figure 7: Confusion matrix plot for target domain data following the non-TL approach. Holding \(\lambda\) at 0.0001, the dimensions of features have to be determined. Figure 10 plots the target domain predictive accuracy against the dimension of the features. For both TCA and JDA, the predictive accuracy rises considerably when the dimension of features increases from one to four and remains roughly constant when using six or more dimensions. Therefore, six was determined to be the optimal dimension for the transformed features in TCA and JDA. Using \(\lambda\) = 0.0001 and holding the transformed feature dimension at six, the TL-based diagnostic algorithm with TCA and JDA significantly improved the target domain prediction accuracy. Compared with the 41.99% predictive accuracy of the non-TL approach, the TL approach increased the target domain predictive accuracy to 97.97% by TCA and 92.82% by JDA. The confusion matrices for both cases are plotted in Figure 11.  
To justify the significant improvement made by the TL-based approach, it is expected that TCA and JDA align the distribution discrepancy between the source and target domains. Two observations support this claim. The first piece of evidence is provided by calculating the MMD distance between the source and target domain features. Before and after applying TCA and JDA to the data, the MMD distance between the source domain features and target domain features was calculated to be 4.9 \(\times\) 10\({}^{-2}\). This distance was reduced to 6.2 \(\times\) 10\({}^{-7}\) by TCA and to 4.6 \(\times\) 10\({}^{-5}\) by JDA, which means that both TCA and JDA significantly reduced the distribution discrepancy. Figure 10: Predictive accuracy of TCA and JDA against dimension of transformed features, when \(\lambda\) = 0.0001. Figure 9: Predictive accuracy of TCA and JDA against value of \(\lambda\), using 6 features. The second confirmation was achieved through visualisation. To visualise the six-dimensional data in two-dimensional plots, t-distributed stochastic neighbour embedding (t-SNE) is employed as the dimensionality reduction technique, because it performs the non-linear mapping of high-dimensional data onto a low-dimensional space, and many relevant studies have reported the successful application of t-SNE to visualise engineering data [14; 33; 34]. Using t-SNE, the source and target domain cases are plotted in Figure 12. The source domain cases are marked with dots, and the target domain cases are marked with crosses. Different colours were used to differentiate cases under each health state. It should be mentioned that the labels in the target domain have only been used to plot the cases, and they are not known by the algorithm. In Figure 12, the elimination of the domain shift is achieved when the dots and crosses of the same colour coincide.