## 4 Result and Analysis
### Non-TL Approach  
To establish a baseline diagnostic accuracy for the TL-based approach, a diagnostic process without TL was first tested. The non-TL approach consisted of two stages. The first stage aims at dimensionality reduction, and the technique chosen uses the minimal-redundancy-maximal-relevance (mRMR) criterion [31]. mRMR selects the most important features among all parameters in the PACK simulation data, ranking them from those with the highest relevance to the labels and the lowest redundancy with other features [31]. Compared to other dimensionality reduction techniques that map data onto a lower-dimensional space, the features selected by mRMR still carry physical significance, and hence, can be physically interpreted. In the second stage, a classifier is required to map each symptom vector (i.e., the vector of a PACK simulation case with features selected by mRMR) to a health state label. In this study, the k-nearest neighbour (k-NN) algorithm was chosen. Figure 5: Distribution of the first principal component value for cases of the same label at 28,000 ft and 41,000 ft, plotted to represent the discrepancy in conditional distribution. \begin{table}
\begin{tabular}{p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt} p{42.7 pt}} \hline \hline
**Domain** & **Altitude (ft)** & **Bleed Temperature (K)** & **Bleed Pressure (kPa)** & **Ram Air Temperature (K)** & **Target Temperature (K)** & **Target Stage Flow Rate (kg\(\cdot\)s\({}^{-1}\))** & **Number of Cases** \\ \hline Source & 28,000 & 469.0 & 193 & 232.7 & 291.2 \(\pm\) 10 & 0.45 \(\pm\) 0.02 & 91 healthy-state \\  & & & & & & & 90 faulty-state \\  & & & & & & & \\ \hline Target & 41,000 & 469.3 & 190 & 216.7 & 298.1 \(\pm\) 10 & 0.42 \(\pm\) 10 & 0.42 \(\pm\) 10 & 91 healthy-state \\  & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of operating condition in source and target domain for developing TL-based diagnosis algorithm. Because there is no label for the target domain data, k-NN cannot be trained by it. The only way to predict target domain labels is by training the k-NN with source domain labelled data and then directly applying the trained k-NN to predict labels in the target domain. In addition, to verify the training of the k-NN, 80% of the source domain data was used for training, and the other 20% was used for validation. After passing the training and validation steps, the k-NN classifier can be used on the target domain data. Figure 6 plots the k-NN prediction accuracy in both the source and target domains, and how it varies with the number of features selected by mRMR. The predictive accuracy of the source domain validation data increased when the input dimension to the k-NN increased from 1 to 4. It maintained at 100% when the input dimension is equal to or above four, which indicates that the minimum dimension should be taken as four to ensure a well-trained k-NN from the source domain. When applying the trained k-NN directly to the target domain at the 41k ft condition, its predictive accuracy increased steadily when the input dimension increased from 1 to 4 and continued to increase from 4 to 6, before remaining roughly constant at input dimensions above 6. Hence, the optimal dimension for mRMR feature selection was chosen as 6, where the predictive accuracy of k-NN on the target domain data was 41.99%. Reducing from 100% in the source domain to 41.99% in the target domain, it is clear that the predictive accuracy of the k-NN trained by the 28k ft cruising data suffers significant degradation when applied to the 41k ft cruising data. The reason for this performance degradation was the distribution discrepancy between the source and target domain data. For instance, from the confusion matrix of target domain data shown in Figure 7, one major source of misclassification is the prediction of a target domain case with a healthy true label as a RI blockage case.