## 4 Case Studies
### An Experimental Case Study on Complex-Shaped Metallic Turbine Blades
#### 4.1.3 Reference MD Space Based on Healthy and Defective Samples  
From the above results, it may be concluded that the use of defective samples to construct MD space has to be avoided. In this section, a two-dimensional reference MD space is constructed on the basis of both healthy and defective samples, partitioning the MD space into four regions of \begin{table}
\begin{tabular}{c c c c} \hline  & **Training Accuracy** & **Testing Accuracy** & **Number of Features** \\  & **(\%)** & **(\%)** & **(-)** \\ \hline
**IMCS\({}_{\text{PSO}}^{\text{BPSO}}\)** & \(87.20\pm 9.29\) & \(79.46\pm 10.11\) & \(5.33\pm 2.90\) \\ \hline \end{tabular}
\end{table} 
Table 2: Classification performance of the algorithm over 20 independent runs using a reference MD space based on defective samples. Figure 7: (**a**) Classification on the training dataset; (**b**) classification on the testing dataset. interest by two linear thresholds, _threshold_ 1 and _threshold_ 2. These two thresholds are learned from the IMCS classifier based on a reference MD space constructed from healthy samples and defective samples, respectively. The classification results of training and testing datasets are shown in Figure 6, where SMD-R1 and SMD-R2 stand for SMD based on the reference MD space constructed by healthy samples and defective samples, respectively. The four areas are referred to as A1, A2, A3, and A4 in Figure 8, providing insight into the reliability of the classification. Areas A1 and A2 indicate \"high\" reliability, and areas A3 and A4 indicate \"low\" reliability. Notably, the samples located in area A1 are all correctly classified as healthy samples. Moreover, most of the misclassifications appear in the areas with \"low\" reliability. Although there is no specific contribution in terms of classification accuracy by considering a two-dimensional MD space, further insight into the classified samples can be gained from the divided areas with different credibility. A significant drop in classification accuracy observed in both training and testing datasets from Table 3 is associated with the two introduced linear thresholds, partitioning the two-dimensional MD space into healthy (A1) and defective (A2, A3, A4) regions. From a more technical point of view, the linear thresholds may pose a strict restriction in classification accuracy. Instead of using linear threshold boundaries, a nonlinear classification boundary is considered to reach maximal discrimination between classes. Random forest (RF) is employed, which is an ensemble classifier composed of multiple decision trees [32]. RF consists of a set of simple tree predictors, in which each predictor can respond to a set of predicted scores. By combining multiple weak classifiers, the final classification scores are voted or averaged. Because of this, RF can yield good classification performance, with high generalization capability; at the same time, it is less sensitive to overfitting issues. In this study, the RF model was built and trained by generating an ensemble of 20 bagged classification trees. The training process was implemented in MATLAB(r) by using the Statistics and Machine Learning toolbox. The training and prediction outcomes, considering the RF approach for threshold determination, are displayed in Figure 9.  
Figure 8: (**a**) Classification on the training dataset; (**b**) classification on the testing dataset. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Training Accuracy** & **Testing Accuracy** & **Number of Features** \\  & **(\%)** & **(\%)** & **(\(\_\))** \\ \hline
**IMCS\({}_{\text{PSO}}^{\text{BPSO}}\)** & \(90.12\pm 1.22\) & \(80.58\pm 2.81\) &
\begin{tabular}{c} SMD-R1: \(5.50\pm 0.73\) \\ SMD-R2: \(5.33\pm 2.90\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Classification performance of **IMCS\({}_{\text{PSO}}^{\text{BPSO}}\)** over 20 independent runs using a reference MD space based on both healthy and defective samples.