## 3 Results and Discussion
### Root Rule Node: Distinguish MS1 Class from the Rest by the First PCA Submodel in SIMCA  
A SIMCA model consists of a collection of PCA submodels, which is typically one for each class in the dataset, but it is also possible to build a PCA submodel on combined classes. In this context, the first PCA submodel with four PCs had been fitted on samples only belonging to the MS1 class; then, it was added to SIMCA. After that, another PCA submodel with four PCs fitted on the rest of the samples that combined all the remaining classes had also been built and added to SIMCA. Finally, these two PCA submodels were assembled in the SIMCA model.  
Both submodels were trained using the same preprocessing method consisting of the first Savitzky-Golay derivative (second polynomial with a filter width of 15) followed by mean centering. The configurations in cross-validation of the Venetian blinds method included 10 splits and one sample per split. Based on the value of RMSECV, the optimal number of principal components (PCs) in these two models was selected, and both were equal to four. The performances of these two submodels in the training and test phases are shown in Table 2.  
Figure 6: Loadings plot on the two principal components of the corresponding PCA model.  
From Table 2, it can be found that during the training phase, the first PCA submodel in SIMCA had misclassified three samples of MS1 as the remaining class; despite this case, this PCA submodel perfectly identified all the members of MS1 in the test phase. It also can be verified in plots featuring the classification results of this PCA submodel both in the training and test phases, as shown in Figure 7.  
In Figure 7a, those members of the MS1 class that had been misclassified as the remaining class were selected and labeled with their filename labels. However, all the members belonging to the MS1 class in the test phase were precisely classified as MS1 in Figure 7b. Furthermore, to incorporate this PCA submodel as the root decision node into the hierarchical model structure, the projection test-based rule that tells how to choose a branch based on the predictions of the given model from the input data needs to be defined.  
This PCA submodel had a specific Q limit that equaled 1.4385e-06, as shown in Figure 8, and according to this Q limit, this model had misclassified three samples of the MS1 class; their labels were just the same with those observed in Figure 7a. That is to say, this defined test condition \"Q >1.4385 \(\times 10^{-6}\)\" would work well to specify which given branch would be selected after the root decision node.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Actual Class**} \\ \cline{2-5}
**Predicted Class** & \multicolumn{2}{c}{**Train**} & \multicolumn{2}{c}{**Test**} \\ \cline{2-5}  & **MS1** & **MS2, MS3, MS4, MS5** & **MS1** & **MS2, MS3, MS4, MS5** \\ \hline MS1 & 87 & 10 & 90 & 0 \\ MS2, MS3, MS4, MS5 & 3 & 229 & 0 & 237 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Confusion matrix of the SIMCA model during its training and test phases.  
Figure 7: Predicted as a member of the MS1 class in training (**a**) and test (**b**) phases by the first PCA submodel in SIMCA.