## 3 Proposed Framework
### PCA-Ca  
The generated PMs from different branches within the Pa structure exhibit a strong correlation for each class. For example, the PMs corresponding to the 'Wheat' class across different branches show similarities. This can also be concluded from the previous articles in the literature, which suggests that different RS data sources often produce similar outcomes [74]. To address this issue, the current study incorporates Principal Component Analysis (PCA) on the PMs of the Pa structure for each class. PCA is a linear orthogonal transformation technique commonly used for high-dimensional datasets [75]. It involves transforming the input feature space into a new space where the features are uncorrelated. In this study, the bands corresponding to the same class from the five branches are stacked (referred to as class-wise arrangement in Figure 3), resulting in seven new PMs (number of target classes), each consisting of five bands (number of branches). PCA is then applied to the PMs of each class, generating seven new collections, each comprising five uncorrelated bands known as principal components. The top 'n' components from each new collection are stacked together to form a probability cube, which is utilized as an input to the Meta-model, referred to as the Ca structure, for classifying different crop types. The value of 'n' is determined through a grid search ranging from 1 to 5 (number of branches) using five-fold cross validation of training data.  
For the selection of the best Meta-model, the same methodology was applied as in the Pa structure. Specifically, the PMs from the five Pa branches were fed to the four mentioned ML models (CART, SVM, RF, and GBT). It is worth noting that PCA was not employed on the PMs at this stage for a better evaluation of its effect on the Meta-model performance. Using the identical methodology as described in the Pa structure, the hyperparameters of each model were determined using a five-fold cross-validation technique and a grid  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Hyper Parameters** & **Grid Search Space** \\ \hline \multirow{3}{*}{CART} & MN & [1–5, step = 1] \\  & MLP & [1–10, step = 2] \\  & G & [1, 5, 10, 100, 1000] \(\times\) 10\({}^{-4}\) \\ \multirow{3}{*}{SVM} & C & [1, 10, 100, 1000, 10,000] \(\times\) 10\({}^{-3}\) \\  & NT & [10, 50, 100, 200, 300] \\ \multirow{2}{*}{RF} & MN & [1–5, step = 1] \\  & VPS & [1–5, step = 1] \\ \multirow{4}{*}{GBT} & NT & [10, 50, 100, 200, 300] \\  & MN & [1–5, step = 1] \\ \multirow{4}{*}{Pa-PCA-Ca} & SH & [1, 10, 100, 1000] \(\times\) 10\({}^{-4}\) \\  & n & [1–5, step = 1] \\ \cline{1-1}  & (number of PCA top components) & [1–5, step = 1] \\ \hline \hline \end{tabular}
\end{table}
Table 4: Hyperparameters optimized in this study.  
search approach with the grid search space outlined in Table 4. The model that achieved the highest average classification accuracy was selected as the Meta-model. Once the best model was chosen, the Meta-model was trained using the entire training dataset, which accounted for 70% of the reference dataset. It is important to emphasize that the Meta-model was also trained using the same training data as the branches.  
This study evaluated various model architectures to demonstrate the superior performance of the proposed methodology. These different model architectures are outlined in Table 5. Model No. 3 represents a conventional approach of crop type classification, where all input FCs (FC\({}_{1}\)-FC\({}_{5}\)) were simply stacked and a single ML model was used for classification. Additionally, to investigate the impact of PCA in traditional approaches, another model architecture was tested (Model No. 4). In this architecture, the top components resulting from the PCA transformation of the input FCs (FC\({}_{1}\)-FC\({}_{5}\)) were fed to a single ML model for the final classification.