## 2 Materials and Methods
### Data Analysis
#### 2.5.2 Soybean Grain Yield Estimation Using Machine Learning  
For yield estimation, methods similar to those published by [PERSON] et al. [26] were adopted. Initially, correlation (r) analysis was performed between SY and all the derived spectral (REF and VI) features. Next, from these features, variables to be used as inputs for the ML and statistical modeling of SY were identified as part of dimensionality reduction to minimize the model's over-fitness and enhance robustness. This was accomplished by first identifying the collinear variables through a principal component analysis (PCA) around two main axes that explained maximum variability in data. Secondly, intercorrelations between the REFs and VIs were evaluated through a pair-wise correlation and variables with strong correlations (>0.95) were identified. The variables depicting largest mean absolute correlation were eliminated and the remaining ones were selected for SY modeling. Following dimensionality reduction, SY was modeled using two statistical models: stepwise linear regression (SLR), partial least-square regression (PLSR); and four ML models: k \begin{table}
\begin{tabular}{l l c} \hline \hline
**Vegetation Index** & **Equation** & **Reference** \\ \hline Green Ratio Vegetation Index (GRVI) & \(\mathrm{NIR/G}\) & [23] \\ Green Chlorophyll Index (GCI) & \((\mathrm{NIR/G})-1\) & [21] \\ Green Leaf Index (GLI) & \(((\mathrm{G-R})+(\mathrm{G-B}))/((2\times G)+\mathrm{R+B})\) & [27] \\ Modified Simple Ratio (MSR) & \(((\mathrm{NIR/R})-1)/(\mathrm{sqrt}(\mathrm{NIR/R})+1)\) & [15] \\ Renormalized Difference Vegetation Index (RDVI) & \((\mathrm{NIR-R})/\mathrm{sqrt}(\mathrm{NIR+R})\) & [25] \\ Transformed Difference Vegetation Index (TDVI) & \(1.5\times((\mathrm{NIR-R})/\mathrm{sqrt}(\mathrm{NIR+R+0.5}))\) & [26] \\ Visible Atmospherically Resistant Index (VARI) & \((\mathrm{G-R})/(\mathrm{G+R-B})\) & [13] \\ Leaf Area Index (LAI) & \(3.618\times\mathrm{EVI}-0.118\) & [26] \\ \hline \hline \end{tabular}  
* The spectral reflectance in red, green, blue, red edge, and near-infrared wavelength ranges are represented by R, G, B, RE, and NIR, respectively.  
\end{table} 
Table 1: _Cont._nearest neighbor (KNN), support vector machine (SVM), random forest (RF), and artificial neural network (ANN). In SLR, the variable with the highest sum of squares of regression is chosen. Next, an additional variable is selected from the remaining variables to develop binary relationships between dependent and independent variables [18; 38]. In this study, forward and backward variable selection and Akaike Information Criterion (AIC) was used for implementing SLR. PLSR offers a multivariate method for utilizing highly correlated (collinear) variables to predict a single response variable utilizing the cross-validation hyperparameter. PLSR functions by preserving relationships between independent and dependent variables (SY) while eliminating multiple linearities between independent variables. RF is a widely adopted ML model for agricultural studies that predicts an output through integrated multiple decision trees. RF is known to efficiently processes large datasets and reduce noises [39; 40]. In this study, RF was initiated with 1000 trees following which best-fit trees were identified in the ranges of 300-400 through a five-fold cross validation i.e., five variables chosen at \"random\" to serve as candidates for every tuning iteration [41]. KNN approximates the relationship between input variables and an output based on the average of observations in the same neighborhood. KNN assigns a contribution weight to each neighbor, to have the closer neighbors contribute more to the average than the farther neighbors do. In this study, KNN was trained through repeated cross validation with three iterations for 30 neighbors, and an optimum tuning length of 20 [42]. SVM iteratively finds a hyperplane in an n-dimensional space that relates the input and output data points.