# Defining the Ideal Phenological Stage for Estimating Corn Yield Using Multispectral Images
## 2 Materials and Methods
### Generation of Prediction and Quality Control Models  
For model generation, we selected samples from the total population of the yield data for each area. We divided each area into grids with 100 equal rectangles. The grids \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & **Equation** & **Reference** & **Contribution** \\ \hline Normalized & & & High correlation \\ difference & & & with yield \\ vegetation index & & & Sensitivity to \\ Green normalized & & & chlorophyll \\ difference & & concentration \\ Ratio vegetation & & Correlation with crop \\ index & & leaf density \\ Soil adjusted & & Minimizes the ground \\ vegetation index & & glare effect \\ Modified soil & & Minimizes the soil effect of \\ adjusted & & the SAVI index \\ vegetation index 1 & & Analysis on \\ Optimized soil & & vegetative stages \\ adjusted vegetation index & & Enhance \\ Enhanced vegetation index & & vegetation with less \\ Triangular vegetation index & & & atmospheric influence \\ Second modified & & Sensitivity to crop leaf \\ triangular vegetation index & & & \\ Chlorophyll & & Increased \\ vegetation index & & sensitivity to chlorophyll \\ Chlorophyll index & & Assits in \\ Green leaf index & & & \\ Triangular greenness index & & Leaf area intensity of the crop \\ Normalized green, red & & & \\ difference index & & \\ \hline \hline \end{tabular}  
* \({}^{1}\) d = \((2\times\wp 820)-8\times(\wp 820-\rho 630)\). \(\rho\) refers to the satellite band and the number refers to the central wavelength of the spectral range. NDVI refers to normalized difference vegetation index. C refers to values as coefficients for atmospheric resistance. X refers to the value to be adjusted for canopy background.  
\end{table} 
Table 1: Equations and references for calculations of vegetation indices derived from original Planet satellite bands. were used to select one point per rectangle. Thus, a total of 100 points were selected from the entire data set, which was homogeneously distributed throughout each study plot (Figure 4). Points closest to the centroid of each rectangle were selected. The data points used for training and validation of the model were divided such that 80 points were randomly selected for model training and 20 points for validation. The surface reflectance band values and the brightness of the multispectral indices were extracted by generating regions of interest using part of the points of the vectorial files. In this manner, each georeferenced yield value was related to the respective digital number in the image. Following point selection, we calculated the yield and the spectral values derived from the images with the respective IVs using the Pearson model. Considering the high quantity of bands and IVs available to estimate yield, we defined criteria with a greater correlation with yield, for which the bands and vegetation indices would be predictor variables for each model. We also justified this step of mining predictor variables as a strategy to mainly avoid overfitting of the generated models, which can occur when the number of parameters used in the regression is high. Thus, we selected the five highest absolute correlation values of the bands/IVs to develop the models. The indices/bands selected varied according to each phenological stage. The following two types of methodologies were used to define the algorithms for yield estimation: (1) nonparametric regression methods, which included random trees (RTs), random forests (RFs), neural networks (NNs) and support vector machines (SVMs). It is important to emphasize that these algorithms were chosen by the application used in research to generate the spectral models for yield crop estimation. This step was developed in the Weka 3.9.5 software. RT, RF, NN, and SVM are learning techniques that use multiple decision trees. These are used to make a validation set from which statistical predictions are generated, which are based on sets of independent variables.