# Defining the Ideal Phenological Stage for Estimating Corn Yield Using Multispectral Images
## 3 Results  
### Exploratory Analysis of the Yield Variable  
Figure 5 presents the histogram of the study area for the yield variable, which is superimposed by a normal distribution curve (Table 3).  
The upper and lower threshold values for the study area are 11,254 kg/ha and 6345 kg/ha, respectively. The first quartile (Q1) has a value of 8115 kg/ha, while the third quartile (Q3) has a value of 9675 kg/ha. Thus, 50% of the production values are comprised in the interval established between Q1 and Q3, which has an interquartile range of 1560 kg/ha. The average yield was equal to 8871 kg/ha, and is greater than the average yield of the state of Minas Gerais in the 2019/2020 harvest, which obtained a value of 5726 kg/ha [39]. The coefficient of variation (CV) was equal to 12.52%, which does not prevent the variable from being estimated. Ref. [40] managed to estimate the yield of the maize crop with significant results, despite having a CV value of 26.8% in vegetative stage 6.  
Analysis of the Validation Metrics between Yield and Phenological Stages and Selection of the Best Model for Estimation  
Phenological stages and the validation metrics used, as well as the indices/flags that obtained the best response with the yield variable for each phenological stage are presented in Table 4.  
Figure 5: Histogram and normal distribution curve of the study area for the corn crop.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Variable** & **Average** & **D.P.** & **Minimum** & **Maximum** & **Q1** & **Median** & **Q3** & **C.V. (\%)** \\ \hline Yield & 8871 & 1110 & 6345 & 11,254 & 8115 & 8808 & 9675 & 12.52 \\ \hline \end{tabular}
\end{table}
Table 3: Exploratory analysis of the yield variable: S.D. = standard deviation, Q1 = quartile 1, Q3 = quartile 3, C.V. = coefficient of variation.  
The yield in the VE vegetative stage can be estimated with the RMSE and MAPE values for both areas, which were lower than 26% and 21%, respectively. The models are more assertive in R1 and VE, where the MAPE and RMSE values are lower than 25% and 23%, respectively.  
Figure 6 also presents MAPE% and RMSE% trends, generated via the models of the yield estimation along the phenological stages.  
The vegetative stage VE begins with a value near the lower threshold of the graph. Throughout the development of the base vegetative stages, there is a growth of values noted in V11. Hereon, a drop in the values of the metrics used is observed as the reproductive stages begin.  
To better represent the variability between the observed and estimated yield values, the scatter plots of the observed versus estimated values for the study area are shown in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Date** & **E.F.** & **Alg.** & **MAPE** & **RMSE** & **Bands/Indexes** \\ \hline
20//2002 & VE & SVM & 6.31 & 9.68 & GLI, VARI, GNDVI, CI-G, CVI \\
13//2003 & V5 & RT & 7.09 & 9.62 & R, B, G, CVI, GNDVI \\
31//2003 & V8 & NN & 7.39 & 9.70 & R, G, GNDVI, CI-G, CVI \\
11//2004 & V11 & NN & 7.95 & 10.32 & CVI, CI-G, RVI, TGI, GNDVI \\
25 April 2020 & R1 & NN & 7.25 & 9.25 & G. RVI, CVI, GNDVI, CI-G \\
13//2005 & R2 & NN & 7.07 & 9.17 & NIR, CI-G, GNDVI, TVI, RVI \\
22 May 2020 & R3 & SVM & 7.30 & 10.53 & EVI, CVI, NGRDI, VARI, TGI \\
05//2006 & R4 & RT & 9.62 & 13.62 & G, GNDVI, CI-G, CVI, MTVI \\
18//2006 & R5 & RF & 9.42 & 12.88 & NGRDI, VARI, TGI, EVI, RVI \\
03//2007 & R6 & RT & 15.91 & 20.96 & MSAVI, MTVI, SAVI, OSAVI, NDVI \\ \hline \end{tabular}
\end{table}
Table 4: Analysis of the model performance according to phenological stage for the study area: neural network (NN), linear regression (LR), support vector machine (SVM), random forest (RF), algorithm (Alg.), root mean square error percentage (RMSE), and mean absolute percentage error (MAPE).  
Figure 6: Root mean square error percentage (RMSE%) and mean absolute percentage error (MAPE%) curves for the study area.  
angular coefficient are once again those with the highest deterministic coefficient, as shown in Table 4, which are in the VE, R2, R5, and R6 stages.