## 2 Materials and Methods
### Models and Analysis
#### Ordinary Least Square and Geographically Weighted Regression  
The ordinary least square (OLS) was used in the first term to explore the relationship between grain yield and the explanatory variables. In OLS the existence of local variation is not taken into account in the regression, hence the regression coefficient remains constant for each variable (Equation (3)). \[y_{i}=\beta_{0}+\sum\
olimits_{k}\beta_{k}\,x_{ik}+\epsilon_{i} \tag{3}\]  
where \(y_{i}\) is the dependent variable (grain yield), \(\beta_{0}\) is the intercept; \(\beta_{k}\) is the coefficient of the independent variable (x\({}_{ik}\)) and the random error is \(\epsilon_{i}\). Alternatively, GWR 4.0 software, developed by [PERSON] et al. (2016), was used. A geographically weighted regression (GWR) model was also calculated in order to explore the relationship between grain yield and the explanatory variables (Equation (4)) taking into account the spatial factor, which is an essential feature when dealing with spatial heterogeneity, something that is especially central to the objectives of this study. GWR uses the least square method given the location as a weighting factor. The optimal bandwidth, namely the optimum number of neighbors, was determined by the Akaike Information Criterion (AIC). A multiple comparison of AIC to find the best bandwidth (the one with the lowest AIC) was computed. For OLS and GWR comparison, an ANOVA was performed to compare the accuracy of both levels. Furthermore, for both OLS and GWR, the spatial autocorrelation (Moran's I) of residuals were tested. \[y_{i}=\beta_{0}(\mu_{i}v_{i})+\sum\
olimits_{k}\beta_{k}(\mu_{i}v_{i})\,x_{ik }+\epsilon_{i} \tag{4}\]  
where \(y_{i}\) is the dependent variable (grain yield), (\(\mu_{i}v_{i}\)) are the coordinates (x,y) at the location \(i\), \(\beta_{0}\) is the intercept, \(\beta_{k}\) is the coefficient of the independent variable (x\({}_{ik}\)) at the specific weighted location, and \(\epsilon_{i}\) is the random error. A geographic variability test was performed in order to determine if the explanatory variables were spatially heterogeneous. In order to do so, two models were developed, one that considers all of the variables (slope, altitude, height, and rainfall) to be spatially heterogeneous, and on the other hand one that considers all of the variables to be spatially homogenous. If the variables were indeed spatially heterogeneous determinants of grain yield, i.e., if the coefficients varied significantly in space, then the AIC size of the second model should be larger and result in a negative diff-criterion value for this test (GWR4 manual) (S  
the correlations yielded between 0.32 and 0.49 for the mean, 0.31 and 0.49 for the median, 0.38 and 0.45 for the minimum, and 0.34 and 0.45 for the maximum. Furthermore, the statistical significance of the correlations at heading was greater than that at both tillering and ripening, which demonstrated lesser (<0.05) or no significance differences in the majority of the cases. At heading, the results from the multilinear stepwise regression (Table 7) show the most suitable combination of vegetation indices in order to estimate grain yield. In this sense, the best-suited equation uses minimum averages of MSAVI pixels and maximum averages of RDVI pixels. VIF was 2.16 for both variables. The histogram of the residuals is shown in Figure 10 in the Appendix A together with the Shapiro-Wilk normality test results, the null hypotheses of which assumes that the data is normal while the alternative assumes it is not. The p value of the test was 0.297 and the statistic 0.982. The resulting multilinear stepwise equation (Table 7) was tested at heading for the studied fields for the following year 2019 using the corresponding calculated vegetation indices and summary statistics. The equations worked accurately as shown in Figure 4, with an R\({}^{2}\) of 0.83 (RSE = 733.1 kg\(\cdot\)ha\({}^{-1}\)).