# A Forecast Model Applied to Monitor Crops Dynamics Using Vegetation Indices (NDVI)
## 3 Results and Discussion
### Type of Maize Aggregate  
Double smoothing and ARIMAX models were applied to estimate the type of maize aggregate. Weather measures were found to be significantly influential in the models, contributing to the explanatory power of the proposed models. Equation (22) presents the  
Figure 10: Forecast values for one-year time horizon given by the estimation models presented in Equations (19)–(21) and least squares and mean squares aggregation models.  
Figure 9: Representation for type 4 maize, estimation model and the residuals given by Equations (20) and (21).  
estimation model for type of maize aggregate considering the interpolation obtained by means of the single exponential smoothing method.  
\[\begin{split} S_{t}&=0.85\,y_{t}+(1-0.85)S_{t-1}\\ D_{t}&=0.85\,S_{t}+(1-0.85)D_{t-1}\end{split} \tag{22}\]  
The Sum of Squared Residuals (SSR) associated value for the model presented in Equation (22) is 1.4163, and the Root Mean Squared Error (RMSE) is 0.157. The associated mean value for the model presented in Equation (22) 0.49451, and the trend is 0.002152. The weight \(\alpha=0.85\), being high, offers more weight to recent data.  
Equation (23) presents the estimation model for type of maize aggregate. The best selected model was ARIMAX(2,0,0)X(0,0,0) considering the smallest AIC value.  
\[Y_{t}=0.34+0.0122\text{R}+0.67Y_{t-1}+0.354Y_{t-2}+\varepsilon_{t} \tag{23}\]  
where \(Y_{t}\) is type of maize aggregate at time \(t\), and R is the radiation. All estimated parameters were significant at the 0.05 level. The Sum of Squared Residuals (SSR) associated value for the model presented in Equation (23) is 0.496374, and the Akaike info criterion (AIC) is \(-1.720534\). We can observe the positive impact of the radiation over type of maize aggregate and a dynamic dependence.  
Ljung Box autocorrelation proof validates the hypothesis of white noise in the residuals of the model, confirming the adequacy and reliability of the estimation model. Figure 11 presents the estimation model as a dynamic explanatory response in time and the residuals of the estimation model given by Equation (23).  
A similar model considering the dynamic structure of the time series and by means of transference functions is presented in Equation (24). The selected model was ARIMAX(2,0,0)x(0,0,0).  
\[Y_{t}=0.022\text{T}+0.0011\text{ P}+0.041\text{H}+0.473Y_{t-1}+0.365Y_{t-2}+ \varepsilon_{t} \tag{24}\]  
where \(Y_{t}\) is type of maize aggregate at time \(t\) and R is radiation. All estimated parameters were significant at the 0.05 level. The Sum of Squared Residuals (SSR) associated value for the model presented in Equation (24) is 0.47194, and the Akaike info criterion (AIC) is \(-1.73\). We can observe the positive impact of the radiation over type of maize aggregate and a dynamic dependence. Ljung Box autocorrelation proof validates the hypothesis of white noise in the residuals of the model, confirming the adequacy and reliability of the estimation model. Figure 11 presents the estimation model as a dynamic explanatory response in time and the residuals of the estimation model given by Equation (24).  
Figure 11: Representation for type of maize aggregate, estimation model and the residuals given by Equations (23) and (24).  
The forecast model is constructed as a combination of models presented in Equations (22)-(24). Therefore, the dynamic structure of the type of maize aggregate time series is included as the effect of average temperature, precipitation, average humidity and radiation. The combination of the models was obtained through least squares and mean squares, with a resulting MSE of 0.091 and 0.011, respectively. Figure 12 presents the forecast values for a one-year time horizon given by the estimation models presented in Equations (22)-(24) and least squares and mean squares aggregation models. The associated MSEs for each estimation model presented in Equations (22)-(24) were 0.026, 0.012 and 0.013. The best forecast model attending to the best MSE value was obtained by least squares.