# A Forecast Model Applied to Monitor Crops Dynamics Using Vegetation Indices (NDVI)
## 3 Results and Discussion
### Type 2 Maize  
Double smoothing, ARIMA and ARIMAX models were applied to estimate the type 2 maize. Weather measures were found to be significantly influential in the models, contributing to the explanatory power of the proposed models. Equation (12) presents the estimation model for type 2 maize considering the interpolation obtained by means of the single exponential smoothing method.  
\[S_{t}=0.86\ y_{t}+(1-0.86)S_{t-1}D_{t}=0.86\ S_{t}+(1-0.86)D_{t-1} \tag{12}\]  
The Sum of Squared Residuals (SSR) associated value for the model presented in Equation (12) is 1.418, and the Root Mean Squared Error (RMSE) is 0.157. The associated mean value for the model presented in Equation (12) is 0.647, and the trend is 0.001. The weight \(\alpha\) = 0.86, being high, offers more weight to recent data.  
Figure 4: Forecast values for one-year time horizon given by the estimation models presented in Equation (7)-(11) and least squares and mean squares aggregation models.  
Considering the dynamic structure of the time series and by means of transference functions, Equation (13) presents the estimation model for type 2 maize. The best selected model was ARIMAX(2,0,0)X(0,0,0) considering the smallest AIC value.  
\[Y_{t}=0.45+0.099\text{T}+0.606Y_{t-1}+0.494Y_{t-2}+\varepsilon_{t} \tag{13}\]  
where \(Y_{t}\) is type 2 maize at time \(t\) and T is the average temperature. All estimated parameters were significant at the 0.05 level. The Sum of Squared Residuals (SSR) associated value for the model presented in Equation (12) is 1.343, and the Akaike info criterion (AIC) is \(-0.72\). We can observe the positive impact of temperature over type 2 maize 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 5 presents the estimation model as a dynamic explanatory response in time and the residuals of the estimation model given by Equation (13).  
A similar model considering the dynamic structure of the time series and by means of transference functions is presented in Equation (14). The selected model was ARIMAX(2,0,0)X(0,0,0).  
\[Y_{t}=0.436+0.01\text{R}+0.602Y_{t-1}+0.499Y_{t-2}+\varepsilon_{t} \tag{14}\]  
where \(Y_{t}\) is type 2 maize 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 (14) is 1.32, and the Akaike info criterion (AIC) is \(-0.73\). We can observe the positive impact of the radiation over type 2 maize 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 5 presents the estimation model as a dynamic explanatory response in time and the residuals of the estimation model given by Equation (14).  
The forecast model is constructed as a combination of models presented in Equations (12)-(14). Therefore, the dynamic structure of the type 2 maize time series is included as the effect of average temperature and radiation. The combination of the models was obtained through least squares and mean squares, with a resulting MSE of 0.016 and 0.017, respectively. Figure 6 presents the forecast values for a one-year time horizon given by the estimation models presented in Equations (12)-(14) and least squares and mean squares aggregation models. The associated MSEs for each estimation model presented in Equations (12)-(14) were 0.026, 0.035 and 0.035. The best forecast model attending to the best MSE value was obtained by least squares.  
Figure 5: Representation for type 2 maize, estimation model and the residuals given by Equations (13) and (14).