# A Forecast Model Applied to Monitor Crops Dynamics Using Vegetation Indices (NDVI)
## 3 Results and Discussion
### Olive  
Double smoothing, ARIMA and ARIMAX models were applied to estimate olive crops. Weather measures were found to be significantly influential in the models, contributing to the explanatory power of the proposed models. Equation (25) presents the estimation model for olive considering the interpolation obtained by means of the single exponential smoothing method. \[ \begin{split} S_{t}&=0.14\ y_{t}+(1-0.14)S_{t-1}\\ D_{t}&=0.14\ S_{t}+(1-0.14)D_{t-1}\end{split}  \tag{25}\]  
The Sum of Squared Residuals (SSR) associated value for the model presented in Equation (25) is 0.354, and the Root Mean Squared Error (RMSE) is 0.078. The associated mean value for the model presented in Equation (25) is 0.503, and the trend is 0.001. The weight \(\alpha\) = 0.14, being low, offers less weight to recent data. Considering the dynamic structure of the time series and by means of transference functions, Equation (26) presents the estimation model for olive crops. The best selected model was ARIMAX(1,0,0)X(0,0,0) considering the smallest AIC value. \[ \begin{split} Y_{t}&=0.011\text{P}+0.06\text{H}+0.0 9\text{R}+0.217Y_{t-1}+\epsilon_{t}\end{split}  \tag{26}\]  
where \(Y_{t}\) is olive crops at time \(t\). All estimated parameters were significant at the 0.05 level. The Sum of Squared Residuals (SSR) associated value for the model presented in Equation (26) is 0.314, and the Akaike info criterion (AIC) is \(-2.18\). We can observe 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 13 presents the estimation model as a dynamic explanatory response in time and the residuals of the estimation model given by Equation (26). Figure 12: Forecast values for one-year time horizon given by the estimation models presented in Equations (22)–(24) and least squares and mean squares aggregation models. The forecast model is constructed as a combination of models presented in Equations (25) and (26). The combination of the models was obtained through least squares and mean squares, with a resulting MSE of 0.003 and 0.003, respectively. Figure 14 presents the forecast values for a one-year time horizon given by the estimation models presented in Equations (25) and (26) and least squares and mean squares aggregation models. The associated MSEs for each estimation model presented in Equations (25) and (26) were 0.005 and 0.006. The best forecast model attending to the best MSE value was obtained by least squares. The model equations present a positive contribution of the atmospheric variables T, P, H and R to the plant's growth. In the case of type 1 and 3 maize crops, the model considers lags in periods \(t=1\), \(t=2\) and \(t=24\) because they have been maintained over several years, whereas for type 2 and 4 maize crops and the aggregated maize crops, the model considers solely lags in periods \(t=1\) and \(t=2\) because years of fallow have been interspersed. The corresponding MSE is quite low for each model, which represents a good accuracy of the model fitting. The trends of all the considered models are near to zero, which implies very few variations in the considered time sample. For maize crops, the variation fluctuates between \(-0.0005\) (Type 1) and 0.003 (Type 1): 0.002 for the aggregated maize crop model and 0.001 for olive crop. The weight obtained from double smoothing models (\(\alpha\in[0.75\):\(0.87]\)) associated with maize crops is positive and near to one, which implies more weight to recent data; this is consistent with the phases of rapid evolution in the development of maize cultivation. In the case of olive trees, since it is a perennial tree crop with less physiological changes, the dependency of recent data is lower (\(\alpha=0.14\)). Figure 14: Forecast values for one-year time horizon given by the estimation models presented in Equations (25) and (26) and least squares and mean squares aggregation models.