# A Forecast Model Applied to Monitor Crops Dynamics Using Vegetation Indices (NDVI)
## 2 Materials and Methodology Applied to the Forecast and Estimation Model
### Forecast Averaging: Prediction Model  
Different forecast methods, from intuitive and subjective to more complex quantitative models, aim to develop accurate predictions. Furthermore, considering that the forecast related to a given magnitude can be developed by different agents and derived from various methods, the consideration of a prediction composed by the combination of several predictions is relevant. Since procedures differ in their philosophy, computational cost, complexity and accuracy, the selection of alternative methods or their combination results in a difficult task, the use of evaluation measures based on forecast errors being usual in practice. The main idea of the model is to use the concept of predictive combination, which has proven to be an effective method in the predictive literature. Several methods can be used to forecast the same variable, and the literature shows that a combination of predictions made through different processes can improve the precision of a single prediction. The idea of a combination of predictions implicitly assumes that the model cannot identify a number of basic processes, and that each prediction method can capture different aspects of information that can be used for prediction, so predictions made with different technologies are the most accurate forecasts.  
Therefore, a combination of predictions derived from different methods usually improves the forecast accuracy. In this study, we combined predictions obtained by the proposed methods by means of forecast averaging. This method has the advantage of combining the relevant information of the time series given the different elements captured by each estimation model. The resulting prediction is expressed as a weighted average of the predictions resulting from the selected relevant estimation models, enriching the final forecast [40; 41; 42]. Among different approaches described in the literature, we selected the weighted average forecast procedure:  
* Least Squares Weights (LS), computed as a regression of predicted values to real values and employing regression coefficients as weights for the prediction combinations [43].
* Mean Squared Error (MSE), assigning higher weight values to predictions associated with the model with smaller average quadratic error. MSE weighting, proposed by [PERSON] (2001), compares the individual forecasts with the actual values over a forecast period [44]. The MSE (Equation (5)) of each forecast is computed and used to form individual forecast weights (Equation (6)): \[\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}\] (5) \[w_{i}=\frac{\frac{1}{\text{MSE}_{i}^{2}}}{\sum_{j=1}^{N}\frac{1 }{\text{MSE}_{j}^{2}}}\] (6)  
\(k\) is used to raise the MSE to different power; \(k=1\) is the most used power, and yields a weight based on the ratio of each forecast's MSE to the total of all the MSEs.  
To forecast procedures, we applied an inside proof-rolling sample, and in order to evaluate the proposed methods, we applied the MSE as a forecast sample measure.