# A Forecast Model Applied to Monitor Crops Dynamics Using Vegetation Indices (NDVI)
## 2 Materials and Methodology Applied to the Forecast and Estimation Model
### Time Series Analysis: Estimation Models  
A useful methodology to study the evolution of time-dependent measures considering a period is time series analysis. This econometric model allows us to combine structural and time dimensions in an estimating model for forecast purposes. The main objective in time series analysis is to capture unobservable heterogeneity, between agents under study as well as in time. Several models are considered, attending to the structural patterns of the time series derived from the different crops in the study. The simplest case can be modelled using the single exponential smoothing method, which is appropriate for series that move randomly above and below a constant mean with no trend and no seasonal patterns, and the error terms associated with the estimation model are uncorrelated. For the different crops considered in the study, we applied a double smoothing fit, appropriated for time series with a linear trend. Equation (2) presents the general mathematical model assumed for the variable in the study as a dynamic explanatory response in time. Let \(y_{t}\) be the observation of time series Y at period \(t\in\{1,\ldots,T\}\), then the single smoothed series \(S_{t}\) and the double smoothed series \(D_{t}\) are given by:  
\[ \begin{array}{l}S_{t}=\alpha\ y_{t}+(1-\alpha)S_{t-1}\\ D_{t}=\alpha\ S_{t}+(1-\alpha)D_{t-1}\end{array}  \tag{2}\]  
where \(\alpha\) is the damping factor (\(0<\alpha\leq 1\)). Equation (3) presents the forecast mathematical model for a new observation at time period \(k\). \[ \begin{array}{l}y_{T+k}=\Big{(}2+\frac{\alpha\ k}{1-\alpha}\Big{)}S_{T}- \Big{(}1+\frac{\alpha\ k}{1-\alpha}\Big{)}D_{T}\\ =\big{(}2S_{T}-D_{T}+\frac{\alpha}{1-\alpha}\big{(}S_{T}-D_{T}\big{)}k\big{)} \end{array}  \tag{3}\]  
where \(\alpha\) is the weight used in the level component of the smoothed estimate. \(\alpha\) is like a moving average of the observations. The weights adjust the level of smoothing by defining the reaction of each component to current conditions. Lower weights offer less weight to recent data, which produces a smoother line. Higher weights give more weight to recent data, which produces a less smooth line. More complex time-dependent structures can be modelled by dynamic regression methods and the composed autoregressive and moving averages method (ARIMA). These methods have been used widely in different application areas due to their flexibility and fitness ability for time-dependent data. ARIMA models were proposed by [PERSON] and [PERSON] (1970) for stationary time series with linear autodependence [37]. They were developed with the aim of estimating and forecasting the behavior of the time data based on the previous performance considering autocorrelation in the error terms associated with the estimation model. These models were improved to include regressive variables known as the autoregressive integrated moving average with explanatory variables (ARIMAX). Equation (4) presents the general ARIMAX model, where \(B\) is the lag operator, \(d\) is the number of differences that must be applied to the series to make it stationary, \(\phi_{i}\) is the parameters of the \(p\) autoregressive part of the model, \(\theta_{i}\) is the parameters of the moving average part, \(x_{j}\) represents the model's exogenous variables, \(\lambda_{j}\) is the parameters of the exogenous part and \(\epsilon_{t}\) is the error terms [37; 38]. The error terms \(\epsilon_{t}\) are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean. \[(1-B)^{d}y_{t}=a+\sum_{i=1}^{p}\phi_{i}y_{t-i}+\sum_{h=1}^{q}\theta_{h} \epsilon_{t-h}+\sum_{j=1}^{r}\lambda_{j}x_{j}+\epsilon_{t} \tag{4}\]In general, these models improve the prediction quality of the general model and allow us to analyze the impact of the explicative variables in the time series. In the present study, weather variables (precipitation, temperature and radiation) were considered as possible explicative measures in the estimation models.