## 2 Materials and Methods
### Modelling Scheme
#### 2.2.2 Model Building  
The study uses multiple indices and combines two techniques in the prediction of drought. The prediction of drought is, for operational purposes, formulated as the prediction of future VCI3M values using the predictor variables presented earlier (Table 2). We focus here on predictions 1-month ahead, while 2-month and 3-month predictions ahead were also tested for the GAM process. Two approaches are combined (Figure 4):  
* A statistical approach-generalized additive models (GAM), and
* Artificial neural networks (ANN). GAM models are used to arrive at the set of models that offer the best predictions. These set of variables are then used to build ANN models. GAM models are thus used for model space reduction in a set-up similar to variables selection prior to subjecting the chosen models to the ANN modelling process. GAM models are reviewed for example in [PERSON] (1999) and a good description of ANNs is provided in [PERSON] and [PERSON] (2014). The modelling process in Figure 4 is automated using scripts written in the free statistical computing R software developed by the R Core Team. The inputs into and outputs out of the R script are provided as comma separated value (csv) files. Each of the models and their parameters are serialized and saved to disk for making predictions on new data. The two modeling techniques (GAM and ANN) are described below in detail. General Additive Models  
The GAM technique was selected because it does not assume linearity between the predictor and the response variables ([PERSON] et al., 2016). In addition, GAM is free form and does not require the ascertainment of the functional form of the relationship and has the ability to model complex non-linear relationships, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**VCI3M Limit Lower** & **VCI3M Limit Upper** & **Description of Class** & **Drought Class** \\ \hline \(\leq\)0 & \(<\)10 & Extreme vegetation deficit & \\
10 & \(<\)20 & Severe vegetation deficit & \\
20 & \(<\)35 & Moderate vegetation deficit & \\
35 & \(<\)50 & Normal vegetation & \\  & & conditions & \\
50 & \(\geq\)100 & Above normal vegetation & \\  & & conditions & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Classification of drought based on vegetation deficit classes following values proposed by [PERSON] (2002), [PERSON] et al. (2016) and [PERSON] et al. (2016). Figure 4: Schema of the modelling process. In the sub-processes (**a**) and (**b**), General Additive Models (GAM) models are used to arrive at the set of variables that offer the best predictions. These combinations of variables are then used to build Artificial Neural Network (ANN) models. GAM are thus used, essentially, as a model variable selection method to the subsequent ANN modelling sub-processes. even in the presence of multiple predictors. This makes GAM models a viable tool for weather-based data modelling which exhibits non-linearity in the relationship between predictors and dependent variables. GAM models are expressed as shown in Equation (1). \[Y=a+f_{1}(x_{1})+f_{2}(x_{2})+\cdots+f_{n}(x_{n})+\varepsilon \tag{1}\]  
where \(a\) is an intercept and \(f\) are smooth functions; \(Y\) is the response function and \(x_{1}\) to \(x_{n}\) are the \(n\) predictor variables. Smoothing functions are either local linear regression (loess) or splines. In practical application, caution is advised, since smoothing generally leads to model overfitting. The model space for the study, as given by Equation (2) is around 2.15 billion. This space would be impractical to navigate in the search for the best predictor model. \[\sum_{r=1}^{31}\frac{n!}{(n-r)!r!}\approx 2.15\;\text{{billion}} \tag{2}\]  
To minimize the space complexity, we make some prior assumptions to avoid the futility of bias-free learning and also follow Occam's razor [45]. First, we assume that a maximum of two variables in the GAM models will give us reasonably simple models while still remaining predictive.