## 2 Material and Methods
### Filling of Missing Data
#### 2.3.1 Multiple Linear Regression (MLR) with Fixed \"Donor\" Sites  
The MLR method for filling missing values, as proposed by [PERSON] (1983) [40], utilizes data from nearby sites as explanatory variables in the regression equation. Formally, for applying the MLR model, one must use at least two independent variables (\"donor\" sites). The rainfall amount to be estimated at the target site (dependent variable) is then related to \(N\) independent variables, as expressed in Equation (1):  
\[Y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+\beta_{N}x_{N} \tag{1}\]  
in which \(Y\) is the estimated value for the dependent variable; \(\beta_{0}\) is the intercept; \(\beta_{1},\beta_{2},\ldots,\beta_{N}\) are regression coefficients; and \(x_{1},x_{2},\ldots,x_{N}\) are the rainfall amounts at the \"donor\" sites (recorded simultaneously to the dependent variables in the calibration dataset). Parameters \(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{N}\) were estimated by means of the least squares method.  
For avoiding overfitting, only those \"donor\" stations with significant linear relationship with the target sites (according to partial F tests at the 5% significance level) are included in the regression equations. Hence, for each target site, model identification proceeds as follows  
* At least a pair of neighbor stations (with the highest levels of linear correlation), which do not have missing values along the period to be filled at the target site, is selected for composing the regression model. We restrict our attention to those \"donor\" stations located within a radius of 100 km of the target site for, at least to some extent, preserving similar climate conditions. More distant gauging stations (at most 150 km) are considered only in those cases in which the aforementioned criteria are not met;
* Other gauging stations are sequentially included in the model according to the linear correlation level with the target site. Then, partial F tests, at the 5% significance level, are performed for assessing whether the inclusion of a given station improves the predictive abilities of the regression model [41].
* The procedure is repeated until the inclusion of any of the remaining gauging stations does not significantly improve the models.  
We note that the multiple regression equation allows for the inclusion of an intercept (or bias term), which enables the translation of the regression hyperplane towards the equality counterpart. In other words, systematic trends of under/overestimation of the MLR equation may be, at least to some extent, corrected by the model. However, the bias term, yet small, may prevent the simulation of null rainfall amounts in the target sites. Moreover, since no missing data are allowed in the \"donor\" sites during the period in which rainfall amounts are to be estimated at the target station, the most linearly correlated sites may not be included in the MRL equation. All steps for deriving the MLR equations are performed in RStudio interface of the R software. The \"donor\" stations for the previously selected target sites are depicted in Figure 2, for the daily time scale, and Figure 3, for the monthly counterpart.