## 5 Case Studies
### Problem Description 2  
The considered database is the one recently described in [27]. It includes 424 experimental results about the compressive strength (\(CS\)) of cement-based mortar specimens and was obtained by collecting 20 published research works on the behavior of cement-based mortars, with and without metakaolin, as well as superplasticizers.  
The candidate explanatory variables are the following: the age of specimen (\(AS\)), the cement grade (\(CG\)), the metakaolin percentage in relation to total binder (\(MK/B\)), the water-to-binder ratio (\(W/B\)), the superplasticizer (\(SP\)), which is the percentage of the addition of superplasticizer in relation to the total binder (\(\%w/w\)), and the binder-to-sand ratio (B/S). The \(AS\) ranges from 0.67 to 91 days, the \(CG\) ranges from 32 to 53.5 MPa, the \(MK/B\) ranges from 0 to 30\(\%w/w\), the \(W/B\) ranges from 0.3 to 0.6 \(w/w\), the \(SP\) ranges from 0 to 2.35\(\%w/w\), the \(B/S\) ranges from 0.33 to 0.51 \(w/w\), and the \(CS\) of specimens ranges from 4.1 to 115.25 MPa (Figure 3).  
The database consists of four input variables that assume values that are always greater than zero. On the contrary, the parameters \(MK/B\) and \(SP\) assume null values.  
Figure 3: Histograms of input and target parameters, database 2.  
Figure 2: Histograms of input and target parameters, database 1.  
These parameters are intended as sensitivity variables aimed at capturing the effects on the compressive strength by their use. Therefore, the \(MK/B\) and \(SP\) parameters cannot be used as explanatory variables since they lack regular dependency on the compressive strength. Based on these observations, the EPR technique has been performed by considering the \(AS\), \(CG\), \(W/B\), and \(B/S\) as explanatory variables. This choice appears reasonable and this is also why in [27] it is demonstrated that the \(MK/B\) and \(SP\) are the parameters with the lowest influence on the CS.  
The amount of input data associated with each specified range of variation reveals a high heterogeneity in the experimental input distributions. Furthermore, the histogram of the cement-based mortar compressive strength does not reflect a perfect Gaussian distribution. The majority of the experimental values range between 10 and 70 MPa, while some observations are far from the median value; therefore, they may be considered as leverage points. These extreme values can affect the regression analysis, causing false estimates.  
A deeper insight into the problem is given in Figure 4, where the relationship between the compressive strength values and each explanatory variable is depicted. Figure 4 shows how the variability of each explanatory variable is limited. In the case of the parameters \(CG\) and \(B/S\), only five different values are observed. Instead, a wider range of variability is observed for the parameters \(AS\) and \(W/B\). The limited range of the explanatory variables may result in a significant dispersion of the observed variable [28]. In addition to that, the presence of leverage points in the dataset may lead to vertical extreme values which affect the accuracy of the regression estimates. To avoid these problems, a robust regression approach is needed to estimate the parameters of the regression model.