# Environmental Impacts of Cement Production: A Statistical Analysis
## 3 Results
### Linear Regression and Variable Selection for GWP  
In this Section two predictive models for GWP are fit. The first model is a linear regression where GWP is the scalar response and the Consumption ICs are the input variables. All variables have been preliminarily normalized to simplify the interpretation. The estimated coefficients are listed in Table 5, together with the related standard deviations (St. dev.) and the corresponding significance for the \(p\)-values associated to the significance test of the model. To develop the second model, an AIC backward selection procedure is then performed on the linear regression, to find the best subset of Consumption ICs to accurately predict GWP, leading to the model described in Table 6.  
Then, a 10-fold cross-validation procedure is performed to compare the two models. The root mean square errors (RMSE) are computed by Equations (5) and (6):  
\[ \begin{split} RMSE_{lin}=(\frac{1}{n}\sum\limits_{i=1}^{n}{(GWP_{i }-\hat{\beta}^{lin}_{ADPt}ADPe_{i}-\hat{\beta}^{lin}_{ADPt}ADPf_{i}-\hat{\beta}^ {lin}_{TRPE}TRPE_{i}-\hat{\beta}^{lin}_{SRM}SRM_{i}}\\ -\hat{\beta}^{lin}_{NRSF}NRSF_{i}-\hat{\beta}^{lin}_{RSFi}RSF_{i}- \hat{\beta}^{lin}_{Water}Water_{i}-\hat{\beta}^{lin}_{EL}EL_{i})^{2}\\ \end{split}  \tag{5}\]  
\[ \begin{split} RMSE_{AIC}=(\frac{1}{n}\sum\limits_{i=1}^{n}{(GWP_{i }-\hat{\beta}^{AIC}_{ADPt}ADPe_{i}-\hat{\beta}^{AIC}_{ADPf_{i}}ADPf_{i}-\hat{ \beta}^{AIC}_{TRPE}TRPE_{i}-\hat{\beta}^{AIC}_{SRM}SRM_{i}}\\ -\hat{\beta}^{AIC}_{NRSF}NRSF_{i}-\hat{\beta}^{AIC}_{RSFi}-\hat{ \beta}^{AIC}_{Water}Water_{i}-\hat{\beta}^{AIC}_{EL}EL_{i})^{2}\\ \end{split}  \tag{6}\]  
for the linear and AIC model, respectively. The sample size \(n=193\) is the number of observations, while \(\left(\hat{\beta}^{lin}_{ADPt^{\prime}}\hat{\beta}^{lin}_{ADPt^{\prime}}, \hat{\beta}^{lin}_{TRPE},\hat{\beta}^{lin}_{SRM},\hat{\beta}^{lin}_{PNKSF}, \hat{\beta}^{lin}_{RSF},\hat{\beta}^{lin}_{Water},\hat{\beta}^{lin}_{E}\right)\), and \(\left(\hat{\beta}^{AIC}_{ADPt^{\prime}}\hat{\beta}^{AIC}_{ADPf^{\prime}}, \hat{\beta}^{AIC}_{TRPE},\hat{\beta}^{AIC}_{SRM},\hat{\beta}^{AIC}_{NRSF}, \hat{\beta}^{AIC}_{RSF}\right)\) are the estimated parameters with the linear model and then selected by the AIC criterion, respectively. Recall that the RMSE is a risk function aimed to measure the discrepancy between the observations \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**ICs**} & \multicolumn{3}{c}{**Linear Regression (GWP vs. Selected Consumption ICs)**} \\ \cline{2-4}  & **Coefficients** & **St. Dev.** & \(p\)**-Value** \\ \hline ADPe & \(-0.140\) & \(0.022\) & *** \\ ADPf & \(1.486\) & \(0.044\) & *** \\ TRPE & \(-0.085\) & \(0.025\) & *** \\ SRM & \(0.527\) & \(0.024\) & ** \\ NRSF & \(0.527\) & \(0.040\) & *** \\ RSF & \(0.558\) & \(0.037\) & *** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Linear regression model summary (after variable selection). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**ICs**} & \multicolumn{3}{c}{**Linear Regression (GWP vs. Consumption ICs)**} \\ \cline{2-4}  & **Coefficients** & **St. Dev.** & \(p\)**-Value** \\ \hline ADPe & \(-0.127\) & \(0.038\) & ** \\ ADPf & \(1.503\) & \(0.077\) & *** \\ TRPE & \(-0.082\) & \(0.026\) & ** \\ SRM & \(0.532\) & \(0.025\) & ** \\ NRSF & \(0.531\) & \(0.052\) & *** \\ RSF & \(0.563\) & \(0.039\) & *** \\ Water & \(-0.017\) & \(0.041\) & \\ El & \(-0.006\) & \(0.043\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Linear regression summary (GWP vs. Consumption ICs). and the corresponding estimated values. In Table 7_RMSE\({}_{lin}\)_ is higher than \(RMSE_{AIC}\), thus the variable reduction produces a more accurate model. Finally, Figure 5 describes the size of each regression slope coefficient, after the variable selection. The highest contribution to GWP is given by ADPf. This result complies with the release of carbon dioxide into the atmosphere by burning of fossil fuels [38; 39; 40].