# Environmental Impacts of Cement Production: A Statistical Analysis
## 2 Data and Methods
The answer to the first question is given by fitting and evaluating a linear regression model linking GWP to the Consumption ICs, (Equation (3)):  
\[ \begin{array}{l}\textit{GWP}=\beta_{ADPe}\;ADPe+\beta_{ADPf}\;ADPf+\beta_{ TRPE}\;TRPE+\beta_{SRM}\;SRM+\beta_{nonRSF}\;nonRSF+\beta_{RSF}\;RSF+\\ \beta_{Water}\;Water+\beta_{EI}\;El+\varepsilon\end{array}  \tag{3}\]  
where \(\varepsilon\) is the so-called noise, a vector of size \(n\) = 193 of independent and identically distributed random variables, while (\(\beta_{ADPe}\), \(\beta_{ADPf}\), \(\beta_{TRPE}\), \(\beta_{SRM}\), \(\beta_{nonRSF}\), \(\beta_{RSF}\), \(\beta_{Water}\), \(\beta_{EI}\)) is the vector of parameters to be estimated [31]. In statistics, linear regression is a widely used approach to establish the relationship between the so-called response (in our case GWP for Sections 3.2 and 3.3, the other Emission ICs in Section 3.4) and a set of explanatory variables (here the Consumption ICs) [32]. The relationship between the response and the explanatory variables is modeled by means of a linear predictor function whose unknown model parameters are estimated from the data. In this work, all the linear models are fit by means of the so-called least squares approach, which minimizes the sum of the squared residuals (the differences between the observed value and the one predicted by the model. In our case, solving the regression problem produces a vector of estimators \((\hat{\beta}_{ADPe},\hat{\beta}_{ADPf},\hat{\beta}_{TRPE},\hat{\beta}_{SRM}, \hat{\beta}_{SRM},\hat{\beta}_{SRM},\hat{\beta}_{Water},\hat{\beta}_{EI})\). The most relevant estimators can be then selected by means of the so-called Akaike Information  
Figure 1: Correlation matrix of ICs. Criterion (AIC) [32]. This procedure results in a model where only statistically significant variables in terms of the highest variances are selected, while the others are iteratively discarded. A model validation is then performed by means of a 10-fold cross validation procedure [31] to assess and compare the accuracy of the two models. Cross validation evaluates the accuracy of a predictive model, estimating its ability to predict new data. In the k-fold cross validation, the original dataset is randomly partitioned into k subsamples of equal size. Then, one subsample (the validation data) is used to test the model obtained by using the remaining k-1 subsamples (the so-called training data). This procedure is then repeated k times and averaged, so that all the observations are used for both validation and testing. The measure of the accuracy of each method is provided by the root mean square error, the risk function which measures the square root of the average squared difference between observations and the estimated values [31]. As far as the second question is concerned, it is now investigated if different types of cement influence GWP in different ways and, consequently, if the statistical model's accuracy can be improved by fitting a separate regression model for each class. These regression models can be expressed as Equation (4):  
\[ \begin{array}{l}GWP=\beta_{ADPei}\;ADPe_{i}+\beta_{ADpfi}\;ADPf_{i}+\beta_{ TRPEi}\;TRPE_{i}+\beta_{SRM;i}\;SRM_{i}+\beta_{nonRSF;i}\;nonRSFi+\beta_{RSF;i}\\ RSFi+\beta_{Water;}\;Water_{i}+\beta_{EI;i}\;El_{i}+\varepsilon_{i}\end{array}  \tag{4}\]  
where \(i\) = I, II, III, IV and solving the regression model produces estimates for the set of parameters \((\beta_{ADPEi},\beta_{ADpfi},\beta_{TRPEi},\beta_{SRM;i},\beta_{nonRSF;i}, \beta_{Water;i},\beta_{EI;i})\), \(i\) = I,..., IV. It is also relevant to check if some types of cement behave differently in terms of the regression models and relevant variables. Again, 10-fold cross validation procedures [31] are used to compare the results and to verify how accurate each predictive model is. Regarding the third question, a multiple linear regression using the full set of Consumption ICs is performed and used to evaluate two alternative models.