## 2 Materials and Methods
### Modeling \(V\)
#### 2.4.3 Procedure 6  
The linear correlation coefficient between \(P_{1}\) = [10 20 30 40] (\(p_{2}\) values from 10% to 40%) and \(C_{sp_{1}}(1:4)\) is \(\mathrm{corr}\left(P_{1},\ C_{sp_{1}}(1:4)\right)\geq 0.98\) for all \(sp_{1}\) cases [60,65,75,85,90]. Between \(P_{2}\) = [50 60 70 80] (\(p_{2}\) values from 50% to 80%) and \(C_{sp_{1}}(5:8)\) are the following: \(corr(P_{2},\ C_{60}(5:8))=-0.5759,\)\(corr(P_{2},\ C_{65}(5:8))=\)0.3794, \(corr(P_{2},\ C_{75}(5:8))=-0.5309,\)\(corr(P_{2},\ C_{85}(5:8))=\)0.2267, and \(corr(P_{2},\ C_{90}(5:8))=-0.6393\). In \(C_{sp_{1}}(1:4)\), the first four values, and in \(C_{sp_{1}}(5:8)\), the last four values of \(C_{sp_{1}}\). There is a strong positive correlation between \(P_{1}\) and \(C_{sp_{1}}(1:4)\) for all \(sp_{1}\) cases. The correlations between \(P_{2}\) and \(C_{sp_{1}}(5:8)\) cases are not that strong. Thus, we modeled (independent variable \(p_{2}\) and dependent variable \(V\)) a vector \(C_{sp_{1}}\) in two parts: the first four points with a linear equation and the last four points with a polynomial equation. When modeling vectors \(C_{sp_{1}}(1:4)\), we obtained parameter \(b_{1}\) (in vector \(P_{b_{1}}\)) and \(b_{2}\) (in vector \(P_{b_{2}}\)) values. The \(r\) between \(X_{1}\) and \(P_{b_{1}}\) is \(-0.8604\), and that between \(X_{1}\) and \(P_{b_{2}}\) is \(0.8160\). \(\overline{E}=0.0141\)\(s\), with \(E_{sp_{1}}\) expressing the error between \(C_{sp_{1}}(1:4)\) and \(\tilde{C}_{sp_{1}}(1:4)\). By modeling \(C_{sp_{1}}(5:8)\), we obtained the \(c_{1}\), \(c_{2}\), and \(c_{3}\) parameter's values (in vectors \(P_{c_{1}},P_{c_{2}},\) and \(P_{c_{3}}\), respectively). We selected for modeling \(C_{sp_{1}}(5:8)\) a polynomial of degree \(n=2\), as \(\overline{E}=0.0434<0.1\), and neither \(n=2\) nor \(n=3\) satisfy \(\left.1\right|<0.60\) for all correlations. Nevertheless, with \(n=2\), \(\mathrm{corr}(P_{c_{1}},X_{1})=\)0.58, \(\mathrm{corr}(P_{c_{2}},X_{1})=\)0.59 and \(\mathrm{corr}(P_{c_{3}},X_{1})=\)0.57 are good approximations. Henceforth, using parameters \(b_{1}\) and \(b_{2}\), we modeled a vector \(\tilde{C}_{sp_{1}}(k_{1})=b_{1}p_{2}+b_{2}\), with \(k_{1}=1\ldots 4\) and \(p_{2}=10\ldots 40\) (see Table 20). With parameters \(c_{1}\), \(c_{2}\), and \(c_{3}\), we modeled a vector \(\tilde{C}_{sp_{1}}(k_{2})=c_{1}p_{2}^{2}+c_{2}p_{2}+c_{3}\), with \(k_{2}=5\ldots 8\) and \(p_{2}=50\ldots 80\) (see Table 21). Vector \(P_{b_{1}}=[0.0409\ldots 0.0233]\) contains \(b_{1}\) parameter values, and vector \(P_{b_{2}}=[-0.4094\ldots-0.1509]\) contains \(b_{2}\) parameter values. Vectors \(P_{c_{1}}\) = [3.5789 \(\times\)\(10^{-5}\ldots\) 0.7276 \(\times\)\(10^{-3}\)], \(P_{c_{2}}\) = [-0.0126 \(\ldots\) -0.1039], and \(P_{c_{3}}\) = [2.1544 \(\ldots\) 4.8104]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Vector** & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{b_{3}}\) & \(\mathbf{b_{4}}\) & \(\mathbf{b_{5}}\) & \(\mathbf{b_{6}}\) \\ \hline \(\tilde{C}_{sp_{1}=60}\) & \(-1.4070\times 10^{-8}\) & \(3.3891\times 10^{-6}\) & \(-0.3069\times 10^{-3}\) & \(0.0122\) & \(-0.1633\) & \(0.6934\) \\ \hline \(\tilde{C}_{65}\) & \(-1.6574\times 10^{-8}\) & \(3.2941\times 10^{-6}\) & \(-0.2344\times 10^{-3}\) & \(0.0069\) & \(-0.0506\) & \(0.0313\) \\ \hline \(\tilde{C}_{75}\) & \(-2.1557\times 10^{-9}\) & \(3.4381\times 10^{-7}\) & \(-2.7155\times 10^{-5}\) & \(0.0012\) & \(0.0028\) & \(-0.0381\) \\ \hline \(\tilde{C}_{85}\) & \(3.1903\times 10^{-8}\) & \(-6.8689\times 10^{-6}\) & \(0.5373\times 10^{-3}\) & \(-0.0188\) & \(0.3114\) & \(-1.5865\) \\ \hline \(\tilde{C}_{90}\) & \(1.8897\times 10^{-8}\) & \(-3.9332\times 10^{-6}\) & \(0.2878\times 10^{-3}\) & \(-0.0090\) & \(0.1463\) & \(-0.7256\) \\ \hline \end{tabular}
\end{table} 
Table 17: Vectors \(\tilde{C}_{sp_{1}}\) parameters’ values (procedure 5).