## 3 Analysis and Discussion of Results
### Correlation and Univariate Regression Analyses  
Developing Pearson's correlation analysis (Figure 3), the positive correlation between UCS and \(v_{p,dry}\) is highlighted, and the negative correlation between UCS and \(n\). It is also observed that the UCS dependent variable does not have a strong linear correlation with the other sampled variables, especially with the \(\rho_{sat}\), \(v_{s,dry}\) and \(v_{s,sat}\) variables, where there is no linear correlation (r = 0). Figure 3: Correlation plot of sampled variables. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Equation**} & \multicolumn{2}{c}{**Predicted UCS (MPa)**} & \multicolumn{2}{c}{**Relative Error (\%)**} \\ \cline{2-5}  & **Mean** & **SD** & **Mean** & **SD** \\ \hline (5) & 101.085 & 9.044 & -9.562 & 16.650 \\ (6) & 55.394 & 0.823 & -65.301 & 35.301 \\ (7) & 79.360 & 15.094 & 16.950 & 23.426 \\ (8) & 115.620 & 7.454 & -20.993 & 15.015 \\ (9) & 62.564 & 5.549 & 46.153 & 27.091 \\ (10) & 121.152 & 0.322 & -24.291 & 16.889 \\ (11) & 105.265 & 13.277 & -13.101 & 14.506 \\ (12) & 118.498 & 8.768 & -22.607 & 15.952 \\ (13) & 189.695 & 2.726 & -51.561 & 11.208 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average obtained UCS values in strength tests and average predicted UCS values, as well as their relative errors according to different authors. Figure 2: Predicted vs. Experimental UCS values. With the purpose of studying the relationship between the independent variables, \(v_{p,dry}\) (hereinafter referred to as \(v_{p}\)) and \(n\), and the dependent variable (UCS), a simple regression model (Equations (14)-(19)) was set up, to model the compression strength as a function of the independent variable (considering the exponential and potential models, after log transformation). Additionally, the curvature is considered in the analytical model, to study its impact on the response variable (Equations (20) and (21)). The mathematical models generated as a function of \(v_{p}\) and \(n\) and the respective goodness-of-fit statistics (R\({}^{2}\)) are presented in Table 4.  
From the analysis of the observed data distributions in Figure 4, and considering the goodness-of-fit statistics presented in Table 4, the best model to explain the UCS as a function of \(v_{p}\) is the quadratic model (R\({}^{2}\) = 0.6804), while the model as a function of n that presents a better fit also is the quadratic model (R\({}^{2}\) = 0.7573). On the other hand, although the adjusted models presented in Table 4 yielded considerably high coefficients of determination, these equations only include one independent variable. Therefore, an analysis that considers a greater number of independent variables that affect the response variable jointly, and that help explain the variability of the response variable has the potential to improve the percentage of variability explained by the analytical model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fitting** & **Analytical Model** & \(p\)**-Value** & **R\({}^{2}\) Statistic** & **Equation** \\ \hline Linear & UCS\(\left(\text{v}_{\text{p}}\right)\) = \(-123.37+41.13\text{v}_{\text{p}}\) & 0.0000 & 0.6028 & (14) \\ Linear & UCS\(\left(\text{n}\right)\) = \(119.36-7.11\text{n}\) & 0.0000 & 0.6788 & (15) \\ Exponential & UCS\(\left(\text{v}_{\text{p}}\right)\) = \(9.1369\text{e}^{0.4379\text{v}_{\text{p}}}\) & 0.0000 & 0.6324 & (16) \\ Exponential & UCS\(\left(\text{n}\right)\) = \(120.6763\text{e}^{-0.0749\text{n}}\) & 0.0000 & 0.7114 & (17) \\ Potential & UCS\(\left(\text{v}_{\text{p}}\right)\) = \(2.2544\text{v}_{\text{p}}^{2.2325}\) & 0.0000 & 0.6167 & (18) \\ Potential & UCS\(\left(\text{n}\right)\) = \(115.32\text{n}^{-0.2116}\) & 0.0000 & 0.7223 & (19) \\ Quadratic & UCS\(\left(\text{v}_{\text{p}}\right)\) = \(843.45-333.25\text{v}_{\text{p}}+36.11\text{v}_{\text{p}}^{2}\) & 0.0000 & 0.6804 & (20) \\ Quadratic & UCS\(\left(\text{n}\right)\) = \(130.81-14.69\text{n}+0.96\text{n}^{2}\) & 0.0000 & 0.7573 & (21) \\ \hline \hline \end{tabular}
\end{table}