## 3 Results and Discussion
### Experimental Correlation between UCS and \(\mathrm{v_{P}}\) in This Study
#### 3.2.1 Simple Regression Analyses  
The obtained simple regression model that could predict UCS beginning from the independent variable (\(\mathrm{v_{P}}\)) and fitted to the values of Table 3 is shown in Equation (17):  
\[\mathrm{UCS}=-448.3+91.10\mathrm{v_{P}}. \tag{17}\]  
Based on variance analysis, the simple regression shown in Figure 2a was suitable to predict UCS. In addition, the \(p\)-value (\(<\)0.05), \(F\)-value (26.89), \(R^{2}\) (70.97%), and the plot of predicted UCS versus experimental UCS (Figure 2b) validated the model, which demonstrates the great normality of the residuals. On the other hand, exponential Equation (18), potential Equation (19), and quadratic Equation (20) models were fitted to the results, and a better fit was obtained compared with the linear simple regression model. The statistics \(p\)-value (\(<\)0.05), \(F\)-value, \(R^{2}\), and errors also validated them (Table 5):  
\[\mathrm{UCS}=0.02866\mathrm{e}^{1.363\mathrm{v_{P}}}, \tag{18}\]  
\[\mathrm{UCS}=0.000111\mathrm{v_{P}^{7.655}}, \tag{19}\]  
\[\mathrm{UCS}=5371-1973\mathrm{v_{P}}+182.9\mathrm{v_{P}^{2}} \tag{20}\]  
Predicted UCS values of the tested specimens were plotted against measured values. The error of each predicted value was represented as the distance to the 1:1 line. A point located in the line 1:1 indicates an exact prediction, while the points located far away from the 1:1 line represent an error. As can be observed (Figure 2b,d,f,h), the obtained points were located surrounding the 1:1 line. Figure 2: _Cont._The checked simple regression models are shown in Figure 2. The quadratic solution (Figure 2g,h) is the analytical model that provided the best fit (goodness-of-fit statistics of all the models are shown in Table 5). On the other hand, the worst fit was obtained by the simple linear regression model (Figure 2a,b), followed by the potential (Figure 2c,d) and the exponential (Figure 2e,f) models. Figure 2: Left: correlation between experimental UCS and P-wave velocity. Right: predicted against experimental UCS values. (**a**,**b**) Linear fit, (**c**,**d**) exponential fit, (**e**,**f**) potential fit, and (**g**,**h**) quadratic fit. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Source** & **Degrees of Freedom** & **SS Adj** & **MS Adj** & **F-Value** & \(p\)**-Value (Significance)** & \(R^{2}\) (\%) \\ \hline Linear & Regression & 1 & 3671.99 & 3671.99 & 26.89 & 0.000 & 70.97 \\ Equation (17) & Error & 11 & 1502.22 & 136.57 & & & \\  & Total & 12 & 5174.21 & & & & \\ \hline Exponential & Regression & 1 & 0.82251 & 0.822506 & 27.76 & 0.000 & 71.62 \\ Equation (18) & Error & 11 & 0.32588 & 0.029626 & & & \\  & Total & 12 & 1.14899 & & & \\ \hline Potential & Regression & 1 & 0.81531 & 0.815308 & 26.93 & 0.000 & 71.00 \\ Equation (19) & Error & 11 & 0.33308 & 0.030280 & & & \\  & Total & 12 & 1.1489 & & & \\ \hline Quadratic & Regression & 1 & 4056.82 & 2028.41 & 18.15 & 0.000 & 78.40 \\ Equation (20) & Error & 11 & 1117.39 & 111.74 & & & \\  & Total & 12 & 5174.21 & & & \\ \hline Multiple & Regression & 2 & 4257.1 & 2128.55 & 23.21 & 0.000 & 82.28 \\ Equation (21) & \(\
u_{p}\) & 1 & 710.6 & 710.62 & 7.75 & 0.019 & \\  & \(n\) & 1 & 585.1 & 585.12 & 6.38 & 0.030 & \\  & Error & 10 & 917.1 & 91.71 & & & \\  & Total & 12 & 5174.2 & & & \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c c c} \hline \hline  & Regression & 2 & 0.9750 & 0.48750 & 28.12 & 0.000 & 84.90 \\ Exponential and & \(\
u_{p}\) & 1 & 0.1440 & 0.14398 & 8.3 & 0.016 & \\ Multiple Exponent & \(n\) & 1 & 0.1525 & 0.15249 & 8.79 & 0.014 & \\ Equation (22) & Error & 10 & 0.1734 & 0.01734 & & & \\  & Total & 12 & 1.1484 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of the studied correlations. Although the simple regression analyses (including exponential and potential models by means of logarithmic transformation) obtained high coefficients of determination, these analytical models only considered one independent variable.