# A Simple Relationship to Estimate Parameters of the Optimum Compaction Point
## 3 Investigation Results  
The correlation between the different parameters was performed using the XLSTAT software [20] and results of this investigation are presented in Table 2. This last table shows that \(w_{Opt}\) is correlated positively to \(w_{L}\), \(w_{P}\), \(I_{P}\) and \(w_{Opt}\) and negatively to \(\gamma_{dmax}\). The \(\gamma_{dmax}\) is correlated negatively to all the other parameters (see Table 2). The highest values in the correlation matrix were used as the starting point to investigate relationships between pairs of parameters. The results obtained suggest that \(\gamma_{dmax}\) and \(w_{Opt}\) can be estimated from \(w_{L}\), \(w_{P}\), \(I_{P}\) and fine-grained content due to their higher correlation (see Table 2). In the following, the number of retained parameters used to establish the mathematical correlations with \(\gamma_{dmax}\) and \(w_{Opt}\) was limited to two (to facilitate the application of the correlations). These parameters correspond to \(w_{L}\), and \(w_{P}\) due to their higher correlation with \(\gamma_{dmax}\) and \(w_{Opt}\) (see Table 2). The Lab Fit software [29] was used to establish the mathematical equations that best relate the various parameters (based on the considerations described above). The equations obtained can be expressed as follows:  
\[\gamma_{dmax}=\text{A}\;(w_{L}w_{P})^{\text{B}} \tag{2}\]  
\[w_{Opt}=\frac{w_{L}}{\text{A}+\text{B}*w_{P}} \tag{3}\]  
The values of the parameters A and B and their uncertainties are presented in Table 3.  
For the different comparison between measured and estimated values for each parameter, r is calculated using Equation (1). A comparison between measured values and estimated values using Equations (2) and (3) are presented in Figures 2 and 3, respectively, for \(\gamma_{dmax}\) and \(w_{Opt}\). It is important to note that for the presented figures, confidence intervals of 95% are also presented. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(w_{L}\) & \(w_{P}\) & \(I_{P}\) & \(w_{Opt}\) & \(\gamma_{dmax}\) & **Fines Content** \\ \hline \(w_{L}\) & \(1\) & & & & & \\ \(w_{P}\) & 0.957 & \(1\) & & & & \\ \(I_{P}\) & 0.804 & 0.942 & \(1\) & & & \\ \(w_{Opt}\) & 0.833 & 0.802 & 0.678 & \(1\) & & \\ \(\gamma_{dmax}\) & \(-0.803\) & \(-0.782\) & \(-0.673\) & \(-0.961\) & \(1\) & \\ Fines content & 0.54 & 0.490 & 0.385 & 0.594 & \(-0.550\) & \(1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation matrix between different parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Parameter A** & **SD of A** & **Parameter B** & **SD of B** \\ \hline \(\gamma_{dmax}\) (\(w_{L}\), \(w_{P}\)) & \(0.4237\times 10^{2}\) & \(0.3285\times 10^{1}\) & \(-0.1159\times 10^{0}\) & \(0.9958\times 10^{-2}\) \\ \(w_{Opt}\) (\(w_{L}\), \(w_{P}\)) & \(0.1733\times 10^{1}\) & \(0.13256\times 10^{0}\) & \(0.9085\times 10^{-2}\) & \(0.1622\times 10^{-2}\) \\ \(w_{Opt}\) (\(\gamma_{dmax}\)) & \(0.8605\times 10^{3}\) & \(0.7594\times 10^{2}\) & \(-0.3264\times 10^{2}\) & \(0.4530\times 10^{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters A and B and their uncertainties (Parameters A and B correspond respectively to the first and second constant in Equations (2)-(4)). For \(\gamma_{dmax}\), one can observe that the correlation between measured and estimated values is very high, and the obtained r is about 0.999, confirming that this parameter can be estimated with high accuracy. For \(w_{Opt}\), one can observe that the correlation between measured and estimated values is in the same range as the one obtained for \(\gamma_{dmax}\). The obtained r is about 0.990 and can be considered as an excellent correlation. Based on the matrix of correlation (see Table 2), where one can observe that the correlation between \(\gamma_{dmax}\) and \(w_{Opt}\) is very high (r = \(-\)0.96), a new equation was proposed to estimate \(w_{opt}\) by using only estimated \(\gamma_{dmax}\).