## 4 Results and Discussion
### Prediction of Initial Chord Elastic Modulus with Ensemble Method  
Recently, studies have been conducted to overcome the nonlinearity of data by applying machine learning (ML) methods, and to improve prediction accuracy and reliability. In this study, ML methods  
Figure 11: Comparison of prediction results with two frequencies. (**a**) Predicted Ei with multiple linear regression (MLR), (**b**) Predicted Ed with Equation (4). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Correlation** & **LT.f1-Ei** & **LT.f2-Ei** & **LT.f3-Ei** & **LT.f4-Ei** & **TR.f1-Ei** & **TR.f2-Ei** & **TR.f3-Ei** \\ \hline Values & 0.9420 & 0.9397 & 0.8629 & 0.7071 & 0.9413 & 0.8992 & 0.7915 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Correlation of frequencies and _Ei_ in LT and TR modes. are applied to overcome the nonlinearity of data that is difficult to predict using basic regression analysis and correction factors. The errors of the initial chord elastic modulus (\(Ei\)) and actual \(Ei\) values as predicted by various combinations of frequencies by the ensemble method are summarized in Tables 10 and 11. As a result of the analysis, when a single frequency was used, MAPE values of 3.90% and 4.31% were found for the longitudinal (LT) and transverse (TR) modes, respectively. In the ensemble method, using only \(f1\) can reduce the error by 1-2% more than in the correction coefficient and MLR method. Thus, it is possible to predict a sufficiently accurate \(Ei\) value using only \(f1\). As the number of frequencies being used increased, the accuracy tended to increase. In the LT/TR modes, maximum MAPE values of 3.51%/3.40% were shown. Through the correlation analysis shown in Table 10 and the predicted accuracy values in Tables 9 and 10, approximate contributions to \(f1\) to \(f4\) can be inferred. However, as they are not exact values, an additional contribution analysis was performed. The relative importance (RI) was calculated using the Statistics and Machine Learning Toolbox of MATLAB R2019b to identify the detailed contribution of each variable from a combination of four frequency variables. Next, an analysis was conducted by considering various variables for the concrete cylinder. The \(Ei\) was predicted by the ML method by adding the mass (\(M\)), length/diameter (\(L\)/\(d\)), area (\(A\)), volume (\(V\)), and density (\(\rho\)) as dimensional variables, with the frequency obtained by the resonance frequency test. It was found that \(M\), \(L\)/\(d\), \(A\), \(V\), and \(\rho\) do not contribute to \(Ei\) prediction, as they have constant values for the same specimen. Additionally, the Ei was predicted with variables such as the curing age (\(day\)), \(W\)/\(B\), frequencies of longitudinal (LT) and transverse (TR) mode, and RIs for each variable, as summarized in Figures 12 and 13. In the LT and TR modes, \(f\)/\(l\) contributed more than 87%, and \(\ell\)2 contributed more than 6%, accounting for most of the ratio. The W/B ratio was expected to have an impact but did not contribute to the actual \(Ei\) prediction. Thus, it is considered that this factor is contained in the main resonance frequencies such as \(f\)/\(l\) and \(f\)/2, such that the elastic modulus and strength can be accurately predicted using only the main resonance frequency, i.e., without \(W\)/\(B\) information. Although the day variable was very small, it contributed partly to the accuracy of initial chord elastic modulus (\(Ei\)) prediction, and the prediction results are shown in Figure 14. When the number of frequencies (including the day) is increased from \(f\)1 to \(f\)4 for the LT mode, the errors are 3.79%, 3.69%, 3.51%, and 3.47% in the ensemble method, respectively, and 3.66%, 3.57%, 3.42%, and 3.31 in ANN method, respectively. Even in the TR mode, the predicted results including the day improved from 0.1% to 0.2%. This means that the frequency obtained by the resonance frequency test includes the day information, and it is sufficient to use only the frequency to predict the \(Ei\).