## 3 Results
### Compressive Strength Modeling  
Experimental relationships between compressive strength and NDT response (i.e., UPV or dynamic modulus) have been explored in past studies included in the literature [13; 14; 15; 16; 19]. The response of selected models from the literature with the data from this study are presented in Table 8 along with the corresponding root-mean-squared error (RMSE). The purpose of the analysis presented in Table 8 was to verify the model form (i.e., non-linearity) between compressive strength and pulse velocity. It should be mentioned that some of these models were developed with data from specific mixtures. For example, [PERSON] et al. [16] developed the relationship for self-consolidating concrete (SCC). The lower RMSEs of the [PERSON] et al. [19] and [PERSON] [15] models indicate that the relationship between compressive strength and UPV is non-linear, thus confirming earlier findings by [PERSON] et al. [37]. In this study, the testing results of the concrete mixtures at different ages were used to develop a multivariate relationship between compressive strength with UPV, dynamic modulus from RFT, and the temperature-time product. As mentioned earlier, past studies also recommended that more robust prediction models may be obtained when the results from multiple testing methods are combined [19; 38]. Linear regression was employed to identify the best fit considering the following: higher R\({}^{2}\), lower \(p\)-value, residual standard error, AIC (Akaike's information criteria), and BIC (Bayesian information criteria). AIC and BIC are indices representing the complexity of a model, with higher AIC and BIC representing higher levels of complexity. Generally, a simple model form may be more desirable and practical in terms of simple computational calculations. Some of the models with higher performances are presented in Table 9.  
Considering all aforementioned factors, Model 4 seems to be a model better representing the data, as shown in Equation (8):  
\[CS=1.36\ ln(MI)-6.76 ln\ (V)+1.29 DM+30.92 \tag{8}\]  
where \(CS\) is compressive strength (MPa); \(MI\) is maturity index (temperature-time factor), \({}^{\circ}\)C-day; \(V\) is UPV, m/s; and \(DM\) is dynamic modulus of elasticity from RFT in GPa. Figure 4: Comparison between observed and predicted shift factors for (**a**) UPV and (**b**) E\({}_{\text{d}}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Study** & **[PERSON] et al. (2019) [19]** & **[PERSON] (2004) [15]** & **[PERSON] et al. (2008) [16]** & **[PERSON] (2000) [13]** &
\begin{tabular}{c} **[PERSON] and** \\ **[PERSON] (2003) [14]** \\ \end{tabular} \\ \hline Model & \(CS=0.7237e^{BV}\) & \(CS=0.0872e^{1.29W}\) & \(\frac{CS}{0.2291V^{2}+5.939V+0.258}\) & \(CS=36.72V-129.077\) & \(CS=33V-109.6\) \\ \hline RMSE & 5.16 & 6.21 & 9.52 & 10.52 & 12.16 \\ \hline \multicolumn{5}{c}{\(CS=\) Compressive strength (MPa), \(V=\) UPV (km/s), \(n=\) 41.} \\ \end{tabular}
\end{table} 
Table 8: RMSE of compressive strength prediction models. Figure 5 presents a comparison between observed and predicted compressive strength values from Equation (8) using the data from the mixtures included in this study. Overall, there is very good agreement between observed and predicted strength values (R\({}^{2}\) = 0.93), and most of the values are within the 95% prediction interval. Finally, it can be observed in Figure 5a that the hypothesis of the randomness of residuals holds. It should be noted that in some cases, larger deviations from the equality between observed and predicted values are encountered. Figure 5a thus reflects sources of variability, such as sample to sample variations of the same concrete mix, testing accuracy, temperature control, and experimental-related factors. Thus, \"master curves\" represent mathematical expressions of experimental data, and any uncertainty in the testing process could cause some variation between experimental and predicted data. Figure 5: _Cont._