# Prediction Models for Mechanical Properties of Cement-Bound Aggregate with Waste Rubber
## 4 Prediction Models  
In order to make the results of the research conducted usable in practice, two prediction models were created by regression analysis. One was designed to predict the compressive strength (f\({}_{\rm c}\)) and the other was designed for the modulus of elasticity (E\({}_{\rm st}\)) prediction. In both models, the UPV was used as a predictor. This analysis was carried out based on raw pairs of data (three pairs for every mixture and curing period) of the UPV-f\({}_{\rm c}\) and UPV-E\({}_{\rm st}\) results. The R programming language was used to build the model. To build acceptable models, the predictors UPV and curing period were used in a linear relationship with the logarithm of the response variable. The curing period was statistically insignificant in the model for f\({}_{\rm c}\) prediction based on the measured UPV values. Hence, the developed model takes only the ultrasonic pulse velocity as an input parameter. This means that the same model can be used for all three  
Figure 6: Impact of cement and rubber on UPV for curing periods of (a) 7 days, (b) 28 days and (c) 90 days. curing periods, simplifying the prediction of \(\rm f_{c}\). The plotted data and the regression line obtained are presented in Figure 7.  
The model is homoscedastic (non-constant variance score test \(p\)-value = 0.84747), and errors are normally distributed (Shapiro-Wilk test \(p\)-value = 0.2124). The 95% confidence intervals for the intercept and UPV coefficient are \((-1.3783653,\)\(-1.1341046)\) and \((0.8035473,\)\(0.8915409)\), respectively. Given that this is a linear log(\(\rm f_{c}\)) model, it must be transformed to obtain an expression for the prediction of \(\rm f_{c}\). The model for \(\rm f_{c}\) is therefore:  
\[\rm f_{c}\ (\rm UPV)=e^{(-1.25623+0.84754\times\rm UPV+\epsilon)}=e^{\epsilon} \ \times e^{(-1.25623+0.84754\times\rm UPV)} \tag{1}\]  
An adjusted coefficient of determination for this model equals 0.9154, which characterizes a very strong relationship. The \(\rm\varepsilon\) is the zero mean model error, with an estimated standard deviation of 0.2414. As the hypothesis of normality was accepted, the mean prediction and prediction intervals were calculated based on the lognormal distribution, with the parameters \(\rm\mu=0\) and \(\rm\sigma^{2}=0.2414^{2}=0.05828\). For instance, the mean prediction can be calculated by the formula:  
\[\rm mean(f_{c}\ (\rm UPV))=1.0296\ \times e^{(-1.25623+0.84754\times\rm UPV)} \tag{2}\]  
The mean prediction line is with the 95%-prediction-interval boundaries presented in Figure 8.  
The situation is more complicated in the case of \(\rm E_{st}\) prediction based on the measured UPV values. The UPV and curing period were statistically significant, so a model with two predictors was developed. This means that the prediction of log(\(\rm E_{st}\)) depends, apart from the UPV, on the duration of the curing period. The model for log(\(\rm E_{st}\)) is also homoscedastic (non-constant variance score test \(p\)-value = 0.38964), but the errors are not normally distributed (Shapiro-Wilk test \(p\)-value = 0.002291). The estimated model coefficients, intercept, UPV and days are \(-0.7216\), \(0.7585\) and \(0.0019\), respectively, with \(p\)-values of 2 \(\times\) 10\({}^{-16}\), 2 \(\times\) 10\({}^{-16}\) and 2.59 \(\times\) 10\({}^{-3}\), respectively. Based on the asymptotic regression theory of 95% confidence intervals for  
Figure 7: UPV-In(\(\rm f_{c}\)) relationship. the intercept, the UPV and days coefficients are (\(-0.8406\), \(-0.6025\)), (0.7132, 0.8038) and (0.0007, 0.0031), respectively. One can see that the longer curing period significantly affects the increase in E\({}_{\text{st}}\). The plotted data and the linear models obtained are presented in Figure 9. As the curing period is also a predictor, the presented lines differ for different values of the curing period. The established model is as follows:  
Figure 8: Mean predictions (blue line) and 2.5-97.5%-prediction-interval boundaries. Figure 9: UPV-E\({}_{\text{st}}\) relationship.