# Prediction of Friction Coefficient Based on 3D Texture Characteristics of Road Surfaces
## 3 Results and Discussion
### Development of a Friction Prediction Model Using Texture Parameters
Using multiple linear regression analysis, the following formula for the \(\mu_{predicted}\) calculation was obtained:  
\[\mu_{predicted}=-11.526+0.138\cdot S_{mr2,MIC}+0.153\cdot S_{pc,MAC} \tag{3}\]  
The outcome of the regression model, along with the comparison between the measured and predicted friction coefficient (\(\mu\)) values, is depicted in Figure 12.  
Table 3 illustrates the outcomes of the statistical evaluation of the prediction model's accuracy and significance. The statistical significance and credibility of the regression model are evident from Figure 12 and Table 3. An \(R^{2}\) value of 0.7809 suggests a strong correlation between the predicted and measured values, while a low _RMSE_ of 0.07946 indicates high accuracy in the model's predictions. The slope close to 1 (0.9722) with a narrow confidence interval further supports the model's reliability. While the overall trend is strong, there is some scatter around the regression line. This indicates variability in the prediction accuracy, with some predicted values deviating from the measured values. There appear to be a few outliers that do not fit the general trend. These points can impact the overall model accuracy and may need further investigation to understand their cause. The dale void volume of the primary surface (\(V_{vv,PS}\)) was another texture parameter included in the prediction model. The hypothesis was that the accuracy and credibility of the regression model could be enhanced by including this parameter, which describes the effect of the drainage properties of the road surface. The equation determined using multiple linear regression analysis for calculating the friction coefficient \(\mu_{predicted}\), by combining the texture parameters \(V_{vv,PS}\) and \(S_{m2,MIC}\), \(S_{pc,MAC}\), is shown below:  
\[\mu_{predicted}=-11.362-0.568\cdot V_{vv,PS}+0.136\cdot S_{m2,MIC}+0.192\cdot S _{pc,MAC} \tag{4}\]  
The outcome of the refined regression model, along with the comparison between the measured and predicted friction coefficient (\(\mu\)) values, is depicted in Figure 13. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Coefficients**} & \multicolumn{2}{c}{**Standard Error**} & \multicolumn{1}{c}{\(p\)**-Value**} & \(r\) & \(R^{2}\) & \(R^{2}_{adjusted}\) & \(F\)**-Value** & **Ftab**ab \\ \hline Intercept & \(a_{0}\) & \(-11.526\) & 0.557 & 3.94 \(\times\) 10\({}^{-45}\) & & & & & \\ \hline \(S_{m2,MIC}\) & \(a_{1}\) & 0.138 & 0.007 & 2.50 \(\times\) 10\({}^{-45}\) & 0.8837 & 0.7809 & 0.814 & 321.740 & 3.058 \\ \hline \(S_{pc,MAC}\) & \(a_{2}\) & 0.153 & 0.021 & 1.86 \(\times\) 10\({}^{-11}\) & & & & & \\ \hline \end{tabular}
\end{table} 
Table 3: Model statistical significance and regression coefficients for texture parameters \(S_{m2,MIC}\), \(S_{pc,MAC}\). Figure 12: Correlation of measured and predicted friction coefficients \(\mu\) vs. \(\mu_{predicted}\). Figure 13 and Table 4 demonstrate the effectiveness and statistical significance of the regression model for predicting the friction coefficient (\(\mu\)). The inclusion of the dale void volume (\(V_{vv,PS}\)) improved the model's accuracy, evidenced by an increase in the coefficient of determination (\(R^{2}\) = 0.8002) and a decrease in the \(RMSE\) (0.07651) in comparison with the model without this parameter. Despite the improvement, there is still some variability and a few outliers which suggest that the model could be further refined. The \(R^{2}\) value of 0.8002 and the \(R^{2}_{adjusted}\) value of 0.826 indicate the model's strong explanatory power. Despite the higher \(p\)-value for \(V_{vv,PS}\), which suggests lower statistical significance compared to \(S_{m2,MIC}\) and \(S_{pc,MAC}\), the overall model remains robust and credible. The \(F\)-value of 234.1 far exceeds the critical value of 2.667, confirming the model's statistical significance. There is a well-founded assumption that the significance of \(V_{vv,PS}\) may increase at higher speeds, enhancing the model's predictive capability.