# 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  
In creating the mathematical model for predicting the friction coefficient \(\mu\), it was assumed that, under constant boundary conditions for measurements on different types of road surfaces, is the arrangement of surface irregularities what influences the friction between the tire and the road most. The shape and distribution of these irregularities, or surface texture, were characterized in the previous section using 3D parameters calculated from the primary, micro, and macrotexture surfaces. A linear regression model, represented by Equation (2), was chosen to determine the prediction formula for the friction coefficient using a combination of various texture characteristics:  
\[\mu_{predicted}=a+b\cdot TP_{1}+c\cdot TP_{2}+\ldots+m\cdot TP_{m} \tag{2}\]  
where \(\mu_{predicted}\) is the predicted value of friction coefficient, \(TP_{1}\), \(TP_{2}\)\(\ldots\)\(TP_{m}\) are texture parameters, \(a\), \(b\), \(c\), \(\ldots\)\(m\) are parameters of the regression model, and \(m\) represents the number of parameters included in the model. The multiple linear regression analysis was conducted using MATLAB software (version R2023a), specifically utilizing the Regression Learner application. This software allowed us to efficiently model the relationships between the texture parameters and the friction coefficient, and to perform cross-validation for evaluating the robustness and accuracy of the models. The model included texture characteristics representing microtexture and macrotexture surfaces, as well. The selection of individual texture characteristics was based on theoretical knowledge of their potential impact on the friction coefficient as well as the results obtained from comparing individual texture parameters with the friction coefficient. Ultimately, the selection and combination of texture parameters were validated through statistical significance tests of individual parameters and the overall regression models. Cross-validation was employed to identify the combination of texture characteristics that best represent the measured friction coefficient values and to determine the model's accuracy. To optimize the number of texture parameters used in the model, \(R^{2}_{adjusted}\) values were calculated in addition to basic statistical indicators (\(R^{2}\) and \(p\)-value). The inclusion of texture parameters in the regression model was guided by the adjusted \(R^{2}\) metric. Parameters were incorporated if both \(R^{2}\) and \(R^{2}_{adjusted}\) values increased with their addition. If \(R^{2}_{adjusted}\) started to decline despite rising \(R^{2}\) values, the inclusion process was stopped. Each independent variable's statistical significance was checked using the \(p\)-value, with those exceeding 0.001 being excluded. The overall model significance was then validated by comparing the calculated \(F\)-value to the critical value, ensuring it surpassed the threshold for statistical validity. Given that the valley material portion \(S_{mr2,MIC}\) showed the strongest correlation with the friction coefficient, this parameter was used as the foundation of the prediction model. Additional texture parameters were progressively tested and evaluated to determine their suitability for inclusion in the model. The formula for predicting friction coefficient \(\mu\) measured by TWO was, after many combinations of different texture parameters, compiled using the valley material portion \(S_{mr2,MIC}\) representing the surface microtexture, and the arithmetic mean peak curvature \(S_{pc,MAC}\) representing the surface macrotexture.