#### 4.2. Properties of Low-Noise Asphalt Mixtures

Seeking to identify the effect of asphalt mixture parameters on noise when driving on asphalt pavement at a speed of 80 km/h, a regression model was developed through analysis of 64 observations. The dependent variable selected for the model was the asphalt pavement noise criterion, CPX80 (dB), the values of which were measured on-site at a speed of 80 km/h.

The following data on the asphalt mixture were used for evaluation of the properties of low-noise asphalt mixture (AC): bitumen content Pb (%), calculated surface area of aggregates SA (m^{2}/kg), density of the mineral aggregate Gsb (kg/m^{3}), effective specific gravity Gse, apparent density of the AC mixture Gmb (kg/m^{3}), maximum asphalt mixture density Gmm (kg/m^{3}), calculated air-void content VA (%), total bitumen content in the AC mixture VB (%), voids in the mineral aggregate VMA (%) and voids in the mineral aggregate filled with bitumen VFB (%).The independent variables for the model were selected after analyzing 19 criteria. These criteria were divided into two groups: factual gradation of asphalt mixture aggregates and data of the asphalt mixture determined in a laboratory. Factual aggregate gradation is described as the percent passing through nine different types of sieves no and mesh size: p0.063—0.063 mm, p0.125—0.125 mm, p0.25—0.25 mm, p0.5—0.5 mm, p1—1.0 mm, p2—2.0 mm, p5.6—5.6 mm, p8—8.0 mm, and p11.2—11.2 mm.

In order to determine the weight of each criterion, the entropy weighting method was applied [

36]. The calculated weights of criteria for SA, VA, Pb, VMA, and VFB are given in

Table 4. According to the criteria weights, the criterion with the largest weight is VA and the criterion with the least weight is Pb. Analysis of the weights of gradation criteria using the entropy weighing method showed that the most dominant were aggregates passing through a sieve size of 2.0 mm (p2) and the least dominant were aggregates passing through a 11.2 mm sieve size (p11.2,

Table 5).

The selection of the variables to be included in a regression model was based on the correlation of the independent variables with a dependent variable. The correlation of variables was determined by calculating the Pearson correlation coefficient:

where (

X_{i},

Y_{i})—a pair of values of the variables X and

Y,

$\left(i=\overline{1{,}_{}n}\right)$,

n—sample size.

A correlation matrix (

Table 6) showed that noise level CPX80 is highly correlated with p8, p11.2, Gsb, Gmb, VA, VMA, and VFB). Therefore, it should be included in the model if they have no intercorrelation among them. According to

Table 6, some of the selected criteria were correlated. In order to find out if these correlations are statistically significant, the hypothesis regarding the significance of correlation coefficients was tested.

Conclusions on the significance of the correlation coefficients were made by observing the

p-values and comparing them with the selected significance level α = 0.05. Correlation of variables was statistically significant when

p < α. Analysis of the data in

Table 7 showed that correlation between the following pairs of variables was not significant: SA with p8, p11,2 and Pb; VA with Pb and Gsb, although the correlation between SA and VA (

r = −0.65) was statistically significant. A statistically significant correlation was observed between SA and VMA, SA and VFB, VA and VMA, VA and VFB, and also between VMA and VFB. It is not possible for the model to contain non-correlated criteria.

#### 4.3. Pavement Acoustic Properties Prediction Model

Based on the correlation matrix, from the 19 criteria, the following criteria were selected to develop the CPX: SA, Pb, VA, VMA, and VFB. The criterion SA was selected since it comprises all the studied criteria of gradation. The air-void content, VA, was calculated using the values of Gmb and Gmm. The bitumen content, Pb, was included in the regression model in order to observe its effect on asphalt pavement noise. Since all five criteria cannot be included in one model due to their intercorrelation, two linear regression models were developed:

To develop the model and to test the assumptions, the

R software was used [

2]. With the help of calculations, the model CPX

_{1} was obtained:

The obtained model, CPX_{1}, demonstrated that the value of CPX was more highly affected by the VA (with a standardized value of –0.845) compared to the SA⋅Pb (with a standardized value of –0.161).

In the second case,

${\mathrm{CPX}}_{2}=f\left(\mathrm{VFB},\mathrm{Pb}\right)$ the following model was obtained:

The obtained model, CPX_{2}, showed that the value of CPX experienced a greater effect from the VFB (with a standardized value of 0.675) compared to the Pb (with a standardized value of –0.225).

The values of assumptions describing the reliability of the models CPX

_{1} and CPX

_{2} are given in

Table 8. The models that were obtained according to the coefficient of determination,

R^{2}, adequately described the testing data; in both cases,

R^{2} ≥ 0.20. The ANOVA test showed that the model variables were statistically significant, in both cases the

p-value was <0.05. Based on the Breusch–Pagan test results, it can be stated that the model CPX

_{1} data were compatible with the homoscedasticity assumption since the

p-value was ≥0.05. In the case of the model, CPX

_{2}, an insignificant heteroskedasticity of the data was observed. Based on the VIF values, neither of the models possess multicollinearity. Measured versus predicted noise of CPX at 80 km/h for developed models CPX

_{1} and CPX

_{2} are given in

Figure 5.

When testing the normality of the residuals according to the Shapiro–Wilk test and also when observing the PPplots (probability–probability plots) (

Figure 6), the assumption on the normality of the residuals was not satisfied. In the case of the model, CPX

_{1}, after implementing the Shapiro–Wilk test the obtained

p-value was 0.00013 (<0.05)—the assumption on the normality of the residuals was not satisfied. In the case of the model, CPX

_{2}, after implementing the Shapiro–Wilk test, the obtained

p-value was 0.018 (<0.05)—the assumption on the normality of the residuals was not satisfied, although the residuals of the model CPX

_{2} were almost normal.

The Bonferroni test showed that the models CPX

_{1} and CPX

_{2} contain outliers. When observing the Cook’s distance plots, it is obvious that none of the Cook‘s distance values are larger than 1, meaning that the studied data contain no outliers. The influence plot shows (

Figure 7) that, in case of both models 8 (SMA 8 TM) and 19 (SMA 8 S), observations of studentized residuals are lower than (–3), meaning that these observations have a relatively high effect on the results of the regression analysis. It was concluded that, generally, the data had no outliers.