# Dynamic Modulus Prediction of a High-Modulus Asphalt Mixture

Authors: Chaohui Wang; Songyuan Tan; Qian Chen; Jiguo Han; Liang Song; Yi Fu
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

Dynamic modulus is a key evaluation index of the high-modulus asphalt mixture, but it is relatively difficult to test and collect its data. The purpose is to achieve the accurate prediction of the dynamic modulus of the high-modulus asphalt mixture and further optimize the design process of the high-modulus asphalt mixture. Five high-temperature performance indexes of high-modulus asphalt and its mixture were selected. The correlation between the above five indexes and the dynamic modulus of the high-modulus asphalt mixture was analyzed. On this basis, the dynamic modulus prediction models of the high-modulus asphalt mixture based on small sample data were established by multiple regression, general regression neural network (GRNN), and support vector machine (SVM) neural network. According to parameter adjustment and cross-validation, the output stability and accuracy of different prediction models were compared and evaluated. The most effective prediction model was recommended. The results show that the SVM model has more significant prediction accuracy and output stability than the multiple regression model and the GRNN model. Its prediction error was 0.98–9.71%. Compared with the other two models, the prediction error of the SVM model declined by 0.50–11.96% and 3.76–13.44%. The SVM neural network was recommended as the dynamic modulus prediction model of the high-modulus asphalt mixture.


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## Body

## 1. Introduction

The dynamic modulus can better reflect the actual stress status of the pavement under different conditions. It is not only an important parameter for the design of the pavement structure but also one of the key evaluation indicators for the mechanical properties of high-modulus asphalt mixtures [1–3]. Some scholars have carried out a series of studies on this. Ma et al. [4] compared and analyzed the dynamic modulus and rutting resistance of the high-modulus modified recycled asphalt mixture and ordinary recycled asphalt mixture. Fu [5] used ultrahigh-molecular-weight polyolefins to prepare the high-modulus asphalt mixture and tested its dynamic modulus. Lu et al. [6] compared and analyzed the dynamic modulus of the high-modulus asphalt mixture and SBS-modified asphalt mixture. Islam et al. [7] studied the influencing factors of the dynamic modulus of the high-modulus asphalt mixture. However, since most of the equipment used in dynamic modulus tests is imported products, there are problems such as expensive equipment and high test costs. Some laboratories do not have the conditions to carry out dynamic modulus tests. This problem can be effectively solved if the relationship between the road performance of high-modulus asphalt or high-modulus asphalt mixture and the dynamic modulus can be established.In recent years, some scholars have researched the prediction of the dynamic modulus of asphalt mixtures. Zhang et al. [8] revised the existing Hirsch dynamic modulus prediction model. Dao et al. [9] established the prediction model of dynamic modulus using a machine learning method based on ensemble boosted trees. Behnood and Mohammadi Golafshani [10] established a prediction model for the dynamic modulus of asphalt mixtures based on biogeography-based optimization. Solatifar et al. [11] used six prediction models to determine the dynamic modulus of the asphalt layer and compared them with the data measured by using the falling weight deflectometer to modify the existing prediction model. To build a new in situ global model, Moussa and Owais [12] established the prediction model of hot-mix asphalt dynamic modulus based on adapting deep convolution learning technology, compared it with the Witczak model and the Hirsch model, and evaluated the accuracy of the model through the dynamic modulus data measured by the laboratory. In summary, the above research is mainly based on the estimation of a large number of sample data. However, the related test cost of dynamic modulus of the high-modulus asphalt mixture is relatively high, the existing research results are scattered, and the amount of data is small. It is difficult to establish a prediction model based on a large number of sample data. Therefore, the establishment of a predictive model for small sample data has more significant value.Therefore, the high-temperature performance-related index data of high-modulus asphalt and asphalt mixtures were collected and analyzed for their correlation with the dynamic modulus of high-modulus asphalt mixtures. Through parameter adjustment and cross-validation, three prediction models of multiple regression, general regression neural network (GRNN), and support vector machine (SVM) were established; the output stability and prediction accuracy of different prediction models were compared and evaluated. The best prediction model of dynamic modulus is recommended to provide a useful reference for the study of dynamic modulus prediction of the high-modulus asphalt mixture.


## 2. Methods

### 2.1. Test Method for Dynamic Modulus of the High-Modulus Asphalt Mixture

The simple performance tester (SPT) recommended by the Strategic Highway Research Program (SHRP) that can be applied with offset sine wave or half-sine wave load is used for dynamic modulus testing of high-modulus asphalt mixtures. A cylindrical specimen of a high-modulus asphalt mixture with a size ofϕ, 100 mm × 150 mm, is used. The test temperature is 15°C. The loading frequency is 10 Hz. The dynamic modulus of the high-modulus asphalt mixture is measured at the condition of unconfined specimens. It is calculated according to the following equation:(1)E∗=σ0ε0,where σ0 is the axial stress value (MPa) and ε0 is the axial strain value (mm/mm).


### 2.2. Data Collection and Standardized Processing

The test data [5, 13] and survey data [2–4, 6–12, 14–18] of the high-modulus asphalt mixture are used as the sample data, as shown in Table 1. Among them, the first 16 groups are used as the training set, and the last 4 groups are used as the test set. The dynamic modulus of the high-modulus asphalt mixture is selected as the output factor (15°C, 10 Hz). High-temperature performance indicators are selected as input factors, including high-modulus asphalt penetration degree (X1), high-modulus asphalt softening point (X2), high-modulus asphalt rutting factor (X3), the stability of the high-modulus asphalt mixture (X4), and the dynamic stability of the high-modulus asphalt mixture (X5).Table 1
Sample data.
No.High-modulus asphalt penetration degree (0.1 mm)High-modulus asphalt softening point (°C)High-modulus asphalt rutting factor (Pa)Stability of the high-modulus asphalt mixture (kN)Dynamic stability of the high-modulus asphalt mixture (times/mm)Dynamic modulus of the high-modulus asphalt mixture (MPa)X1X2X3X4X5Y121.062.15.5014.01731119,702214.766.07.3612.50684828,442320.465.15.7011.90789526,479423.759.93.8113.10283823,893514.068.06.5016.40413622,711624.061.04.9012.30281220,737719.964.65.4018.30725417,580836.063.75.3914.50478915,065936.063.75.3914.20495014,2851042.188.76.3114.12728811,7631126.085.55.8112.36567520,8951232.057.712.3014.14623713,9051359.056.02.2212.32484611,9821436.065.05.1213.57412615,6741538.062.55.4614.01425516,8791639.060.04.9212.31525016,9531748.053.11.0011.12258314,9061835.058.91.808.70324721,8351930.060.92.5210.69114317,5052019.964.65.4023.90487414,654When constructing the neural network prediction model, in order to avoid problems such as the large difference between the sample data and the failure of the network to converge or the extension of the training time, the sample data are standardized according to the following equation:(2)Xi=X−XminXmax−Xmin,where Xi is the standardized data, X is the sample data, Xmin is the minimum value of the sample data, and Xmax is the maximum value of the sample data.


### 2.3. Neural Network Prediction Method

#### 2.3.1. General Regression Neural Network

The general regression neural network is a 4-layer radial basis function neural network. Compared with the traditional radial basis function neural network, it has more advantages in terms of nonlinear mapping ability and learning rate. The general regression neural network is highly fault tolerant and robust. At present, it is widely used in processing unstable data [19, 20]. The GRNN consists of four layers of neurons, which are the input layer, pattern layer, summation layer, and output layer. According to the nonlinear regression theory and the joint probability density function, the GRNN model can be expressed as the following equation:(3)Y^X=∑i=1nYiexp−X−XiTX−Xi/2s2∑i=1nexp−X−XiTX−Xi/2s2,where Y^X is the network prediction output and σ is the width coefficient of the Gaussian function, namely, smooth factor.The radial basis function is the theoretical basis of the GRNN. Its value depends only on the real-valued function of the distance from the origin. The function that satisfies equation (4) is the radial basis function. SPREAD is the expansion coefficient of the radial basis function. A reasonable expansion coefficient can make the radial basis neuron respond to the interval covered by the input vector to enhance the network’s ability to approximate samples.(4)Rxp−ci=exp12s2xp−ci2,where xp−ci is the Euclidean norm, ci is the center of the Gaussian function, and σ is the variance of the Gaussian function.


#### 2.3.2. Support Vector Machine Neural Network

The local extremum problem that cannot be avoided in the traditional neural network method is theoretically solved by the support vector machine, and the actual problem is converted into the high-dimensional feature space by the nonlinear method, which ingeniously solves the problem caused by the dimensionality. Support vector machines are widely used in the fields of pattern classification and nonlinear regression [21, 22]. The support vector regression algorithm is mainly based on the kernel function algorithm. This research uses the radial basis kernel function shown in equation (5) to make the original algorithm nonlinear. The general support vector machine regression model can be expressed as equation (6).(5)Kx,xi=exp−12σ2x−xi2,12σ2>0,where x−xi is the Euclidean norm, xi is the center of the Gaussian function, and σ is the variance of the Gaussian function.(6)Yi=w×ϕx+b.Assuming that the training set isN-dimensional and a certain sample point is xi,yi, the support vector machine regression model with loss function measurement can be expressed as follows:(7)min︸α⌣,α^12∑i=1,j=1mα^i−α⌣iα^j−α⌣jKij+∑i=1mε−yiα^i+ε+yiα⌣i,(8)s.t.∑i=1mα^i−α⌣i=00<α^i<C0<α⌣i<Ci=1,2,…,m,where α^i,α⌣i>0 are the Lagrangian coefficient, Kij is the kernel function, C is the fixed parameter, and ε>0 is the loss limit (constant).For objective function equation (5), the corresponding α^i and α⌣i are found. Then, the regression model coefficients w and b are found.


### 2.4. Error Evaluation Method

In order to evaluate various error indicators of different models [23], mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE) were used as error criteria, 0 according to equations (9)–(11).(9)MAE=1n∑i=1nY^−iYi,(10)MAPE=1n∑i=1nY^i−YiYi,(11)RMSE=1n∑i=1nY^i−Yi2,where n is the number of test samples, Y^i is the predicted value, and Yi is the true value.


## 2.1. Test Method for Dynamic Modulus of the High-Modulus Asphalt Mixture

The simple performance tester (SPT) recommended by the Strategic Highway Research Program (SHRP) that can be applied with offset sine wave or half-sine wave load is used for dynamic modulus testing of high-modulus asphalt mixtures. A cylindrical specimen of a high-modulus asphalt mixture with a size ofϕ, 100 mm × 150 mm, is used. The test temperature is 15°C. The loading frequency is 10 Hz. The dynamic modulus of the high-modulus asphalt mixture is measured at the condition of unconfined specimens. It is calculated according to the following equation:(1)E∗=σ0ε0,where σ0 is the axial stress value (MPa) and ε0 is the axial strain value (mm/mm).


## 2.2. Data Collection and Standardized Processing

The test data [5, 13] and survey data [2–4, 6–12, 14–18] of the high-modulus asphalt mixture are used as the sample data, as shown in Table 1. Among them, the first 16 groups are used as the training set, and the last 4 groups are used as the test set. The dynamic modulus of the high-modulus asphalt mixture is selected as the output factor (15°C, 10 Hz). High-temperature performance indicators are selected as input factors, including high-modulus asphalt penetration degree (X1), high-modulus asphalt softening point (X2), high-modulus asphalt rutting factor (X3), the stability of the high-modulus asphalt mixture (X4), and the dynamic stability of the high-modulus asphalt mixture (X5).Table 1
Sample data.
No.High-modulus asphalt penetration degree (0.1 mm)High-modulus asphalt softening point (°C)High-modulus asphalt rutting factor (Pa)Stability of the high-modulus asphalt mixture (kN)Dynamic stability of the high-modulus asphalt mixture (times/mm)Dynamic modulus of the high-modulus asphalt mixture (MPa)X1X2X3X4X5Y121.062.15.5014.01731119,702214.766.07.3612.50684828,442320.465.15.7011.90789526,479423.759.93.8113.10283823,893514.068.06.5016.40413622,711624.061.04.9012.30281220,737719.964.65.4018.30725417,580836.063.75.3914.50478915,065936.063.75.3914.20495014,2851042.188.76.3114.12728811,7631126.085.55.8112.36567520,8951232.057.712.3014.14623713,9051359.056.02.2212.32484611,9821436.065.05.1213.57412615,6741538.062.55.4614.01425516,8791639.060.04.9212.31525016,9531748.053.11.0011.12258314,9061835.058.91.808.70324721,8351930.060.92.5210.69114317,5052019.964.65.4023.90487414,654When constructing the neural network prediction model, in order to avoid problems such as the large difference between the sample data and the failure of the network to converge or the extension of the training time, the sample data are standardized according to the following equation:(2)Xi=X−XminXmax−Xmin,where Xi is the standardized data, X is the sample data, Xmin is the minimum value of the sample data, and Xmax is the maximum value of the sample data.


## 2.3. Neural Network Prediction Method

### 2.3.1. General Regression Neural Network

The general regression neural network is a 4-layer radial basis function neural network. Compared with the traditional radial basis function neural network, it has more advantages in terms of nonlinear mapping ability and learning rate. The general regression neural network is highly fault tolerant and robust. At present, it is widely used in processing unstable data [19, 20]. The GRNN consists of four layers of neurons, which are the input layer, pattern layer, summation layer, and output layer. According to the nonlinear regression theory and the joint probability density function, the GRNN model can be expressed as the following equation:(3)Y^X=∑i=1nYiexp−X−XiTX−Xi/2s2∑i=1nexp−X−XiTX−Xi/2s2,where Y^X is the network prediction output and σ is the width coefficient of the Gaussian function, namely, smooth factor.The radial basis function is the theoretical basis of the GRNN. Its value depends only on the real-valued function of the distance from the origin. The function that satisfies equation (4) is the radial basis function. SPREAD is the expansion coefficient of the radial basis function. A reasonable expansion coefficient can make the radial basis neuron respond to the interval covered by the input vector to enhance the network’s ability to approximate samples.(4)Rxp−ci=exp12s2xp−ci2,where xp−ci is the Euclidean norm, ci is the center of the Gaussian function, and σ is the variance of the Gaussian function.


### 2.3.2. Support Vector Machine Neural Network

The local extremum problem that cannot be avoided in the traditional neural network method is theoretically solved by the support vector machine, and the actual problem is converted into the high-dimensional feature space by the nonlinear method, which ingeniously solves the problem caused by the dimensionality. Support vector machines are widely used in the fields of pattern classification and nonlinear regression [21, 22]. The support vector regression algorithm is mainly based on the kernel function algorithm. This research uses the radial basis kernel function shown in equation (5) to make the original algorithm nonlinear. The general support vector machine regression model can be expressed as equation (6).(5)Kx,xi=exp−12σ2x−xi2,12σ2>0,where x−xi is the Euclidean norm, xi is the center of the Gaussian function, and σ is the variance of the Gaussian function.(6)Yi=w×ϕx+b.Assuming that the training set isN-dimensional and a certain sample point is xi,yi, the support vector machine regression model with loss function measurement can be expressed as follows:(7)min︸α⌣,α^12∑i=1,j=1mα^i−α⌣iα^j−α⌣jKij+∑i=1mε−yiα^i+ε+yiα⌣i,(8)s.t.∑i=1mα^i−α⌣i=00<α^i<C0<α⌣i<Ci=1,2,…,m,where α^i,α⌣i>0 are the Lagrangian coefficient, Kij is the kernel function, C is the fixed parameter, and ε>0 is the loss limit (constant).For objective function equation (5), the corresponding α^i and α⌣i are found. Then, the regression model coefficients w and b are found.


## 2.3.1. General Regression Neural Network

The general regression neural network is a 4-layer radial basis function neural network. Compared with the traditional radial basis function neural network, it has more advantages in terms of nonlinear mapping ability and learning rate. The general regression neural network is highly fault tolerant and robust. At present, it is widely used in processing unstable data [19, 20]. The GRNN consists of four layers of neurons, which are the input layer, pattern layer, summation layer, and output layer. According to the nonlinear regression theory and the joint probability density function, the GRNN model can be expressed as the following equation:(3)Y^X=∑i=1nYiexp−X−XiTX−Xi/2s2∑i=1nexp−X−XiTX−Xi/2s2,where Y^X is the network prediction output and σ is the width coefficient of the Gaussian function, namely, smooth factor.The radial basis function is the theoretical basis of the GRNN. Its value depends only on the real-valued function of the distance from the origin. The function that satisfies equation (4) is the radial basis function. SPREAD is the expansion coefficient of the radial basis function. A reasonable expansion coefficient can make the radial basis neuron respond to the interval covered by the input vector to enhance the network’s ability to approximate samples.(4)Rxp−ci=exp12s2xp−ci2,where xp−ci is the Euclidean norm, ci is the center of the Gaussian function, and σ is the variance of the Gaussian function.


## 2.3.2. Support Vector Machine Neural Network

The local extremum problem that cannot be avoided in the traditional neural network method is theoretically solved by the support vector machine, and the actual problem is converted into the high-dimensional feature space by the nonlinear method, which ingeniously solves the problem caused by the dimensionality. Support vector machines are widely used in the fields of pattern classification and nonlinear regression [21, 22]. The support vector regression algorithm is mainly based on the kernel function algorithm. This research uses the radial basis kernel function shown in equation (5) to make the original algorithm nonlinear. The general support vector machine regression model can be expressed as equation (6).(5)Kx,xi=exp−12σ2x−xi2,12σ2>0,where x−xi is the Euclidean norm, xi is the center of the Gaussian function, and σ is the variance of the Gaussian function.(6)Yi=w×ϕx+b.Assuming that the training set isN-dimensional and a certain sample point is xi,yi, the support vector machine regression model with loss function measurement can be expressed as follows:(7)min︸α⌣,α^12∑i=1,j=1mα^i−α⌣iα^j−α⌣jKij+∑i=1mε−yiα^i+ε+yiα⌣i,(8)s.t.∑i=1mα^i−α⌣i=00<α^i<C0<α⌣i<Ci=1,2,…,m,where α^i,α⌣i>0 are the Lagrangian coefficient, Kij is the kernel function, C is the fixed parameter, and ε>0 is the loss limit (constant).For objective function equation (5), the corresponding α^i and α⌣i are found. Then, the regression model coefficients w and b are found.


## 2.4. Error Evaluation Method

In order to evaluate various error indicators of different models [23], mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE) were used as error criteria, 0 according to equations (9)–(11).(9)MAE=1n∑i=1nY^−iYi,(10)MAPE=1n∑i=1nY^i−YiYi,(11)RMSE=1n∑i=1nY^i−Yi2,where n is the number of test samples, Y^i is the predicted value, and Yi is the true value.


## 3. Results and Discussion

### 3.1. Correlation Analysis

Based on commonly used elementary functions, SPSS data analysis software was used to establish the best unary regression modelY = fi (Xi) (i = 1–5) between the dependent variable Y and each independent variable (Xi) [24]. The correlation coefficient (R) was compared and analyzed, and an appropriate unary regression model was selected, as shown in Table 2.Table 2
Unary regression model and coefficients.
Independent variableUnary regression modelCorrelation coefficient (R)X1Y = 35502.845 − 786.361X1 + 6.488X120.849X2Y = −172489.497 + 5362.898X2 − 36.713X220.564X3Y = 5211.658 + 4034.071X3 − 265.446X320.417X4Y = 140874.692 − 16206.612X4 + 524.359X420.433X5Y = 45591.796 − 11.178X5 − 0.001X520.505Table2 shows that when the univariate regression model is adopted, the correlation coefficients between the independent and the dependent variables in the descending order are penetration degree (X1) > softening point (X2) > dynamic stability (X5) > stability (X4) > rutting factor (X3). The independent variable with the greatest correlation with the dynamic modulus (Y) is X1, and the correlation coefficient is 0.849. The correlation coefficient between X2 and Y is 0.567, while the correlation coefficient between X3, X4, and X5 and Y is relatively small. It can be seen that the correlation coefficients between the independent variables and the dynamic modulus are all below 0.9, indicating that the accuracy of the unary regression model is not ideal, and it is necessary to further establish a multiple regression model.To comprehensively analyze the correlation between high-temperature performance indexes and dynamic modulus of high-modulus asphalt and its mixture, the binary, ternary, quaternary, and five-variable regression models based on five independent variables, respectively, were established, and the correlation coefficients of 26 regression models were calculated, as shown in Table3.Table 3
The correlation coefficient.
No.Independent variableCorrelation coefficient1X1, X20.8572X1, X30.8913X1, X40.9604X1, X50.8905X2, X30.5926X2, X40.7757X2, X50.7768X3, X40.6679X3, X50.63810X4, X50.78211X1, X2, X30.91512X1, X2, X40.97013X1, X2, X50.93314X1, X3, X40.96915X1, X3, X50.94116X1, X4, X50.98017X2, X3, X40.82218X2, X3, X50.95919X2, X4, X50.75620X3, X4, X50.90721X1, X2, X3, X40.99822X1, X2, X3, X50.98623X1, X2, X4, X50.99324X1, X3, X4, X50.99625X2, X3, X4, X50.99926X1, X2, X3, X4, X50.999Table3 shows that the correlation coefficient of the binary regression model is 0.592 when it is the lowest. The correlation coefficient of the ternary regression model is basically above 0.9. The correlation coefficient in the quaternary regression model is between 0.986 and 0.999. The correlation coefficient of the five-element regression model is 0.999. This shows that the correlation between independent variables and dependent variables in the multiple regression prediction models increases with the increase in the number of independent variables.


### 3.2. Multiple Regression Model

A prediction model with a determination coefficient (R2) greater than 0.9 (Table 4) is selected to verify its output stability and prediction accuracy. Test set data is predicted, and the results are shown in Figure 1.Table 4
Regression prediction model (R2＞0.9).
No.Prediction modelR21Y = 93423.05 − 1189.723X1 − 6110.695X4 + 25.437X1X4 + 6.944X12 + 142.752X420.9212Y = 88639.004 − 884.509X1 + 843.517X2 − 10920.476X4 − 5.577X1X2 + 39.228X1X4 + 58.286X2X4 + 5.413X12 − 9.927X22 + 164.529X420.9413Y = 50908.774 − 955.491X1 + 3123.979X3 − 1626.14X4 − 10.104X1X3 + 18.105X1X4 − 263.252X3X4 + 5.36X12 + 29.575X32 + 49.024X420.9394Y = 110656.631 − 1280.02X1 − 10467.593X4 + 4.517X5 + 55.535X1X4 − 0.039X1X5 − 0.41X4X5 + 6.147X12 + 360.134X420.9605Y = 103658.4 − 1433.557X2 − 30437.188X3 + 13.253X5 + 440.443X2X3 − 0.633X2X5 + 2.309X3X5 + 17.679X22 − 586.769X32 + 0.001X520.9206Y = 62018.428 − 1990.383X1 − 2597.742X4 − 68.109X1X2 + 554.347X1X3 + 0.996X1X4 + 253.463X2X3 + 304.297X2X4 − 2727.888X3X4 + 50.018X12 − 23.046X22 + 375.436X32 − 127.912X420.9967Y = −68635.135 + 2082.956X1+3823.682X2 − 33.416X5 − 80.07X1X2 + 184.77X1X3 + 0.15X1X5 − 83.988X2X3 + 3.29X3X5 + 22.793X12 − 6.952X22 − 1246.046X32 + 0.001X520.9728Y = 325615.554 − 1532.608X1 − 6637.453X2 − 13748.372X4 + 24.671X5 + 35.92X1X2 − 5.982X1X4 − 0.294X1X5 − 1.074X4X5 + 5.743X12 + 38.3X22 + 665.09X420.9869Y = 69989.91 − 6102.336X3 − 4491.458X4 + 3.572X5 + 72.615X1X3 − 53.917X1X4 − 0.099X1X5 + 1.374X3X5 − 0.592X4X5 + 9.073X12 − 272.144X32 + 286.187X420.99210Y = 205353.815 − 3885.777X2 − 19395.397X3 − 5926.33X4 + 16.572X5 + 287.339X2X3 − 87.746X2X4 − 0.453X2X5 + 2.579X3X5 − 0.965X4X5 + 40.402X22 − 822.101X32 + 598.769X42 + 0.001X520.99811Y = −26254.691 − 89.279X2 + 8888.267X4 − 1.841X5 − 30.222X1X2 + 253.913X1X3 − 72.27X1X4 − 0.033X1X5 + 165.611X2X3 − 1441.992X3X4 + 0.921X3X5 − 0.527X4X5 + 24.933X12 − 180.823X32 + 107.771X420.998Figure 1
The prediction error of the multiple regression model: (a) binary regression; (b) ternary regression; (c) quaternary regression; (d) five-element regression.
(a)(b)(c)(d)Figure1 shows that when using the multiple regression model to predict, except for the penetration degree, rutting factor, and stability as independent variables, the prediction error of the no. 3 regression model is between 1.48 and 21.67%. The prediction errors of the other multiple regression models are much greater than those of the no. 3 model, which indicates that the multiple regression model has low prediction accuracy when predicting the dynamic modulus of the high-modulus asphalt mixture, the error dispersion is large, and the model is unstable. Multiple regression prediction models with a prediction error of no more than 100% are selected, and the MAE, MAPE, and RMSE of different multiple regression prediction models are compared, as shown in Figure 2.Figure 2
Multiple regression prediction error comparisons.Figure2 shows that the MAE, MAPE, and RMSE values of the no. 3 prediction model established with penetration degree, rutting factor, and stability as independent variables are 8.57%, 8.02%, and 12.17%, which are significantly smaller than the other four groups of prediction models. It shows that the no. 3 model has more advantages in predicting the dynamic modulus of the high-modulus asphalt mixture, and the model output is more stable. The predicted value of the no. 3 prediction model is smaller than the actual value. If the predicted value meets the specification requirements in physical engineering, the actual value must meet the use requirements.


### 3.3. Neural Network Prediction Model

With the rapid development of computer science in recent years, neural networks have been widely used in the field of road engineering. A large number of domestic and foreign scholars use the BP neural network, RBF neural network, and general regression neural network for prediction [25–29]. To solve the aforementioned problems in the multiple regression model, a neural network is used to establish a prediction model to further improve the accuracy of the prediction.


#### 3.3.1. GRNN Prediction Model

The number of neurons in the input layer in the GRNN neural network is equal to the dimension of the input vector. The number of neurons in the pattern layer is the number of learning samples, which are fully connected to the input layer. There are two neuron nodes in the summation layer. The first node is the arithmetic summation of the output of each neuron in the pattern layer, and the second node is the weighted summation of the output of each neuron node in the pattern layer. The number of neurons in the output layer is equal to the number of sample outputs. A suitable spreading coefficient (SPREAD) can obtain a better prediction result, and its value range is usually 0.1–2. Using the cyclic cross-validation method, the spreading coefficient of the GRNN model is determined according to the mean absolute error (MAE) of the dynamic modulus prediction, as shown in Figure3.Figure 3
Optimal spreading coefficient.Figure3 shows that the spreading coefficient is 1.1 when the MAE is the smallest. Therefore, a 5 × 20 × 2 × 1 GRNN prediction model with a spreading coefficient of 1.1 is constructed, and the prediction effect is shown in Figure 4.Figure 4
The prediction error of the GRNN model.Figure4 shows that the prediction error of the GRNN model is between 4.74 and 23.15%, indicating that the output stability and prediction accuracy of the GRNN prediction model are poor. The reason is that the GRNN has higher requirements for the training data itself. A stable, reliable, and accurate prediction model can only be obtained under the premise of training with a large amount of sample data.


#### 3.3.2. SVM Prediction Model

The output stability and prediction accuracy of the multiple regression model are poor, and the GRNN prediction model has higher requirements for the sample data. These two prediction models are not suitable for small sample data prediction. Based on this, the support vector machine algorithm is introduced to establish the prediction model of the dynamic modulus of the high-modulus asphalt mixture. The support vector machine theory is established based on the idea of structural risk minimization. It is theoretically complete, and the algorithm is simple. Compared with other prediction methods, the support vector machine is more suitable for solving small sample learning problems.The RBF kernel function is used, and the training set and the test set are randomly selected to improve the stability of the prediction model and reduce the error caused by the system and artificial allocation of the sample data. Since the penalty parameter and the kernel function parameter in the SVM model do not give a definite value range, the optimal penalty parameter and the kernel function parameter value are between −0.1 and 0.1 through cross-validation. In order to obtain the optimal penalty parameters and kernel function parameters quickly and accurately, the optimal penalty parameter is determined to be 0.03 by the meshing method based on cross-validation, and then the optimal kernel function parameter is determined according to the MAE value. The forecasted effect is shown in Figures5 and 6.Figure 5
Optimization of penalty parameters and kernel function parameters.Figure 6
SVM model prediction error.Figure5 shows that when the MAE value is the smallest, the penalty parameter and the kernel function parameter are both 0.03. According to Figure 6, the maximum absolute value of the prediction error of the SVM model is 2013 MPa, the minimum absolute value is 215 MPa, and the prediction error is between 0.98 and 9.71%. It shows that the prediction accuracy of the SVM model is higher than that of the GRNN model, and the model output is stable and reliable.


### 3.4. Comparative Analysis of Prediction Results of Different Models

Using MAE, MAPE, and RMSE as evaluation indicators, the prediction errors among multiple regression models, GRNN models, and SVM models (Figure7) are compared and analyzed to clarify the accuracy and output stability of different prediction models.Figure 7
Comparison of prediction errors of different models.Figure7 shows that the prediction error of the GRNN model is larger than that of the prediction models established by the other two methods. The reason is that the amount of sample data required to train the GRNN is huge, and the sample data itself are relatively high. The MAE, MAPE, and RMSE values of the SVM dynamic modulus prediction model are 6.04%, 6.14%, and 6.84%. Compared with the MAE, MAPE, and RMSE values of the multiple regression prediction models, they are reduced by 2.53%, 1.88%, and 5.33%. The MAE, MAPE, and RMSE values of the GRNN prediction model are reduced by 8.84%, 8.03%, and 10.78%, indicating that the SVM prediction model is superior to the other two prediction models in terms of prediction accuracy and output stability. The SVM model is more suitable for solving the prediction problem in the case of a small sample.In order to test the prediction accuracy of the SVM prediction model and the prediction advantage in a small sample, the comparison with the previous research results is shown in Table5.Table 5
Comparison of different research results.
No.Prediction modelRMSE1SVM14942Traditional Witczak model [17]96643BP neural network model [17]1880It can be seen from Table5 that compared with the prediction results of the traditional Witczak model and the BP neural network model, the prediction accuracy of the SVM model established in this research is higher. Therefore, the SVM model is more suitable for the prediction of the dynamic modulus of high-modulus asphalt mixtures.


## 3.1. Correlation Analysis

Based on commonly used elementary functions, SPSS data analysis software was used to establish the best unary regression modelY = fi (Xi) (i = 1–5) between the dependent variable Y and each independent variable (Xi) [24]. The correlation coefficient (R) was compared and analyzed, and an appropriate unary regression model was selected, as shown in Table 2.Table 2
Unary regression model and coefficients.
Independent variableUnary regression modelCorrelation coefficient (R)X1Y = 35502.845 − 786.361X1 + 6.488X120.849X2Y = −172489.497 + 5362.898X2 − 36.713X220.564X3Y = 5211.658 + 4034.071X3 − 265.446X320.417X4Y = 140874.692 − 16206.612X4 + 524.359X420.433X5Y = 45591.796 − 11.178X5 − 0.001X520.505Table2 shows that when the univariate regression model is adopted, the correlation coefficients between the independent and the dependent variables in the descending order are penetration degree (X1) > softening point (X2) > dynamic stability (X5) > stability (X4) > rutting factor (X3). The independent variable with the greatest correlation with the dynamic modulus (Y) is X1, and the correlation coefficient is 0.849. The correlation coefficient between X2 and Y is 0.567, while the correlation coefficient between X3, X4, and X5 and Y is relatively small. It can be seen that the correlation coefficients between the independent variables and the dynamic modulus are all below 0.9, indicating that the accuracy of the unary regression model is not ideal, and it is necessary to further establish a multiple regression model.To comprehensively analyze the correlation between high-temperature performance indexes and dynamic modulus of high-modulus asphalt and its mixture, the binary, ternary, quaternary, and five-variable regression models based on five independent variables, respectively, were established, and the correlation coefficients of 26 regression models were calculated, as shown in Table3.Table 3
The correlation coefficient.
No.Independent variableCorrelation coefficient1X1, X20.8572X1, X30.8913X1, X40.9604X1, X50.8905X2, X30.5926X2, X40.7757X2, X50.7768X3, X40.6679X3, X50.63810X4, X50.78211X1, X2, X30.91512X1, X2, X40.97013X1, X2, X50.93314X1, X3, X40.96915X1, X3, X50.94116X1, X4, X50.98017X2, X3, X40.82218X2, X3, X50.95919X2, X4, X50.75620X3, X4, X50.90721X1, X2, X3, X40.99822X1, X2, X3, X50.98623X1, X2, X4, X50.99324X1, X3, X4, X50.99625X2, X3, X4, X50.99926X1, X2, X3, X4, X50.999Table3 shows that the correlation coefficient of the binary regression model is 0.592 when it is the lowest. The correlation coefficient of the ternary regression model is basically above 0.9. The correlation coefficient in the quaternary regression model is between 0.986 and 0.999. The correlation coefficient of the five-element regression model is 0.999. This shows that the correlation between independent variables and dependent variables in the multiple regression prediction models increases with the increase in the number of independent variables.


## 3.2. Multiple Regression Model

A prediction model with a determination coefficient (R2) greater than 0.9 (Table 4) is selected to verify its output stability and prediction accuracy. Test set data is predicted, and the results are shown in Figure 1.Table 4
Regression prediction model (R2＞0.9).
No.Prediction modelR21Y = 93423.05 − 1189.723X1 − 6110.695X4 + 25.437X1X4 + 6.944X12 + 142.752X420.9212Y = 88639.004 − 884.509X1 + 843.517X2 − 10920.476X4 − 5.577X1X2 + 39.228X1X4 + 58.286X2X4 + 5.413X12 − 9.927X22 + 164.529X420.9413Y = 50908.774 − 955.491X1 + 3123.979X3 − 1626.14X4 − 10.104X1X3 + 18.105X1X4 − 263.252X3X4 + 5.36X12 + 29.575X32 + 49.024X420.9394Y = 110656.631 − 1280.02X1 − 10467.593X4 + 4.517X5 + 55.535X1X4 − 0.039X1X5 − 0.41X4X5 + 6.147X12 + 360.134X420.9605Y = 103658.4 − 1433.557X2 − 30437.188X3 + 13.253X5 + 440.443X2X3 − 0.633X2X5 + 2.309X3X5 + 17.679X22 − 586.769X32 + 0.001X520.9206Y = 62018.428 − 1990.383X1 − 2597.742X4 − 68.109X1X2 + 554.347X1X3 + 0.996X1X4 + 253.463X2X3 + 304.297X2X4 − 2727.888X3X4 + 50.018X12 − 23.046X22 + 375.436X32 − 127.912X420.9967Y = −68635.135 + 2082.956X1+3823.682X2 − 33.416X5 − 80.07X1X2 + 184.77X1X3 + 0.15X1X5 − 83.988X2X3 + 3.29X3X5 + 22.793X12 − 6.952X22 − 1246.046X32 + 0.001X520.9728Y = 325615.554 − 1532.608X1 − 6637.453X2 − 13748.372X4 + 24.671X5 + 35.92X1X2 − 5.982X1X4 − 0.294X1X5 − 1.074X4X5 + 5.743X12 + 38.3X22 + 665.09X420.9869Y = 69989.91 − 6102.336X3 − 4491.458X4 + 3.572X5 + 72.615X1X3 − 53.917X1X4 − 0.099X1X5 + 1.374X3X5 − 0.592X4X5 + 9.073X12 − 272.144X32 + 286.187X420.99210Y = 205353.815 − 3885.777X2 − 19395.397X3 − 5926.33X4 + 16.572X5 + 287.339X2X3 − 87.746X2X4 − 0.453X2X5 + 2.579X3X5 − 0.965X4X5 + 40.402X22 − 822.101X32 + 598.769X42 + 0.001X520.99811Y = −26254.691 − 89.279X2 + 8888.267X4 − 1.841X5 − 30.222X1X2 + 253.913X1X3 − 72.27X1X4 − 0.033X1X5 + 165.611X2X3 − 1441.992X3X4 + 0.921X3X5 − 0.527X4X5 + 24.933X12 − 180.823X32 + 107.771X420.998Figure 1
The prediction error of the multiple regression model: (a) binary regression; (b) ternary regression; (c) quaternary regression; (d) five-element regression.
(a)(b)(c)(d)Figure1 shows that when using the multiple regression model to predict, except for the penetration degree, rutting factor, and stability as independent variables, the prediction error of the no. 3 regression model is between 1.48 and 21.67%. The prediction errors of the other multiple regression models are much greater than those of the no. 3 model, which indicates that the multiple regression model has low prediction accuracy when predicting the dynamic modulus of the high-modulus asphalt mixture, the error dispersion is large, and the model is unstable. Multiple regression prediction models with a prediction error of no more than 100% are selected, and the MAE, MAPE, and RMSE of different multiple regression prediction models are compared, as shown in Figure 2.Figure 2
Multiple regression prediction error comparisons.Figure2 shows that the MAE, MAPE, and RMSE values of the no. 3 prediction model established with penetration degree, rutting factor, and stability as independent variables are 8.57%, 8.02%, and 12.17%, which are significantly smaller than the other four groups of prediction models. It shows that the no. 3 model has more advantages in predicting the dynamic modulus of the high-modulus asphalt mixture, and the model output is more stable. The predicted value of the no. 3 prediction model is smaller than the actual value. If the predicted value meets the specification requirements in physical engineering, the actual value must meet the use requirements.


## 3.3. Neural Network Prediction Model

With the rapid development of computer science in recent years, neural networks have been widely used in the field of road engineering. A large number of domestic and foreign scholars use the BP neural network, RBF neural network, and general regression neural network for prediction [25–29]. To solve the aforementioned problems in the multiple regression model, a neural network is used to establish a prediction model to further improve the accuracy of the prediction.


### 3.3.1. GRNN Prediction Model

The number of neurons in the input layer in the GRNN neural network is equal to the dimension of the input vector. The number of neurons in the pattern layer is the number of learning samples, which are fully connected to the input layer. There are two neuron nodes in the summation layer. The first node is the arithmetic summation of the output of each neuron in the pattern layer, and the second node is the weighted summation of the output of each neuron node in the pattern layer. The number of neurons in the output layer is equal to the number of sample outputs. A suitable spreading coefficient (SPREAD) can obtain a better prediction result, and its value range is usually 0.1–2. Using the cyclic cross-validation method, the spreading coefficient of the GRNN model is determined according to the mean absolute error (MAE) of the dynamic modulus prediction, as shown in Figure3.Figure 3
Optimal spreading coefficient.Figure3 shows that the spreading coefficient is 1.1 when the MAE is the smallest. Therefore, a 5 × 20 × 2 × 1 GRNN prediction model with a spreading coefficient of 1.1 is constructed, and the prediction effect is shown in Figure 4.Figure 4
The prediction error of the GRNN model.Figure4 shows that the prediction error of the GRNN model is between 4.74 and 23.15%, indicating that the output stability and prediction accuracy of the GRNN prediction model are poor. The reason is that the GRNN has higher requirements for the training data itself. A stable, reliable, and accurate prediction model can only be obtained under the premise of training with a large amount of sample data.


### 3.3.2. SVM Prediction Model

The output stability and prediction accuracy of the multiple regression model are poor, and the GRNN prediction model has higher requirements for the sample data. These two prediction models are not suitable for small sample data prediction. Based on this, the support vector machine algorithm is introduced to establish the prediction model of the dynamic modulus of the high-modulus asphalt mixture. The support vector machine theory is established based on the idea of structural risk minimization. It is theoretically complete, and the algorithm is simple. Compared with other prediction methods, the support vector machine is more suitable for solving small sample learning problems.The RBF kernel function is used, and the training set and the test set are randomly selected to improve the stability of the prediction model and reduce the error caused by the system and artificial allocation of the sample data. Since the penalty parameter and the kernel function parameter in the SVM model do not give a definite value range, the optimal penalty parameter and the kernel function parameter value are between −0.1 and 0.1 through cross-validation. In order to obtain the optimal penalty parameters and kernel function parameters quickly and accurately, the optimal penalty parameter is determined to be 0.03 by the meshing method based on cross-validation, and then the optimal kernel function parameter is determined according to the MAE value. The forecasted effect is shown in Figures5 and 6.Figure 5
Optimization of penalty parameters and kernel function parameters.Figure 6
SVM model prediction error.Figure5 shows that when the MAE value is the smallest, the penalty parameter and the kernel function parameter are both 0.03. According to Figure 6, the maximum absolute value of the prediction error of the SVM model is 2013 MPa, the minimum absolute value is 215 MPa, and the prediction error is between 0.98 and 9.71%. It shows that the prediction accuracy of the SVM model is higher than that of the GRNN model, and the model output is stable and reliable.


## 3.3.1. GRNN Prediction Model

The number of neurons in the input layer in the GRNN neural network is equal to the dimension of the input vector. The number of neurons in the pattern layer is the number of learning samples, which are fully connected to the input layer. There are two neuron nodes in the summation layer. The first node is the arithmetic summation of the output of each neuron in the pattern layer, and the second node is the weighted summation of the output of each neuron node in the pattern layer. The number of neurons in the output layer is equal to the number of sample outputs. A suitable spreading coefficient (SPREAD) can obtain a better prediction result, and its value range is usually 0.1–2. Using the cyclic cross-validation method, the spreading coefficient of the GRNN model is determined according to the mean absolute error (MAE) of the dynamic modulus prediction, as shown in Figure3.Figure 3
Optimal spreading coefficient.Figure3 shows that the spreading coefficient is 1.1 when the MAE is the smallest. Therefore, a 5 × 20 × 2 × 1 GRNN prediction model with a spreading coefficient of 1.1 is constructed, and the prediction effect is shown in Figure 4.Figure 4
The prediction error of the GRNN model.Figure4 shows that the prediction error of the GRNN model is between 4.74 and 23.15%, indicating that the output stability and prediction accuracy of the GRNN prediction model are poor. The reason is that the GRNN has higher requirements for the training data itself. A stable, reliable, and accurate prediction model can only be obtained under the premise of training with a large amount of sample data.


## 3.3.2. SVM Prediction Model

The output stability and prediction accuracy of the multiple regression model are poor, and the GRNN prediction model has higher requirements for the sample data. These two prediction models are not suitable for small sample data prediction. Based on this, the support vector machine algorithm is introduced to establish the prediction model of the dynamic modulus of the high-modulus asphalt mixture. The support vector machine theory is established based on the idea of structural risk minimization. It is theoretically complete, and the algorithm is simple. Compared with other prediction methods, the support vector machine is more suitable for solving small sample learning problems.The RBF kernel function is used, and the training set and the test set are randomly selected to improve the stability of the prediction model and reduce the error caused by the system and artificial allocation of the sample data. Since the penalty parameter and the kernel function parameter in the SVM model do not give a definite value range, the optimal penalty parameter and the kernel function parameter value are between −0.1 and 0.1 through cross-validation. In order to obtain the optimal penalty parameters and kernel function parameters quickly and accurately, the optimal penalty parameter is determined to be 0.03 by the meshing method based on cross-validation, and then the optimal kernel function parameter is determined according to the MAE value. The forecasted effect is shown in Figures5 and 6.Figure 5
Optimization of penalty parameters and kernel function parameters.Figure 6
SVM model prediction error.Figure5 shows that when the MAE value is the smallest, the penalty parameter and the kernel function parameter are both 0.03. According to Figure 6, the maximum absolute value of the prediction error of the SVM model is 2013 MPa, the minimum absolute value is 215 MPa, and the prediction error is between 0.98 and 9.71%. It shows that the prediction accuracy of the SVM model is higher than that of the GRNN model, and the model output is stable and reliable.


## 3.4. Comparative Analysis of Prediction Results of Different Models

Using MAE, MAPE, and RMSE as evaluation indicators, the prediction errors among multiple regression models, GRNN models, and SVM models (Figure7) are compared and analyzed to clarify the accuracy and output stability of different prediction models.Figure 7
Comparison of prediction errors of different models.Figure7 shows that the prediction error of the GRNN model is larger than that of the prediction models established by the other two methods. The reason is that the amount of sample data required to train the GRNN is huge, and the sample data itself are relatively high. The MAE, MAPE, and RMSE values of the SVM dynamic modulus prediction model are 6.04%, 6.14%, and 6.84%. Compared with the MAE, MAPE, and RMSE values of the multiple regression prediction models, they are reduced by 2.53%, 1.88%, and 5.33%. The MAE, MAPE, and RMSE values of the GRNN prediction model are reduced by 8.84%, 8.03%, and 10.78%, indicating that the SVM prediction model is superior to the other two prediction models in terms of prediction accuracy and output stability. The SVM model is more suitable for solving the prediction problem in the case of a small sample.In order to test the prediction accuracy of the SVM prediction model and the prediction advantage in a small sample, the comparison with the previous research results is shown in Table5.Table 5
Comparison of different research results.
No.Prediction modelRMSE1SVM14942Traditional Witczak model [17]96643BP neural network model [17]1880It can be seen from Table5 that compared with the prediction results of the traditional Witczak model and the BP neural network model, the prediction accuracy of the SVM model established in this research is higher. Therefore, the SVM model is more suitable for the prediction of the dynamic modulus of high-modulus asphalt mixtures.


## 4. Conclusion

(1)
Based on penetration degree, rutting factor, and stability, a ternary regression prediction model was established, and its MAE, MAPE, and RMSE values were 8.57%, 8.02%, and 12.17%. Compared with other multiple regression models, this model has better prediction accuracy.(2)
The prediction error of the GRNN model was large and exceeds 14%, which was not suitable for the prediction of small sample data.(3)
The SVM model has better prediction accuracy and output stability and has advantages in processing small sample data. Compared with the multiple regression model and the GRNN model, the prediction error of the SVM model was reduced by 0.50–11.96% and 3.76–13.44%.(4)
In this study, only the RBF kernel function was used. In the future, the influence of different kernel functions on the accuracy of SVM model prediction will be studied, and the SVM model parameters will be further optimized to improve the prediction accuracy. Besides, more high-modulus asphalt and mixture performance index data should be collected in the future to improve the predictive ability of the model.



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*Source: 9944415-2021-06-09.xml*