# A Machine Learning Model for Predicting Noise Limits of Motor Vehicles in UNECE R51 Regulations
## 4 Prediction of Acceleration Noise Limits by Using the Proposed BPNN Model  
### Validation of Noise Limit Outputs  
Seventeen datasets, including all vehicle categories, were calculated and simulated by using the proposed BPNN model in this study. To improve the representativeness of data samples, the dividevec function was used to randomly sort the original data, and randomly extracted 70% of the data as the training data, 15% as the validation data, and the other 15% as the test data. In other words, the training data, validation data and test data were counted as 11, 3, 3, respectively. The predicted results by using the proposed BPNN model are shown in Table 4. For comparison, the quadratic regression model was also used, and its predicted results are also given in the same time, in which the least squares method were used to fit the first six historic noise limits to obtain a quadratic regression model, and then the predicted value for the variable set was calculated. In addition, the percentage error (PE) can be determined by  
\[PE=\frac{(\textit{CNL}-PV)}{\textit{CNL}}\times 100\% \tag{7}\]  
where, _CNL_ is the current noise limit in UNECE regulations, and \(PV\) is predicted value by using the proposed BPNN model or the quadratic regression model, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Machine** & \multirow{2}{*}{**Kernel Function**} & \multirow{2}{*}{**MSE**} & \multirow{2}{*}{**Model Type**} \\
**Learning** & & & \\ \hline \multirow{6}{*}{SVM} & Linear & 0.663 & \\  & Quadratic & 0.599 & \\  & Cubic & 0.997 & \\  & Fine Gaussian & 8.65 & Kernel scale: 0.61 \\  & Medium Gaussian & 1.59 & Kernel scale: 2.4 \\  & Coarse Gaussian & 2.02 & Kernel scale: 9.8 \\ \hline \multirow{6}{*}{GPR} & Rational Quadratic & 0.331 & \\  & Squared Exponential & 0.312 & \\  & Matern 5/2 & 0.524 & \\  & Exponential & 0.876 & \\ \hline \multirow{2}{*}{ET} & Boosted Trees & 23.6 & Minimum leaf size: 8; Number of learners: 30; \\  & Bagged Trees & 14.8 & Minimum leaf size: 8; Number of learners: 30 \\ \hline \multirow{2}{*}{RT} & Fine Tree & 4.76 & Minimum leaf size: 4 \\  & Medium Tree & 14.4 & Minimum leaf size: 12 \\  & Coarse Tree & 14.4 & Minimum leaf size: 36 \\ \hline \multirow{2}{*}{BPNN} & Levenberg-Marquardt & \multirow{2}{*}{0.444} & Hidden layer units: 12 \\  & algorithm & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of MSE values for five machine learning models. According to the data, shown in Table 4, the following research findings can be obtained:  
1. For the training data, the predicted values by using the proposed BPNN model with network learning is almost the same as the values of the current noise limits except for one case for N3 with a percentage error of 1.81%. The quadratic regression model extracted the predicted values with much higher errors from \(-\)5.44% to 1.73%. It demonstrated the proposed BPNN model was accurate. 2. As for those validation data and test data, since there was no network learning, the predicted values have much larger errors compared with those training data. However, the maximum deviation is still smaller than the results obtained by the quadratic regression model in most cases. Again, the proposed BPNN model provides a better accuracy than the quadratic regression model in the prediction of noise limits during vehicle acceleration. 3. The proposed BPNN model can predict noise limits for all learned categories of vehicles, which are only related to historic values of the noise limits of a vehicle, but not depending on its category. Differently, the quadratic regression model is strongly related to both historic values of noise limits of a vehicle and its vehicle category. 4. The proposed BPNN model has a migration ability on the prediction of noise limits and can be used to predict the backward trend of noise limits defined in the UNECE regulations. If the inputs of the BPNN model are moved one unit backward, it provides predicted values of noise limits in the next revisions of the UNECE regulations.