# 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
Furthermore, a regression analysis of the outputs of the proposed BPNN model was conducted, and the results are shown in Figure 3. The dots in Figure 3 are the abscissa as the vehicle acceleration noise limits as the expected outputs, and the ordinates are the corresponding data points of the predicted limits by using the BPNN model. The dotted line is the ideal curve assuming that the actual output is equal to the expected output; and the solid line is the data point regression analysis and gives the linear regression equation and correlation coefficient. Table 5 provides the correlations of the results from the proposed BPNN and those from its regression analysis. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Categories of Vehicle**} & \multirow{2}{*}{**CNL**} & \multicolumn{3}{c}{**BPNN Model**} & \multicolumn{3}{c}{**Quadratic Regression Model**} & \\ \cline{3-10}  & & \multicolumn{1}{c}{**Data**} & \multicolumn{1}{c}{**Classification**} & \multicolumn{1}{c}{**PV**} & \multicolumn{1}{c}{**PE**} & \multicolumn{1}{c}{**Prediction Model**} & \multicolumn{1}{c}{**PV**} & \multicolumn{1}{c}{**PE**} \\ \hline \multirow{3}{*}{M1} & 1 & 68 & Test data & 68.34 & \(-\)0.50\% & y = 0.0714x2 \({}^{-}\) 2.9857x \(+\) 85.2 & 67.80 & 0.30\% \\  & 2 & 69 & Training data & 69.00 & 0.00\% & y = 0.1429x2 \({}^{-}\) 3.2571x \(+\) 85.4 & 69.60 & \(-\)0.86\% \\  & 3 & 71 & Validation data & 70.96 & 0.05\% & y = 0.2857x2 \({}^{-}\) 3.8000x \(+\) 85.8 & 73.20 & \(-\)3.10\% \\ \hline \multirow{3}{*}{M2} & 1 & 69 & Training data & 69.00 & 0.00\% & y = 0.2321x2 \({}^{-}\) 4.4821x \(+\) 88.5 & 68.50 & 0.73\% \\  & 2 & 71 & Training data & 71.00 & 0.00\% & y = 0.3750x2 \({}^{-}\) 5.0250x \(+\) 88.9 & 72.10 & \(-\)1.55\% \\  & 3 & 72 & Training data & 72.00 & 0.00\% & y = 0.3750x2 \({}^{-}\) 5.5679x \(+\) 93.3 & 72.70 & \(-\)0.97\% \\  & 4 & 72 & Validation data & 71.09 & 1.26\% & y = 0.2321x2 \({}^{-}\) 4.9964x \(+\) 95.3 & 71.70 & 0.42\% \\ \hline \multirow{3}{*}{M3} & 1 & 73 & Training data & 73.00 & 0.00\% & y = 0.4464x2 \({}^{-}\) 5.8393x \(+\) 93.5 & 74.50 & \(-\)2.05\% \\  & 2 & 76 & Test data & 75.61 & 0.51\% & y = 0.4464x2 \({}^{-}\) 5.8107x \(+\) 95.9 & 77.10 & \(-\)1.45\% \\  & 3 & 77 & Training data & 77.00 & 0.00\% & y = 0.5000x2 \({}^{-}\) 5.8714x \(+\) 95.8 & 79.20 & \(-\)2.86\% \\ \hline \multirow{3}{*}{N1} & 1 & 69 & Training data & 69.00 & 0.00\% & y = 0.1071x2 \({}^{-}\) 3.4357x \(+\) 87.4 & 68.60 & 0.58\% \\  & 2 & 71 & Training data & 71.00 & 0.00\% & y = 0.1071x2 \({}^{-}\) 2.9786x \(+\) 86.8 & 71.20 & \(-\)0.28\% \\ \hline \multirow{3}{*}{N2} & 1 & 74 & Test data & 74.15 & \(-\)0.21\% & y = 0.2321x2 \({}^{-}\) 4.5393x \(+\) 93.7 & 73.30 & 0.95\% \\  & 2 & 75 & Training data & 75.00 & 0.00\% & y = 0.0893x2 \({}^{-}\) 3.2821x \(+\) 92.3 & 73.70 & 1.73\% \\ \hline \multirow{3}{*}{N3} & 1 & 76 & Training data & 76.00 & 0.00\% & y = 0.3750x2 \({}^{-}\) 5.0821x \(+\) 94.1 & 76.90 & \(-\)1.18\% \\  & 2 & 77 & Validation data & 77.00 & 0.00\% & y = 0.5357x2 \({}^{-}\) 6.3214x \(+\) 98.0 & 79.99 & \(-\)3.90\% \\ \cline{1-1}  & 3 & 79 & Training data & 77.57 & 1.81\% & y = 0.6964x2 \({}^{-}\) 7.0750x \(+\) 98.7 & 83.30 & \(-\)5.44\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparisons on prediction values extracted from the proposed BPNN model and quadratic regression model.