Table 8. Engine Torque Prediction with Different Fitting Methods.
| fitting
result (validationa) |
||||||
|---|---|---|---|---|---|---|
| fitting methods | hyperparameters | RMSE | R2 | MSE | MAE | |
| GPR introduced in this study | combined kernel defined in this study | 1.7381 | 1.00 | 3.0211 | 1.0077 | |
| linear regression | linear | preset: linear robust option: off | 11.34 | 1.00 | 128.59 | 8.1743 |
| interaction linear | preset: interactions linear robust option: off | 7.7276 | 1.00 | 59.715 | 4.2302 | |
| robust linear | preset: robust linear robust option: on | 11.786 | 0.99 | 138.91 | 7.9663 | |
| decision tree | fine tree | minimum leaf size: 4 surrogate decision splits: off | 6.1395 | 1.00 | 37.694 | 1.6206 |
| medium tree | minimum leaf size: 12 surrogate decision splits: off | 6.0582 | 1.00 | 36.702 | 1.672 | |
| coarse tree | minimum leaf size: 36 surrogate decision splits: off | 6.6141 | 1.00 | 43.747 | 1.8936 | |
| boosted trees | minimum leaf size: 8preset: boosted trees | 14.932 | 0.99 | 222.96 | 11.591 | |
| bagged tree | minimum leaf size: 8preset: bagged trees | 5.149 | 1.00 | 26.512 | 1.341 | |
| SVM | linear SVM | kernel function: linear kernel scale: automatic | 12.409 | 0.99 | 153.99 | 9.7683 |
| quadratic SVM | kernel function: quadratic kernel scale: automatic | 10.492 | 1.00 | 110.08 | 8.0499 | |
| fine Gaussian SVM | kernel function: Gaussian kernel scale: 0.97 | 12.908 | 0.99 | 166.62 | 9.952 | |
| medium Gaussian SVM | kernel function: Gaussian kernel scale: 3.9 | 10.949 | 1.00 | 119.88 | 8.7303 | |
| coarse Gaussian SVM | kernel function: Gaussian kernel scale: 15 | 10.394 | 1.00 | 108.03 | 7.6224 | |
| neural network | narrow neural network | number of fully connected layers: 1; first layer size: 10;activation: ReLu | 7.1358 | 1.00 | 50.92 | 3.8654 |
| medium neural network | number of fully connected layers: 1; first layer size: 25;activation: ReLu | 6.1125 | 1.00 | 37.362 | 2.8717 | |
| bilayered neural network | number of fully connected layers: 2; first layer size: 10; second layer size: 10;activation: ReLu | 6.69 | 1.00 | 44.756 | 3.326 | |
| trilayered neural network | number of fully connected layers: 3; first layer size: 10; second layer size: 10; third layer size: 10;activation: ReLu | 12.728 | 0.99 | 162 | 6.4575 | |
a
Validation data are 5% randomly selected from the training dataset.