# Estimation of Combustion Parameters from Engine Vibrations Based on Discrete Wavelet Transform and Gradient Boosting

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Work

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## 3. Methods

#### 3.1. Combustion Parameters

#### 3.2. Sensor Signals Window

#### 3.3. Discrete Wavelet Transform and Feature Extraction

#### 3.4. Extreme Gradient Boosting (XGBoost) Regression and Feature Importance (FI)

## 4. Results

#### 4.1. PFP Regression and FI

#### 4.2. MFB50 Regression and FI

#### 4.3. Summary

**Table 3.**Summary of regression results for all selected combustion parameters of introduced approach and comparison method.

DWT + XGBoost | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Engine 1 | Engine 2 | |||||||||||

Validation | Test | Validation | Test | |||||||||

Target | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} |

PFP | $3.98$ | $3.00$ | $0.98$ | $5.69$ | $4.35$ | $0.96$ | $3.87$ | $2.97$ | $0.87$ | $3.7$ | $2.89$ | $0.82$ |

MFB10 | $1.38$ | $1.09$ | $0.83$ | $1.43$ | $1.14$ | $0.77$ | $0.84$ | $0.66$ | $0.53$ | $0.82$ | $0.65$ | $0.48$ |

MFB50 | $0.79$ | $0.55$ | $0.97$ | $0.89$ | $0.63$ | $0.95$ | $0.71$ | $0.54$ | $0.80$ | $0.66$ | $0.52$ | $0.74$ |

MFB90 | $1.42$ | $1.05$ | $0.96$ | $1.43$ | $1.09$ | $0.79$ | $1.63$ | $1.20$ | $0.91$ | $1.42$ | $1.10$ | $0.74$ |

Time/Frequency + XGBoost | ||||||||||||

Engine 1 | Engine 2 | |||||||||||

Validation | Test | Validation | Test | |||||||||

Target | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | R^{2} |

PFP | $6.25$ | $4.77$ | $0.96$ | $7.73$ | $6.06$ | $0.92$ | $5.24$ | $4.12$ | $0.76$ | $4.99$ | $3.95$ | $0.67$ |

MFB10 | $1.70$ | $1.34$ | $0.73$ | $1.90$ | $1.50$ | $0.59$ | $0.97$ | $0.76$ | $0.39$ | $0.96$ | $0.76$ | $0.30$ |

MFB50 | $1.78$ | $1.24$ | $0.86$ | $2.03$ | $1.44$ | $0.71$ | $1.01$ | $0.77$ | $0.60$ | $0.90$ | $0.71$ | $0.51$ |

MFB90 | $2.60$ | $1.88$ | $0.86$ | $3.01$ | $2.17$ | $0.69$ | $1.91$ | $1.47$ | $0.63$ | $1.82$ | $1.40$ | $0.56$ |

## 5. Discussion

#### 5.1. KS Position

#### 5.2. Towards a Theoretic Explanation of FI

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANN | Artificial neural network |

AC | Approximation coefficients |

CA | Crank angle |

CWT | Continuous wavelet transfrom |

DC | Detailed coefficients |

DWT | Discrete wavelet transform |

ECU | Engine control unit |

FI | Feature importance |

IMEP | Indicated mean effective pressure |

KS | Knock sensor |

MAE | Mean absolute error |

MDI | Mean decrease in impurity |

MFB10 | Mass fraction burned 10% |

MFB50 | Mass fraction burned 50% |

MFB90 | Mass fraction burned 90% |

OPs | Operating points |

PFP | Peak firing pressure |

PS | Pressure sensor |

RMSE | Root mean square error |

RMS | Root mean square |

${R}^{2}$ | Coefficient of determination |

SCE | Single cylinder engine |

SHAP | Shapley additive explanations |

XGBoost | Extreme gradient boosting |

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**Figure 1.**Schematic diagram of experimental setup with marked positions for in-cylinder PS (cyan) and KS (orange).

Metric | Haar | Db4 | Sym4 | Coif6 |
---|---|---|---|---|

MAE | 3.00 | 3.97 | 3.94 | 4.31 |

RMSE | 3.98 | 5.12 | 5.11 | 5.57 |

${R}^{2}$ | 0.98 | 0.97 | 0.97 | 0.97 |

Description | Name | Equations |
---|---|---|

1. Index of minimum | arg_min | ${\varphi}_{min}=argmin\left(c\right)$ |

2. Index of maximum | arg_max | ${\varphi}_{max}=argmax\left(c\right)$ |

3. Variance | var | ${\sigma}^{2}={\sum}_{i=1}^{n}(\frac{{({c}_{i}-\overline{c})}^{2}}{n-1}$) |

4. Maximum | max | ${c}_{max}=max\left(c\right)$ |

5. Minimum | min | ${c}_{min}=min\left(c\right)$ |

6. Maximum gradient | max_grad | $\frac{\partial c}{\partial \varphi}{|}_{max}=max\left(\frac{\partial c}{\partial \varphi}\right)$ |

7. Mean difference | mean_diff | $\overline{d}=\overline{{c}_{i+1}-{c}_{i}}$ |

8. Root mean square | rms | ${c}_{rms}=\frac{\sqrt{{\sum}_{i=1}^{n}{{c}_{i}}^{2}}}{n}$ |

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**MDPI and ACS Style**

Kefalas, A.; Ofner, A.B.; Pirker, G.; Posch, S.; Geiger, B.C.; Wimmer, A.
Estimation of Combustion Parameters from Engine Vibrations Based on Discrete Wavelet Transform and Gradient Boosting. *Sensors* **2022**, *22*, 4235.
https://doi.org/10.3390/s22114235

**AMA Style**

Kefalas A, Ofner AB, Pirker G, Posch S, Geiger BC, Wimmer A.
Estimation of Combustion Parameters from Engine Vibrations Based on Discrete Wavelet Transform and Gradient Boosting. *Sensors*. 2022; 22(11):4235.
https://doi.org/10.3390/s22114235

**Chicago/Turabian Style**

Kefalas, Achilles, Andreas B. Ofner, Gerhard Pirker, Stefan Posch, Bernhard C. Geiger, and Andreas Wimmer.
2022. "Estimation of Combustion Parameters from Engine Vibrations Based on Discrete Wavelet Transform and Gradient Boosting" *Sensors* 22, no. 11: 4235.
https://doi.org/10.3390/s22114235