# Fuel Consumption Prediction Models Based on Machine Learning and Mathematical Methods

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Data Description and Preprocessing

## 4. Fuel Consumption Prediction Models

#### 4.1. White-Box Model

#### 4.2. Black-Box Model

- (1)
- Strong interpretability: The tree-based algorithms can provide each feature’s importance and have strong interpretability. In comparison, ANN is brutal in explaining the contribution of each feature.
- (2)
- Fast training speed: RF and XGBoost can be processed in parallel, and the training speed is fast. In contrast, ANN requires iterative optimization of the backpropagation algorithm, and the training speed is relatively slow.
- (3)
- Strong robustness: RF and XGBoost are relatively robust to outliers and noise and are not easily affected by extreme values. In contrast, ANN is sensitive to outliers and noise and requires data preprocessing.
- (4)
- Can handle high-dimensional data: Random Forest and XGBoost can handle high-dimensional data very well, and the problem of dimensionality disaster is challenging. In contrast, ANN requires dimensionality reduction when dealing with high-dimensional data.

#### 4.2.1. RF Model

#### 4.2.2. Xgboost Model

#### 4.2.3. Hyperparameter Optimization (HPO)

#### 4.3. Black-Box Model Cleaning Method

## 5. Results

#### 5.1. HPO Results

#### 5.2. Model Evaluation Results

^{2}and the small value of MAE show that the confidence level is high and equals 0.99.

#### 5.3. Model Prediction Results

## 6. Discussion and Future Work

## 7. Conclusions

^{2}is less than 0.01, showing that the benefit by hyperparameter optimization is insignificant. Additionally, there is a similarity between the RF and XGBoost models; both are based on the decision tree. We will continue to build more models to explore different results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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V (knots) | Wind${}_{\mathit{s}}$ (knots) | Wind${}_{\mathit{d}}$ (${}^{\circ}$) | D${}_{\mathit{a}}$ (m) | D${}_{\mathit{f}}$ (m) | T (m) | Wave${}_{\mathit{h}}$ (m) | Wave${}_{\mathit{d}}$ (${}^{\circ}$) | FCR (tons/h) | |
---|---|---|---|---|---|---|---|---|---|

Count | 147,845 | 147,845 | 147,845 | 147,845 | 147,845 | 147,845 | 147,845 | 147,845 | 147,845 |

Mean | 13.14 | 9.42 | 84.49 | 18.80 | 17.23 | 1.57 | 0.37 | 192.57 | 2.78 |

Std | 1.92 | 5.23 | 44.46 | 2.92 | 4.22 | 1.44 | 0.25 | 30.53 | 0.97 |

Min | 10.00 | 0.00 | 0.00 | 11.83 | 7.17 | −0.64 | 0.12 | 133.20 | 0.81 |

25% | 11.5 | 5.40 | 54.70 | 19.11 | 16.30 | 0.65 | 0.17 | 200.02 | 2.03 |

50% | 13.20 | 8.60 | 79.50 | 20.30 | 19.46 | 0.91 | 0.24 | 204.49 | 2.77 |

75% | 14.90 | 11.90 | 112.00 | 20.40 | 19.69 | 2.29 | 0.35 | 211.72 | 3.51 |

Max | 16.80 | 25.60 | 180.00 | 20.70 | 20.03 | 5.55 | 0.85 | 218.20 | 4.98 |

Models | Execution Time | Resources |
---|---|---|

RF | 9.15 h | 8-core M2 CPU, 24GB RAM |

XGBoost | 15.04 h |

XGBoost | RF | |
---|---|---|

MSE | 0.0022 | 0.0074 |

RMSE | 0.0467 | 0.0861 |

MAE | 0.0308 | 0.0494 |

MAPE | 1.3268 | 2.0949 |

${R}^{2}$ | 0.9977 | 0.9922 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|

RF | 0.9923 | 0.9922 | 0.9920 | 0.9926 | 0.9924 | 0.9923 | 0.9918 | 0.9922 | 0.9921 | 0.9917 |

XGBoost | 0.9976 | 0.9978 | 0.9978 | 0.9976 | 0.9976 | 0.9976 | 0.9978 | 0.9975 | 0.9979 | 0.9979 |

XGBoost | |
---|---|

MSE | 0.0018 |

RMSE | 0.0423 |

MAE | 0.0296 |

MAPE | 1.7461 |

${R}^{2}$ | 0.9954 |

No. | T (m) | ${\mathbf{D}}_{\mathit{a}}$ (m) | ${\mathbf{D}}_{\mathit{f}}$ (m) | ${\mathbf{Wind}}_{\mathit{d}}$ (${}^{\circ}$) | ${\mathbf{Wind}}_{\mathit{s}}$ (knots) | ${\mathbf{Wave}}_{\mathit{h}}$ (m) | ${\mathbf{Wave}}_{\mathit{d}}$ (${}^{\circ}$) |
---|---|---|---|---|---|---|---|

1 | 4 | 22 | 18 | 56.3 | 5.8 | 0.12 | 30.7 |

2 | 4 | 22 | 18 | 128.5 | 13.6 | 0.54 | 45.9 |

3 | 4 | 22 | 18 | 48.9 | 12.1 | 0.20 | 78.9 |

4 | 4 | 22 | 18 | 162.3 | 14.6 | 0.25 | 59.6 |

5 | 4 | 22 | 18 | 146.3 | 18.3 | 0.38 | 153.9 |

6 | 4 | 22 | 18 | 65.4 | 21.3 | 0.48 | 153.8 |

7 | 4 | 22 | 18 | 175.3 | 22.9 | 0.67 | 198.8 |

8 | 4 | 22 | 18 | 170.3 | 16.3 | 0.83 | 37.3 |

9 | 4 | 22 | 18 | 140.3 | 15.2 | 0.79 | 67.2 |

10 | 4 | 22 | 18 | 100.6 | 10.6 | 0.25 | 87.9 |

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**MDPI and ACS Style**

Xie, X.; Sun, B.; Li, X.; Olsson, T.; Maleki, N.; Ahlgren, F.
Fuel Consumption Prediction Models Based on Machine Learning and Mathematical Methods. *J. Mar. Sci. Eng.* **2023**, *11*, 738.
https://doi.org/10.3390/jmse11040738

**AMA Style**

Xie X, Sun B, Li X, Olsson T, Maleki N, Ahlgren F.
Fuel Consumption Prediction Models Based on Machine Learning and Mathematical Methods. *Journal of Marine Science and Engineering*. 2023; 11(4):738.
https://doi.org/10.3390/jmse11040738

**Chicago/Turabian Style**

Xie, Xianwei, Baozhi Sun, Xiaohe Li, Tobias Olsson, Neda Maleki, and Fredrik Ahlgren.
2023. "Fuel Consumption Prediction Models Based on Machine Learning and Mathematical Methods" *Journal of Marine Science and Engineering* 11, no. 4: 738.
https://doi.org/10.3390/jmse11040738