# A Data-Light and Trajectory-Based Machine Learning Approach for the Online Prediction of Flight Time of Arrival

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Contributions of the Paper

#### 1.3. Organization of the Paper

## 2. Methodology

#### 2.1. Definition

**Definition 1:**

#### 2.2. Overall Framework for ETA_TAB and ELDT Prediction

#### 2.3. Trajectory Point Reconstruction and Matching

#### 2.3.1. Trajectory Point Reconstruction

#### 2.3.2. Trajectory Matching

#### 2.4. Trajectory Prediction

#### 2.4.1. LSTM Neural Network

#### 2.4.2. LSTM Network for Trajectory Prediction

#### 2.5. Ground Speed Prediction

Algorithm1: GBM Algorithm | |

1. | $\mathrm{Initialization}:{F}_{0}\left(\chi \right)={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{\rho}\sum _{i=1}^{N}L({y}_{i},\rho )$ |

2. | $\mathrm{For}m=1$$\mathrm{to}M$ do: |

3. | $\hspace{1em}\hspace{1em}{a}_{m}=\underset{a,\mathit{\beta}}{\mathrm{argmin}}\sum _{i=1}^{N}{[-{\left[\frac{\partial L\left(y,F\left({\chi}_{i}\right)\right)}{\partial F\left({\chi}_{i}\right)}\right]}_{F\left(\chi \right)={F}_{m-1}\left(\chi \right)}-\beta h({\chi}_{i};a)]}^{2}$ |

4. | $\hspace{1em}\hspace{1em}{\rho}_{m}=\underset{\rho}{\mathrm{argmin}}L(y,{F}_{m-1}\left(\chi \right)+\rho h(\chi ;{a}_{m}\left)\right)$ |

5. | $\hspace{1em}\hspace{1em}{F}_{m}\left(\chi \right)={F}_{m-1}\left(\chi \right)+v{\rho}_{m}h(\chi ;{a}_{m})$ |

6. | End for |

#### 2.6. ETA_TAB and ELDT Prediction

## 3. Demonstration of the Prediction Approach

#### 3.1. Data

#### 3.2. Trajectory Matching

#### 3.3. Trajectory Prediction

#### 3.4. Ground Speed Prediction

#### 3.5. ETA_TAB and ELDT Prediction

#### 3.5.1. ETA_TAB Prediction

#### 3.5.2. ELDT Prediction

#### Sampling Terminal Approach Time

#### Results

## 4. Summary and Further Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Ground Speeds of FLT1 and FLT2

## References

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**Figure 3.**Illustration of how to identify close enough points on a historical flight trajectory (${T}_{2}$) when calculating the similarity score of trajectory ${T}_{2}$ to trajectory ${T}_{1}$.

**Figure 4.**The architecture of LSTM neural networks [33].

**Figure 5.**The architecture of the proposed trajectory prediction approach using LSTM: (

**a**) training; (

**b**) inference.

**Figure 6.**Illustration of the four steps performed in the construction layer to generate $\left({\widehat{\lambda}}_{c+1},{\widehat{\phi}}_{c+1}\right)$ using $({\widehat{\lambda}}_{c+1}^{0},{\widehat{\phi}}_{c+1}^{0})$ from the dense layer, (${\lambda}_{c},{\phi}_{c}$), and the best-matched historical trajectory.

**Figure 8.**Illustration of when the iterative trajectory point prediction stops (when a trajectory point reaches the boundary of the destination terminal airspace or falls within the boundary).

**Figure 12.**Flight trajectory records for the median- and long-haul airport pairs: (

**a**) flight trajectory records of UA1497 from DEN-SFO from 1 January to 30 June 2020; (

**b**) flight trajectory records of UA2166 from ORD-SFO from 1 January to 30 June 2020.

**Figure 13.**The current flight’s trajectory (in solid pink) with the best-matched historical flight trajectory (in dashed blue) for FLT1.

**Figure 14.**The current flight’s trajectory (in solid pink) with the best-matched historical flight trajectory (in dashed blue) for FLT2.

**Figure 17.**Distributions of the distance between the predicted and actual trajectory points (the two points have the same order position in their respective trajectory point sequences) for FLT1 (left) and FLT2 (right), predicting at 25th, 50th, and 75th GCD percentiles.

**Figure 18.**Boxplot of ETA_TABs using the proposed approach compared with two baseline point estimates for 30 randomly picked flights on DEN-SFO, at 25th, 50th, and 75th percentile GCD.

**Figure 19.**Boxplot of ETA_TABs using the proposed approach compared with two baseline point estimates for 30 randomly picked flights on ORD-SFO, at 25th, 50th, and 75th percentile GCD.

**Figure 20.**Illustration of flight terminal approach times: (

**a**) terminal approach time of B777 series flights for DEN-SFO with entry from east of SFO TRACON; (

**b**) terminal approach time of A320 series flights for ORD-SFO with entry from east of SFO TRACON.

Airport Pair | Current Flight | Current Position | Best-Matched Historical Flight | Similarity Score | Matching Time (min) |
---|---|---|---|---|---|

DEN-SFO | FLT1 (UA1497(4/20/20)) | 25th GCD percentile | UA1497 (4/19/20) | 1.00 | 0.01 |

50th GCD percentile | UA1497 (4/19/20) | 1.00 | 0.04 | ||

75th GCD percentile | UA1497 (4/19/20) | 1.00 | 0.09 | ||

ORD-SFO | FLT2 (UA2166(5/9/20)) | 25th GCD percentile | UA2166 (3/3/20) | 1.00 | 0.28 |

50th GCD percentile | UA2166 (1/19/20) | 1.00 | 0.66 | ||

75th GCD percentile | UA2166 (1/19/20) | 0.99 | 1.05 |

Methods | Features | Training Data | Validation Data | ||
---|---|---|---|---|---|

DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | ||

LSTM | Longitude (degree) | 1.04 × 10^{−4} | 3.22 × 10^{−4} | 2.52 × 10^{−4} | 4.09 × 10^{−4} |

Latitude (degree) | 2.60 × 10^{−5} | 6.86 × 10^{−5} | 1.36 × 10^{−5} | 4.43 × 10^{−4} | |

${d}_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}^{\mathrm{G}\mathrm{C}\mathrm{D}}$ (nm) | 0.33 | 0.84 | 0.10 | 1.37 | |

RNN | Longitude (degree) | 4.18 × 10^{−3} | 5.11 × 10^{−3} | 1.42 × 10^{−3} | 7.06 × 10^{−3} |

Latitude (degree) | 6.31 × 10^{−3} | 4.10 × 10^{−3} | 3.06 × 10^{−3} | 3.93 × 10^{−3} | |

${d}_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}^{\mathrm{G}\mathrm{C}\mathrm{D}}$ (nm) | 1.71 | 4.29 | 1.30 | 3.35 | |

GRU | Longitude (degree) | 2.87 × 10^{−4} | 7.68 × 10^{−4} | 2.93 × 10^{−4} | 8.14 × 10^{−4} |

Latitude (degree) | 5.29 × 10^{−5} | 3.46 × 10^{−4} | 2.12 × 10^{−4} | 7.09 × 10^{−4} | |

${d}_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}}^{\mathrm{G}\mathrm{C}\mathrm{D}}$ (nm) | 0.85 | 1.59 | 0.79 | 1.41 |

**Table 3.**Averaged RMSE (in nm/h) overall multiple trajectory predictions of the predicted ground speed for points on the predicted remaining trajectory for FLT1 and FLT 2 at different GCD percentiles.

GBM | Random Forest | |||
---|---|---|---|---|

FLT1 | FLT2 | FLT1 | FLT2 | |

25th GCD percentile | 7.36 | 10.01 | 7.63 | 12.04 |

50th GCD percentile | 9.41 | 12.98 | 9.42 | 13.97 |

75th GCD percentile | 9.83 | 13.56 | 10.4 | 15.29 |

**Table 4.**$\mathsf{\Delta}{t}_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}$ of FLT1 and FLT2 using the proposed approach and two baseline estimates, at the three GCD percentiles.

25th GCD Percentile | 50th GCD Percentile | 75th GCD Percentile | ||||
---|---|---|---|---|---|---|

DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | |

Proposed approach | −0.43 | −0.30 | 0.28 | 2.75 | −0.91 | 0.31 |

Best match | 6.07 | −27.83 | 1.62 | −12.63 | 7.97 | −12.63 |

Average | 11.17 | 12.54 | 11.17 | 12.54 | 11.17 | 12.54 |

**Table 5.**Root mean square of $\mathsf{\Delta}{t}_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}$ values of 30 randomly picked flights for each of the two ODs using the proposed approach and two alternative estimates, at the three GCD percentiles.

25th GCD Percentile | 50th GCD Percentile | 75th GCD Percentile | ||||
---|---|---|---|---|---|---|

DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | |

Proposed approach | 1.54 | 2.02 | 1.36 | 1.97 | 1.03 | 1.79 |

Best match | 9.55 | 9.84 | 9.55 | 8.37 | 9.38 | 8.58 |

Average | 7.88 | 9.68 | 7.88 | 9.68 | 7.88 | 9.68 |

**Table 6.**$\mathsf{\Delta}{t}_{\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}}$ based on ELDT from the proposed approach and four alternative estimates, for FLT1 and FLT2 at the three GCD percentiles.

25th GCD Percentile | 50th GCD Percentile | 75th GCD Percentile | ||||
---|---|---|---|---|---|---|

DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | |

Proposed approach—LSTM | −0.98 | −3.74 | 0.04 | −0.79 | −0.83 | 1.27 |

Best match | 6.08 | −27.70 | −0.20 | −10.07 | 7.15 | −10.07 |

Average | 11.17 | 16.15 | 11.17 | 16.15 | 11.17 | 16.15 |

Proposed approach—RNN | −1.23 | −4.65 | 0.54 | −1.67 | −1.65 | 1.93 |

Proposed approach—GRU | −1.09 | −4.02 | 0.06 | −1.04 | −0.98 | 1.31 |

Based on en-route flight time from the flight plan | 0 | 7.00 | 0 | 7.00 | 0 | 7.00 |

Using the scheduled landing time from the flight plan | 2.10 | −3.00 | 2.10 | −3.00 | 2.10 | −3.00 |

**Table 7.**Root mean square of $\mathsf{\Delta}{\mathrm{t}}_{\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}}$ values of 30 randomly picked flights for each airport pair based on ELDT from the proposed approach and four alternative estimates at the three GCD percentiles.

25th GCD Percentile | 50th GCD Percentile | 75th GCD Percentile | ||||
---|---|---|---|---|---|---|

DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | DEN-SFO | ORD-SFO | |

Proposed approach-LSTM | 2.78 | 4.76 | 2.65 | 4.54 | 2.08 | 3.96 |

Best match | 9.23 | 15.04 | 10.68 | 21.36 | 8.69 | 16.54 |

Average | 7.46 | 9.09 | 7.46 | 9.09 | 7.46 | 9.09 |

Proposed approach-RNN | 3.62 | 6.01 | 3.95 | 5.65 | 2.97 | 5.64 |

Proposed approach-GRU | 3.21 | 5.14 | 3.32 | 5.02 | 2.41 | 4.72 |

Based on en-route flight time in the flight plan | 3.02 | 7.34 | 3.02 | 7.34 | 3.02 | 7.34 |

Using the scheduled landing time in the flight plan | 28.53 | 10.16 | 28.53 | 10.16 | 28.53 | 10.16 |

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## Share and Cite

**MDPI and ACS Style**

Zheng, Z.; Zou, B.; Wei, W.; Tian, W.
A Data-Light and Trajectory-Based Machine Learning Approach for the Online Prediction of Flight Time of Arrival. *Aerospace* **2023**, *10*, 675.
https://doi.org/10.3390/aerospace10080675

**AMA Style**

Zheng Z, Zou B, Wei W, Tian W.
A Data-Light and Trajectory-Based Machine Learning Approach for the Online Prediction of Flight Time of Arrival. *Aerospace*. 2023; 10(8):675.
https://doi.org/10.3390/aerospace10080675

**Chicago/Turabian Style**

Zheng, Zhe, Bo Zou, Wenbin Wei, and Wen Tian.
2023. "A Data-Light and Trajectory-Based Machine Learning Approach for the Online Prediction of Flight Time of Arrival" *Aerospace* 10, no. 8: 675.
https://doi.org/10.3390/aerospace10080675