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A Geometric Approach to Study Aircraft Trajectories: The Benefits of OpenSky Network ADS-B Data^{ †}

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## Abstract

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## 1. Introduction

## 2. What Is the Shape of a Trajectory?

- The choice of landmark is very subjective. How many landmarks are relevant to summarize the shape of an aircraft trajectory? Should this number depend on the origin–destination we consider? Would landmarks based on flight phases be enough to capture the shape of a trajectory?
- Scientific landmarks are not available in raw data. Both Eurocontrol and OpenSky Network data sources do not include expert knowledge.

## 3. Trajectories as Parameterized Curves in ${\mathbb{R}}^{3}$

#### 3.1. Interpolation: From Raw Data to Curves

#### 3.2. The Square-Root Velocity Function (SRVF) Captures Shape

#### 3.3. How to Deal with Rotation and Re-Parameterization

#### 3.4. Distance between Two Shapes, Geodesic Path

## 4. Application

#### 4.1. Data

#### 4.1.1. R&D Eurocontrol

#### 4.1.2. Retrieving ADS-B Flights Thanks to OpenSky Network

#### 4.2. Interpolation

#### 4.3. How to Compute the Geodesic Path in Practice

`fdasrvf`package maintained by J. Derek Tucker. Computation of the geodesic path for a given flight is illustrated in Figure 3.

#### 4.4. Geodesic Distance

#### 4.5. Hierarchical Clustering

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sun, J.; Basora, L.; Olive, X.; Strohmeier, M.; Schafer, M.; Martinovic, I.; Lenders, V. OpenSky Report 2022: Evaluating Aviation Emissions Using Crowdsourced Open Flight Data. In Proceedings of the 2022 IEEE/AIAA 41st Digital Avionics Systems Conference (DASC), Portsmouth, VA, USA, 18–22 September 2022; p. 8. [Google Scholar]
- Krauth, T.; Morio, J.; Olive, X.; Figuet, B.; Monstein, R. Synthetic Aircraft Trajectories Generated with Multivariate Density Models. Eng. Proc.
**2021**, 13, 7. [Google Scholar] [CrossRef] - Olive, X.; Sun, J.; Lafage, A.; Basora, L. Detecting Events in Aircraft Trajectories: Rule-Based and Data-Driven Approaches. Proceedings
**2020**, 59, 8. [Google Scholar] [CrossRef] - Ramsay, J.O.; Silverman, B.W. Functional Data Analysis, 2nd ed.; Springer series in statistics; Springer: New York, NY, USA, 2005. [Google Scholar]
- Ferraty, F.; Vieu, P. Nonparametric Functional Data Analysis: Theory and Practice; Springer: New York, NY, USA, 2006. [Google Scholar] [CrossRef]
- Puechmorel, S.; Delahaye, D. 4D trajectories: A functional data perspective. In Proceedings of the 2007 IEEE/AIAA 26th Digital Avionics Systems Conference, Dallas, TX, USA, 21–25 October 2007. [Google Scholar] [CrossRef] [Green Version]
- Nicol, F. Statistical Analysis of Aircraft Trajectories: A Functional Data Analysis Approach. In Proceedings of the Alldata 2017, The Third International Conference on Big Data, Small Data, Linked Data and Open Data, Venice, Italy, 23–27 April 2017; p. 51. [Google Scholar]
- Jarry, G.; Delahaye, D.; Nicol, F.; Feron, E. Aircraft atypical approach detection using functional principal component analysis. J. Air Transp. Manag.
**2020**, 84, 101787. [Google Scholar] [CrossRef] [Green Version] - Su, J.; Kurtek, S.; Klassen, E.; Srivastava, A. Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance. Ann. Appl. Stat.
**2014**, 8, 530–552. [Google Scholar] [CrossRef] [Green Version] - Kendall, D.G. Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces. Bull. Lond. Math. Soc.
**1984**, 16, 81–121. [Google Scholar] [CrossRef] [Green Version] - Dryden, I.L.; Mardia, K.V. Statistical Shape Analysis; Wiley: Hoboken, NJ, USA, 1998. [Google Scholar]
- Kendall, D.G. The diffusion of shape. Adv. Appl. Probab.
**1977**, 9, 428–430. [Google Scholar] [CrossRef] - Mardia, K.V.; Dryden, I.L. The Statistical Analysis of Shape Data. Biometrika
**1989**, 76, 271–281. [Google Scholar] [CrossRef] - O’Higgins, P.; Dryden, I.L. Sexual dimorphism in hominoids: Further studies of craniofacial shape differences in Pan, Gorilla and Pongo. J. Hum. Evol.
**1993**, 24, 183–205. [Google Scholar] [CrossRef] - Dryden, I.L.; Hirst, J.D.; Melville, J.L. Statistical Analysis of Unlabeled Point Sets: Comparing Molecules in Chemoinformatics. Biometrics
**2007**, 63, 237–251. [Google Scholar] [CrossRef] [PubMed] - Czogiel, I.; Dryden, I.L.; Brignell, C.J. Bayesian matching of unlabeled marked point sets using random fields, with an application to molecular alignment. Ann. Appl. Stat.
**2011**, 5, 2603–2629. [Google Scholar] [CrossRef] [Green Version] - Sun, J.; Ellerbroek, J.; Hoekstra, J.M. Large-Scale Flight Phase Identification from ADS-B Data Using Machine Learning Methods. In Proceedings of the 7th International Conference on Research in Air Transportation, Paris, France, 14–16 September 2016. [Google Scholar]
- Liu, D.; Xiao, N.; Zhang, Y.; Peng, X. Unsupervised Flight Phase Recognition with Flight Data Clustering based on GMM. In Proceedings of the 2020 IEEE International Instrumentation and Measurement Technology Conference (I2MTC), Dubrovnik, Croatia, 25–28 May 2020; pp. 1–6. [Google Scholar] [CrossRef]
- Srivastava, A.; Klassen, E.; Joshi, S.H.; Jermyn, I.H. Shape Analysis of Elastic Curves in Euclidean Spaces. IEEE Trans. Pattern Anal. Mach. Intell.
**2011**, 33, 1415–1428. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, J.T. Analysis of Variance for Functional Data, 1st ed.; Chapman and Hall/CRC: London, UK, 2013. [Google Scholar]
- Srivastava, A.; Klassen, E.P. Functional and Shape Data Analysis, 1st ed.; Springer: New York, NY, USA, 2016. [Google Scholar]
- Tucker, J.D.; Wu, W.; Srivastava, A. Generative models for functional data using phase and amplitude separation. Comput. Stat. Data Anal.
**2013**, 61, 50–66. [Google Scholar] [CrossRef]

**Figure 1.**Linear interpolation of Eurocontrol trajectories (top) and ADS-B (bottom) versions for the flights departing from Toulouse–Blagnac (LFBO) and landing at Paris–Orly (LFPO) in 2019.

**Figure 2.**Zoom on Paris–Orly (LFPO), where runway-displaced thresholds (DTHR) are indicated by red dots. Linear interpolation of ADS-B trajectories (left) and Eurocontrol versions (right) for the flights departing from Toulouse–Blagnac (LFBO) and landing at Paris–Orly (LFPO) in 2019.

**Figure 3.**Linear interpolation of a given flight between Toulouse–Blagnac (LFBO) and Paris–Orly (LFPO) in 2019. Both its ADS-B and Eurocontrol versions are represented.

**Figure 4.**One possible representation of the shortest path from the Eurocontrol version to the ADS-B version in the quotient space $\mathcal{S}$ thanks to the computation of $\widehat{\alpha}(\tau )$ for 10 values of $\tau $.

**Figure 5.**Histogram of the geodesic distances between Eurocontrol and ADS-B versions of the flights departing from Toulouse–Blagnac (LFBO) and landing at Paris–Orly (LFPO) in 2019.

**Figure 6.**Two clusters of trajectories (orange and blue) based on the geodesic distance. The sample is of size 100 and is randomly drawn from flights going from Toulouse–Blagnac (LFBO) to Paris–Orly (LFPO) in 2019.

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**MDPI and ACS Style**

Perrichon, R.; Gendre, X.; Klein, T.
A Geometric Approach to Study Aircraft Trajectories: The Benefits of OpenSky Network ADS-B Data. *Eng. Proc.* **2022**, *28*, 6.
https://doi.org/10.3390/engproc2022028006

**AMA Style**

Perrichon R, Gendre X, Klein T.
A Geometric Approach to Study Aircraft Trajectories: The Benefits of OpenSky Network ADS-B Data. *Engineering Proceedings*. 2022; 28(1):6.
https://doi.org/10.3390/engproc2022028006

**Chicago/Turabian Style**

Perrichon, Rémi, Xavier Gendre, and Thierry Klein.
2022. "A Geometric Approach to Study Aircraft Trajectories: The Benefits of OpenSky Network ADS-B Data" *Engineering Proceedings* 28, no. 1: 6.
https://doi.org/10.3390/engproc2022028006