{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T06:56:53Z","timestamp":1780642613558,"version":"3.54.1"},"reference-count":43,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,11]],"date-time":"2020-11-11T00:00:00Z","timestamp":1605052800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U1833115"],"award-info":[{"award-number":["U1833115"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this work, a deep Gaussian process (DGP) based framework is proposed to improve the accuracy of predicting flight trajectory in air traffic research, which is further applied to implement a probabilistic conflict detection algorithm. The Gaussian distribution is applied to serve as the probabilistic representation for illustrating the transition patterns of the flight trajectory, based on which a stochastic process is generated to build the temporal correlations among flight positions, i.e., Gaussian process (GP). Furthermore, to deal with the flight maneuverability of performing controller\u2019s instructions, a hierarchical neural network architecture is proposed to improve the modeling representation for nonlinear features. Thanks to the intrinsic mechanism of the GP regression, the DGP model has the ability of predicting both the deterministic nominal flight trajectory (NFT) and its confidence interval (CI), denoting by the mean and standard deviation of the prediction sequence, respectively. The CI subjects to a Gaussian distribution, which lays the data foundation of the probabilistic conflict detection. Experimental results on real data show that the proposed trajectory prediction approach achieves higher prediction accuracy compared to other baselines. Moreover, the conflict detection approach is also validated by a obtaining lower false alarm and more prewarning time.<\/jats:p>","DOI":"10.3390\/a13110293","type":"journal-article","created":{"date-parts":[[2020,11,11]],"date-time":"2020-11-11T19:08:28Z","timestamp":1605121708000},"page":"293","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":44,"title":["A Deep Gaussian Process-Based Flight Trajectory Prediction Approach and Its Application on Conflict Detection"],"prefix":"10.3390","volume":"13","author":[{"given":"Zhengmao","family":"Chen","sequence":"first","affiliation":[{"name":"College of Computer Science, Sichuan University, Chengdu 610065, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongyue","family":"Guo","sequence":"additional","affiliation":[{"name":"College of Computer Science, Sichuan University, Chengdu 610065, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7194-5023","authenticated-orcid":false,"given":"Yi","family":"Lin","sequence":"additional","affiliation":[{"name":"College of Computer Science, Sichuan University, Chengdu 610065, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,11]]},"reference":[{"key":"ref_1","unstructured":"Lin, Y., Guo, D., Zhang, J., Chen, Z., and Yang, B. (2020). A Unified Framework for Multilingual Speech Recognition in Air Traffic Control Systems. IEEE Trans. Neural Networks Learn. Syst., 1\u201313."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"105113","DOI":"10.1016\/j.ast.2019.04.021","article-title":"Deep learning based short-term air traffic flow prediction considering temporal\u2013spatial correlation","volume":"93","author":"Lin","year":"2019","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4572","DOI":"10.1109\/TITS.2019.2940992","article-title":"A Real-Time ATC Safety Monitoring Framework Using a Deep Learning Approach","volume":"21","author":"Lin","year":"2020","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1631\/FITEE.1700224","article-title":"An algorithm for trajectory prediction of flight plan based on relative motion between positions","volume":"19","author":"Lin","year":"2018","journal-title":"Front. Inf. Technol. Electron. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"De Leege, A., van Paassen, M., and Mulder, M. (2013, January 19\u201322). A Machine Learning Approach to Trajectory Prediction. Proceedings of the AIAA Guidance, Navigation, and Control (GNC) Conference, Boston, MA, USA.","DOI":"10.2514\/6.2013-4782"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.trc.2016.10.018","article-title":"Prediction of aircraft performances based on data collected by air traffic control centers","volume":"73","author":"Hrastovec","year":"2016","journal-title":"Transp. Res. Part C Emerg. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Shi, Z., Xu, M., Pan, Q., Yan, B., and Zhang, H. (2018, January 8\u201313). LSTM-based Flight Trajectory Prediction. Proceedings of the 2018 International Joint Conference on Neural Networks, Rio de Janeiro, Brazil.","DOI":"10.1109\/IJCNN.2018.8489734"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Tang, X., Zhou, L., Shen, Z., and Tang, M. (2015, January 24\u201327). 4D Trajectory Prediction of Aircraft Taxiing Based on Fitting Velocity Profile. Proceedings of the 2015 Cota International Conference Transportation Professionals, Beijing, China.","DOI":"10.1061\/9780784479292.001"},{"key":"ref_9","unstructured":"Chen, Z. (2012). Theory and Method of Airspace Management, Science Press. [1st ed.]."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hashemi, S.M., Botez, R.M., and Grigorie, T.L. (2020). New Reliability Studies of Data-Driven Aircraft Trajectory Prediction. Aerospace, 7.","DOI":"10.3390\/aerospace7100145"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Prevost, C.G., Desbiens, A., and Gagnon, E. (2007, January 9\u201313). Extended Kalman Filter for State Estimation and Trajectory Prediction of a Moving Object Detected by an Unmanned Aerial Vehicle. Proceedings of the 2007 American Control Conference, New York, NY, USA.","DOI":"10.1109\/ACC.2007.4282823"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Yan, H., Huang, G., Wang, H., and Shu, R. (2013, January 7\u20138). Application of unscented kalman filter for flying target tracking. Proceedings of the 2013 International Conference on Information Science and Cloud Computing, Guangzhou, China.","DOI":"10.1109\/ISCC.2013.10"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Chan, Y.T., Hu, A.G.C., and Plant, J.B. (1979). A Kalman Filter Based Tracking Scheme with Input Estimation. IEEE Trans. Aerosp. Electron. Syst., 237\u2013244.","DOI":"10.1109\/TAES.1979.308710"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Schultz, C., Thipphavong, D., and Erzberger, H. (2012, January 13\u201316). Adaptive Trajectory Prediction Algorithm for Climbing Flights. Proceedings of the AIAA Guidance, Navigation, and Control Conference, Minneapolis, MN, USA.","DOI":"10.2514\/6.2012-4931"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"15","DOI":"10.2514\/1.58508","article-title":"Adaptive Algorithm to Improve Trajectory Prediction Accuracy of Climbing Aircraft","volume":"36","author":"Thipphavong","year":"2013","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_16","unstructured":"Wang, Z., and Delahaye, D. (2017). Short-Term 4D Trajectory Prediction Using Machine Learning Methods, Seventh SESAR Innovation Days 2017; SESAR Project."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"105628","DOI":"10.1016\/j.aap.2020.105628","article-title":"Automated traffic incident detection with a smaller dataset based on generative adversarial networks","volume":"144","author":"Lin","year":"2020","journal-title":"Accid. Anal. Prev."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.ast.2018.09.020","article-title":"Probabilistic aircraft trajectory prediction in cruise flight considering ensemble wind forecasts","volume":"82","author":"Franco","year":"2018","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_19","unstructured":"Jeung, H., Shen, H.T., and Zhou, X. (2007, January 3\u20137). Mining Trajectory Patterns Using Hidden Markov Models. Proceedings of the International Conference on Data Warehouse Knowledge Discovery, Regensburg, Germany."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1338","DOI":"10.1109\/TIP.2009.2039664","article-title":"Trajectory classification using switched dynamical hidden markov models","volume":"19","author":"Nascimento","year":"2010","journal-title":"IEEE Trans. Image Process."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wiest, J., Hoffken, M., Kresel, U., and Dietmayer, K. (2012, January 3\u20137). Probabilistic trajectory prediction with Gaussian mixture models. Proceedings of the 2012 IEEE Intelligent Vehicles Symposium, Alcala de Henares, Spain.","DOI":"10.1109\/IVS.2012.6232277"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.2514\/1.G002383","article-title":"Probabilistic Approach to Conformance Monitoring Using Gaussian Processes","volume":"40","author":"Yan","year":"2017","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_23","first-page":"1545","article-title":"A 4D Trajectory Prediction Model Based on the BP Neural Network","volume":"29","author":"Wu","year":"2019","journal-title":"J. Intell. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"113246","DOI":"10.1016\/j.dss.2020.113246","article-title":"Bayesian neural networks for flight trajectory prediction and safety assessment","volume":"131","author":"Zhang","year":"2020","journal-title":"Decis. Support Syst."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Altche, F., and de la Fortelle, A. (2017, January 16\u201319). An LSTM network for highway trajectory prediction. Proceedings of the 2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), Yokohama, Japan.","DOI":"10.1109\/ITSC.2017.8317913"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Nivison, S.A., and Khargonekar, P.P. (2017, January 24\u201326). Development of a robust deep recurrent neural network controller for flight applications. Proceedings of the 2017 American Control Conference (ACC), Seattle, WA, USA.","DOI":"10.23919\/ACC.2017.7963784"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Zhao, Z., Zeng, W., Quan, Z., Chen, M., and Yang, Z. (2019, January 6\u20138). Aircraft trajectory prediction using deep long short-term memory networks. Proceedings of the CICTP 2019 Transportation China\u201419th COTA International Conference Transports Professtional, Nanjing, China.","DOI":"10.1061\/9780784482292.012"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Pang, Y., Xu, N., and Liu, Y. (2019, January 23\u201326). Aircraft Trajectory Prediction using LSTM Neural Network with Embedded Convolutional Layer. Proceedings of the Annual Conference of the PHM Society, Scottsdale, AZ, USA.","DOI":"10.36001\/phmconf.2019.v11i1.849"},{"key":"ref_29","unstructured":"Liu, Y., and Hansen, M. (2019). Predicting Aircraft Trajectories: A Deep Generative Convolutional Recurrent Neural Networks Approach. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Hoel, P.G., Port, S.C., Stone, C.J., and Holley, R. (1973). Introduction to Stochastic Processes. IEEE Trans. Syst. Man. Cybern., 533.","DOI":"10.1109\/TSMC.1973.4309295"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1363","DOI":"10.1007\/s10994-018-5723-3","article-title":"Deep Gaussian Process autoencoders for novelty detection","volume":"107","author":"Domingues","year":"2018","journal-title":"Mach. Learn."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"588","DOI":"10.2514\/2.4081","article-title":"Conflict Probability for Free Flight","volume":"20","author":"Paielli","year":"1997","journal-title":"J. Guid. Control. Dyn."},{"key":"ref_33","unstructured":"Damianou, A.C., and Lawrence, N.D. (May, January 29). Deep Gaussian Processes. Proceedings of the 16th International Conference Artificial Intelligence and Statistics, Scottsdale, AZ, USA."},{"key":"ref_34","unstructured":"Thang, B., Daniel, H.-L., Yingzhen, L., Jos\u00e9, H.-L., and Richard, T. (2016, January 20\u201322). Deep Gaussian Processes for Regression using Approximate Expectation Propagation. Proceedings of the International Conference on Machine Learning, New York, NY, USA."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1142\/S0129065704001899","article-title":"Gaussian Processes for Machine Learning","volume":"14","author":"Seeger","year":"2004","journal-title":"Int. J. Neural Syst."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Rasmussen, C.E. (2004). Gaussian Processes in Machine Learning. Summer School on Machine Learning, Springer.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"ref_37","first-page":"243","article-title":"Learning for larger datasets with the Gaussian process latent variable model","volume":"2","author":"Lawrence","year":"2007","journal-title":"J. Mach. Learn. Res. Proc. Track."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Zheng, Y. (2011). Computing with Spatial Trajectories, Springer.","DOI":"10.1007\/978-1-4614-1629-6"},{"key":"ref_39","first-page":"246","article-title":"Approach for 4-D Trajectory Management Based on HMM and Trajectory Similarity","volume":"27","author":"Lin","year":"2019","journal-title":"J. Mar. Sci. Technol."},{"key":"ref_40","unstructured":"Michalis, T., and Neil, L. (2010, January 13\u201315). Bayesian Gaussian Process Latent Variable Model. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, Sardinia, Italy."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1109\/6979.898224","article-title":"A Probabilistic Approach to Aircraft Conflict Detection","volume":"1","author":"Prandini","year":"2000","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.2307\/2965438","article-title":"Markov Chain Monte Carlo in Practice","volume":"92","author":"Kass","year":"1997","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_43","unstructured":"Zhang, Y., and Edgar, T.F. (2008, January 11\u201313). A robust Dynamic Time Warping algorithm for batch trajectory synchronization. Proceedings of the 2008 American Control Conference, Seattle, WA, USA."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/11\/293\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:32:01Z","timestamp":1760178721000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/11\/293"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,11]]},"references-count":43,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["a13110293"],"URL":"https:\/\/doi.org\/10.3390\/a13110293","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,11]]}}}