{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T01:48:12Z","timestamp":1783648092532,"version":"3.55.0"},"reference-count":43,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T00:00:00Z","timestamp":1614902400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010198","name":"Ministerio de Econom\u00eda, Industria y Competitividad, Gobierno de Espa\u00f1a","doi-asserted-by":"publisher","award":["TEC2017-88048-C2-2-R"],"award-info":[{"award-number":["TEC2017-88048-C2-2-R"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Certain difficulties in path forecasting and filtering problems are based in the initial hypothesis of estimation and filtering techniques. Common hypotheses include that the system can be modeled as linear, Markovian, Gaussian, or all at one time. Although, in many cases, there are strategies to tackle problems with approaches that show very good results, the associated engineering process can become highly complex, requiring a great deal of time or even becoming unapproachable. To have tools to tackle complex problems without starting from a previous hypothesis but to continue to solve classic challenges and sharpen the implementation of estimation and filtering systems is of high scientific interest. This paper addresses the forecast\u2013filter problem from deep learning paradigms with a neural network architecture inspired by natural language processing techniques and data structure. Unlike Kalman, this proposal performs the process of prediction and filtering in the same phase, while Kalman requires two phases. We propose three different study cases of incremental conceptual difficulty. The experimentation is divided into five parts: the standardization effect in raw data, proposal validation, filtering, loss of measurements (forecasting), and, finally, robustness. The results are compared with a Kalman filter, showing that the proposal is comparable in terms of the error within the linear case, with improved performance when facing non-linear systems.<\/jats:p>","DOI":"10.3390\/s21051805","type":"journal-article","created":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T05:03:15Z","timestamp":1614920595000},"page":"1805","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Forecasting Nonlinear Systems with LSTM: Analysis and Comparison with EKF"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3476-6261","authenticated-orcid":false,"given":"Juan Pedro","family":"Llerena Ca\u00f1a","sequence":"first","affiliation":[{"name":"Applied Artificial Intelligence Group (GIAA), Carlos III University of Madrid, 28270 Madrid, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1768-2688","authenticated-orcid":false,"given":"Jes\u00fas","family":"Garc\u00eda Herrero","sequence":"additional","affiliation":[{"name":"Applied Artificial Intelligence Group (GIAA), Carlos III University of Madrid, 28270 Madrid, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7484-7357","authenticated-orcid":false,"given":"Jos\u00e9 Manuel","family":"Molina L\u00f3pez","sequence":"additional","affiliation":[{"name":"Applied Artificial Intelligence Group (GIAA), Carlos III University of Madrid, 28270 Madrid, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,5]]},"reference":[{"key":"ref_1","unstructured":"\u00c5str\u00f6m, K., and Wittenmark, B. (2013). Computer-Controlled Systems: Theory and Design, Courier Corporation."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lewis, F.L., Vrabie, D.L., and Syrmos, V.L. (2012). Syrmos, \u201cOptimal Control\u201d, John Wiley & Sons.","DOI":"10.1002\/9781118122631"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.sigpro.2017.01.001","article-title":"Gaussian filters for parameter and state estimation: A general review of theory and recent trends","volume":"135","author":"Afshari","year":"2017","journal-title":"Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Musoff, H., and Zarchan, P. (2009). Fundamentals of Kalman Filtering: A Practical Approach, American Institute of Aeronautics and Astronautics. [3rd ed.].","DOI":"10.2514\/4.867200"},{"key":"ref_5","unstructured":"Welch, G., and Bishop, G. (August, January 30). An Introduction to the Kalman Filter. Proceedings of the SIGGRAPH, Boston, MA, USA."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1080\/00423114.2017.1337914","article-title":"Kalman and particle filtering methods for full vehicle and tyre identification","volume":"56","author":"Bogdanski","year":"2017","journal-title":"Veh. Syst. Dyn."},{"key":"ref_7","first-page":"1895","article-title":"Extended Kalman Filter Based Nonlinear Model Predictive Control","volume":"1","author":"Lee","year":"1993","journal-title":"Am. Control Conf."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.inffus.2020.06.003","article-title":"Real evaluation for designing sensor fusion in UAV platforms","volume":"63","author":"Molina","year":"2020","journal-title":"Inf. Fusion"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.jprocont.2011.10.006","article-title":"Robust stability of nonlinear model predictive control based on extended Kalman filter","volume":"22","author":"Huang","year":"2012","journal-title":"J. Process. Control."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1109\/JSEN.2018.2873357","article-title":"Kalman Filtering Framework-Based Real Time Target Tracking in Wireless Sensor Networks Using Generalized Regression Neural Networks","volume":"19","author":"Jondhale","year":"2018","journal-title":"IEEE Sens. J."},{"key":"ref_11","unstructured":"Wan, E., and Van Der Merwe, R. The unscented Kalman filter for nonlinear estimation. Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373)."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4961","DOI":"10.1080\/03610926.2015.1096390","article-title":"A New Robust Kalman Filter for Filtering the Microstructure Noise","volume":"46","author":"Tsai","year":"2016","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Alahi, A., Goel, K., Ramanathan, V., Robicquet, A., Fei-Fei, L., and Savarese, S. (2016, January 27\u201330). Social LSTM: Human Trajectory Prediction in Crowded Spaces. Proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, USA.","DOI":"10.1109\/CVPR.2016.110"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Coskun, H., Achilles, F., DiPietro, R., Navab, N., and Tombari, F. (2017, January 22\u201329). Long Short-Term Memory Kalman Filters: Recurrent Neural Estimators for Pose Regularization. Proceedings of the 2017 IEEE International Conference on Computer Vision (ICCV), Venice, Italy.","DOI":"10.1109\/ICCV.2017.589"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5915","DOI":"10.1016\/j.eswa.2015.03.023","article-title":"Artificial Intelligence techniques applied as estimator in chemical process systems\u2014A literature survey","volume":"42","author":"Ali","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"106682","DOI":"10.1016\/j.petrol.2019.106682","article-title":"Time-series well performance prediction based on Long Short-Term Memory (LSTM) neural network model","volume":"186","author":"Song","year":"2020","journal-title":"J. Pet. Sci. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Park, S.H., Kim, B., Kang, C.M., Chung, C.C., and Choi, J.W. (2018, January 26\u201330). Sequence-to-Sequence Prediction of Vehicle Trajectory via LSTM Encoder-Decoder Architecture. Proceedings of the 2018 IEEE Intelligent Vehicles Symposium (IV), Changshu, China.","DOI":"10.1109\/IVS.2018.8500658"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2922","DOI":"10.1016\/j.egypro.2019.01.952","article-title":"Short-Term Load Forecasts Using LSTM Networks","volume":"158","author":"Muzaffar","year":"2019","journal-title":"Energy Procedia"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/978-3-030-57802-2_15","article-title":"An Approach to Forecasting and Filtering Noise in Dynamic Systems Using LSTM Architectures","volume":"1268","author":"Llerena","year":"2021","journal-title":"Adv. Intell. Syst. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Hjorts\u00f8, M.A., and Wolenski, P. (2018). Some Ordinary Differential Equations. Linear Mathematical Models in Chemical Engineering. Proceedings of the NeurIPS 2018 Conference, World Scientific.","DOI":"10.1142\/11003"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.neucom.2016.11.099","article-title":"Modelling engineering systems using analytical and neural techniques: Hybridization","volume":"271","author":"Sierra","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.jcp.2019.06.056","article-title":"Deep learning of dynamics and signal-noise decomposition with time-stepping constraints","volume":"396","author":"Rudy","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_23","unstructured":"Raissi, M., Perdikaris, P., and Karniadakis, G.E. (2018). Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Arsene, C.T., Hankins, R., and Yin, H. (2019, January 2\u20136). Deep Learning Models for Denoising ECG Signals. Proceedings of the 2019 27th European Signal Processing Conference (EUSIPCO), A Coru\u00f1a, Spain.","DOI":"10.23919\/EUSIPCO.2019.8902833"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13638-019-1605-z","article-title":"A novel approach to workload prediction using attention-based LSTM encoder-decoder network in cloud environment","volume":"2019","author":"Zhu","year":"2019","journal-title":"EURASIP J. Wirel. Commun. Netw."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","article-title":"Long Shortterm Memory","volume":"9","author":"Hochreiter","year":"1997","journal-title":"Neural Comput."},{"key":"ref_27","first-page":"4171","article-title":"BERT: Pre-training of deep bidirectional transformers for language understanding","volume":"Volume 1","author":"Devlin","year":"2019","journal-title":"Proceedings of the NAACL HLT 2019\u20142019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies;"},{"key":"ref_28","unstructured":"Lechner, M., and Hasani, R. (2020). Learning Long-Term Dependencies in Irregularly-Sampled Time Series. arXiv."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2222","DOI":"10.1109\/TNNLS.2016.2582924","article-title":"LSTM: A Search Space Odyssey","volume":"28","author":"Greff","year":"2017","journal-title":"IEEE Trans. Neural Networks Learn. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1049\/iet-its.2016.0208","article-title":"LSTM network: A deep learning approach for short-term traffic forecast","volume":"11","author":"Zhao","year":"2017","journal-title":"IET Intell. Transp. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Wang, Y., Huang, M., Zhu, X., and Zhao, L. (2016, January 1\u20135). Attention-based LSTM for Aspect-level Sentiment Classification. Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, Austin, TX, USA.","DOI":"10.18653\/v1\/D16-1058"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.eswa.2019.04.038","article-title":"Novel volatility forecasting using deep learning\u2013Long Short Term Memory Recurrent Neural Networks","volume":"132","author":"Liu","year":"2019","journal-title":"Expert Syst. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2099","DOI":"10.1109\/TITS.2017.2749262","article-title":"Energy-efficient timely transportation of long-haul heavy-duty trucks","volume":"19","author":"Deng","year":"2016","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1016\/j.scitotenv.2019.05.288","article-title":"A novel optimal-hybrid model for daily air quality index prediction considering air pollutant factors","volume":"683","author":"Wu","year":"2019","journal-title":"Sci. Total. Environ."},{"key":"ref_35","first-page":"331","article-title":"Learning and predicting sequential tasks using recurrent neural networks and multiple model filtering","volume":"1\u20135","author":"Ravichandar","year":"2016","journal-title":"AAAI Fall Symp. Tech. Rep."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Graves, A. (2012). Supervised Sequence Labelling. Complex Networks & Their Applications IX, Springer.","DOI":"10.1007\/978-3-642-24797-2_2"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1162\/neco.1996.8.3.643","article-title":"The Effects of Adding Noise During Backpropagation Training on a Generalization Performance","volume":"8","author":"An","year":"1996","journal-title":"Neural Comput."},{"key":"ref_38","unstructured":"Neelakantan, A., Vilnis, L., Le, Q.V., Sutskever, I., Kaiser, L., Kurach, K., and Martens, J. (2015). Adding Gradient Noise Improves Learning for Very Deep Networks. arXiv."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Graves, A., Mohamed, A.-R., and Hinton, G. (2013, January 26\u201331). Speech recognition with deep recurrent neural networks. Proceedings of the 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, BC, Canada.","DOI":"10.1109\/ICASSP.2013.6638947"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1162\/neco.1995.7.1.108","article-title":"Training with Noise is Equivalent to Tikhonov Regularization","volume":"7","author":"Bishop","year":"1995","journal-title":"Neural Comput."},{"key":"ref_41","unstructured":"Kingma, D.P., and Ba, J. (2015, January 5\u20138). Adam: A method for stochastic optimization. Proceedings of the International Conference Learn. Represent. (ICLR), San Diego, CA, USA."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/S1064827594276424","article-title":"The MATLAB ODE Suite","volume":"18","author":"Shampine","year":"1997","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_43","unstructured":"Beale, M.H., Hagan, M.T., and Demuth, H.B. (2013). Neural Network Toolbox TM User \u2019 s Guide R2013b, Mathworks Inc."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/5\/1805\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:33:20Z","timestamp":1760160800000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/5\/1805"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,5]]},"references-count":43,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["s21051805"],"URL":"https:\/\/doi.org\/10.3390\/s21051805","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,5]]}}}