{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T12:06:12Z","timestamp":1782475572992,"version":"3.54.5"},"reference-count":57,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,18]],"date-time":"2023-08-18T00:00:00Z","timestamp":1692316800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["427 61790552"],"award-info":[{"award-number":["427 61790552"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The variational Bayesian method solves nonlinear estimation problems by iteratively computing the integral of the marginal density. Many researchers have demonstrated the fact its performance depends on the linear approximation in the computation of the variational density in the iteration and the degree of nonlinearity of the underlying scenario. In this paper, two methods for computing the variational density, namely, the natural gradient method and the simultaneous perturbation stochastic method, are used to implement a variational Bayesian Kalman filter for maneuvering target tracking using Doppler measurements. The latter are collected from a set of sensors subject to single-hop network constraints. We propose a distributed fusion variational Bayesian Kalman filter for a networked maneuvering target tracking scenario and both of the evidence lower bound and the posterior Cram\u00e9r\u2013Rao lower bound of the proposed methods are presented. The simulation results are compared with centralized fusion in terms of posterior Cram\u00e9r\u2013Rao lower bounds, root-mean-squared errors and the 3\u03c3 bound.<\/jats:p>","DOI":"10.3390\/e25081235","type":"journal-article","created":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T01:33:11Z","timestamp":1692581591000},"page":"1235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Variational Bayesian Algorithms for Maneuvering Target Tracking with Nonlinear Measurements in Sensor Networks"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8521-8516","authenticated-orcid":false,"given":"Yumei","family":"Hu","sequence":"first","affiliation":[{"name":"Xi\u2019an Aeronautics Computing Technique Research Institute, AVIC, Xi\u2019an 710069, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Quan","family":"Pan","sequence":"additional","affiliation":[{"name":"School of Automation, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"Key Laboratory of Information Fusion Technology, Ministry of Education, Xi\u2019an 710072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bao","family":"Deng","sequence":"additional","affiliation":[{"name":"Xi\u2019an Aeronautics Computing Technique Research Institute, AVIC, Xi\u2019an 710069, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9563-7391","authenticated-orcid":false,"given":"Zhen","family":"Guo","sequence":"additional","affiliation":[{"name":"System Design Institute of Hubei Aerospace Technology Academy, Wuhan 430040, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Menghua","family":"Li","sequence":"additional","affiliation":[{"name":"Xi\u2019an Aeronautics Computing Technique Research Institute, AVIC, Xi\u2019an 710069, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lifeng","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TSMCB.2010.2040733","article-title":"A game theory approach to target tracking in sensor networks","volume":"41","author":"Gu","year":"2011","journal-title":"IEEE Trans. Syst. Man Cybern. B Cybern."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1109\/JIOT.2015.2475639","article-title":"A novel wireless sensor network frame for urban transportation","volume":"2","author":"Hu","year":"2015","journal-title":"IEEE Internet Things J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1109\/TII.2015.2471263","article-title":"Experimental link quality characterization of wireless sensor networks for underground monitoring","volume":"11","author":"Silva","year":"2015","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_4","first-page":"7","article-title":"Distributed consensus-based Kalman filter estimation and control of formation flying spacecraft: Simulation and validation","volume":"37","author":"Vu","year":"2013","journal-title":"Biulleten Eksp. Biol. I Meditsiny"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1109\/MSP.2006.1657816","article-title":"Distributed fusion in sensor networks","volume":"23","author":"Cetin","year":"2006","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_6","first-page":"21","article-title":"Distributed estimation over a low-cost sensor network: A review of state-of-the-art","volume":"54","author":"He","year":"2020","journal-title":"IEEE Trans. Syst. Man Cybern. B Cybern."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6403","DOI":"10.1109\/TII.2019.2955931","article-title":"Linear fusion estimation for range-only target tracking with nonlinear transformation","volume":"16","author":"Yang","year":"2020","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.inffus.2019.02.009","article-title":"Second-order statistics analysis and comparison between arithmetic and geometric average fusion: Application to multi-sensor target tracking","volume":"52","author":"Li","year":"2019","journal-title":"Inf. Fusion"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1109\/TAES.2005.1468742","article-title":"Decision fusion rules in multi-hop wireless sensor networks","volume":"41","author":"Lin","year":"2005","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1109\/JPROC.1997.554211","article-title":"Distributed fusion architectures and algorithms for target tracking","volume":"85","author":"Liggins","year":"1997","journal-title":"Proc. IEEE"},{"key":"ref_11","first-page":"2932","article-title":"Distributed and centralized fusion estimation from multiple sensors with Markovian delays","volume":"219","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/j.sigpro.2006.06.017","article-title":"Minimum variance generalized state estimators for multiple sensors with different delay rates","volume":"87","author":"Hounkpevi","year":"2007","journal-title":"Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2045","DOI":"10.1016\/j.sigpro.2009.04.007","article-title":"Least-squares linear filtering using observations coming from multiple sensors with one- or two-step random delay","volume":"89","year":"2009","journal-title":"Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2094","DOI":"10.1016\/j.ijleo.2015.11.129","article-title":"Measurement bootstrapping Kalman filter","volume":"127","author":"Hu","year":"2016","journal-title":"Opt.\u2014J. Light Electron. Opt."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2442","DOI":"10.1109\/JSEN.2017.2789239","article-title":"Multi-sensor information filtering with information-based sensor selection and outlier rejection","volume":"18","author":"Bhuvana","year":"2018","journal-title":"IEEE Sens. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4149","DOI":"10.1109\/JSEN.2019.2898106","article-title":"Decentralized kernel-based localization in wireless sensor networks using belief functions","volume":"19","author":"Alshamaa","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1109\/TSMC.2018.2883706","article-title":"A bank of decentralized extended information filters for target tracking in event-triggered WSNs","volume":"50","author":"Yang","year":"2020","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Stamatescu, G., Stamatescu, I., Dragana, C., and Popescu, D. (2015, January 24\u201326). Large scale heterogeneous monitoring system with decentralized sensor fusion. Proceedings of the 2015 IEEE 8th International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS), Warsaw, Poland.","DOI":"10.1109\/IDAACS.2015.7340690"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1109\/TSP.2015.2493979","article-title":"Distributed variational Bayesian algorithms over sensor networks","volume":"64","author":"Hua","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1109\/LSP.2014.2313177","article-title":"Consensus Bernoulli filter for distributed detection and tracking using multi-static doppler shifts","volume":"21","author":"Guldogan","year":"2014","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8669","DOI":"10.1109\/TVT.2015.2508456","article-title":"Consensus-based distributed mixture Kalman filter for maneuvering target tracking in wireless sensor networks","volume":"65","author":"Yu","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1580","DOI":"10.1109\/TCYB.2018.2805717","article-title":"Adaptive consensus-based distributed target tracking with dynamic cluster in sensor networks","volume":"49","author":"Zhang","year":"2019","journal-title":"IEEE Trans. Cybern."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1109\/TSP.2011.2172431","article-title":"Some relations between extended and unscented Kalman filters","volume":"60","author":"Gustafsson","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1109\/9.847726","article-title":"A new method for nonlinear transformation of means and covariances in filters and estimates","volume":"45","author":"Julier","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1109\/9.855552","article-title":"Gaussian filters for nonlinear filtering problems","volume":"45","author":"Ito","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1109\/TAC.2009.2019800","article-title":"Cubature Kalman filters","volume":"54","author":"Arasaratnam","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3204","DOI":"10.1109\/TSP.2015.2423266","article-title":"Generalized iterated Kalman filter and its performance evaluation","volume":"63","author":"Hu","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1016\/j.ast.2019.06.009","article-title":"Unscented Kalman filter state estimation for manipulating unmanned aerial vehicles","volume":"92","author":"Khamseh","year":"2019","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1111\/j.1467-9868.2009.00736.x","article-title":"Particle Markov Chain Monte Carlo methods","volume":"72","author":"Andrieu","year":"2010","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1109\/JPROC.2007.893250","article-title":"An overview of existing methods and recent advances in sequential Monte Carlo","volume":"95","author":"Cappe","year":"2007","journal-title":"Proc. IEEE"},{"key":"ref_31","unstructured":"Merwe, R.V.D., Doucet, A., Freitas, N.D., and Wan, E. (2000, January 1). The unscented particle filter. Proceedings of the Advances in Neural Information Processing Systems 13 (NIPS 2000), Denver, CO, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1111\/1365-2478.12876","article-title":"Markov Chain Monte Carlo algorithms for target-oriented and interval-oriented amplitude versus angle inversions with non-parametric priors and non-linear forward modellings","volume":"68","author":"Aleardi","year":"2020","journal-title":"Geophys. Prospect."},{"key":"ref_33","unstructured":"Doucet, A., Freitas, N.D., Murphy, K., and Russell, S. (1990, January 27\u201329). Rao-Blackwellised particle filtering for dynamic Bayesian networks. Proceedings of the Sixth Annual Conference on Uncertainty in Artificial Intelligence, Cambridge, MA, USA."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"109160","DOI":"10.1016\/j.automatica.2020.109160","article-title":"Nonlinear estimation based on conversion-sample optimization","volume":"121","author":"Lan","year":"2020","journal-title":"Automatica"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1016\/j.conengprac.2012.04.003","article-title":"Nonlinear Bayesian state estimation: A review of recent developments","volume":"20","author":"Patwardhan","year":"2012","journal-title":"Control Eng. Pract."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.ymssp.2015.11.008","article-title":"Particle filter-based prognostics: Review, discussion and perspectives","volume":"72","author":"Jouin","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1109\/JBHI.2017.2780879","article-title":"Smartphone orientation estimation algorithm combining Kalman Filter with gradient descent","volume":"22","author":"Yean","year":"2017","journal-title":"IEEE J. Biomed. Health Inform."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1016\/j.automatica.2020.109034","article-title":"Modified Kalman filtering based multi-step-length gradient iterative algorithm for ARX models with random missing outputs","volume":"118","author":"Chen","year":"2020","journal-title":"Automatica"},{"key":"ref_39","first-page":"1303","article-title":"Stochastic variational inference","volume":"14","author":"Hoffman","year":"2013","journal-title":"J. Mach. Learn. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4425","DOI":"10.1016\/j.jfranklin.2017.04.003","article-title":"Gradient iterative algorithm for dual-rate nonlinear systems based on a novel particle filter","volume":"354","author":"Chen","year":"2017","journal-title":"J. Frankl. Inst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1162\/089976698300017746","article-title":"Natural gradient works efficiently in learning","volume":"10","author":"Amari","year":"1998","journal-title":"Neural Comput."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Amari, S.I. (2016). Information Geometry and Its Applications, Springer.","DOI":"10.1007\/978-4-431-55978-8"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2036","DOI":"10.1109\/TAES.2018.2881352","article-title":"Globally valid posterior Cram\u00e9r-Rao bound for tThree-dimensional bearings-only filtering","volume":"55","author":"Schmitt","year":"2019","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2930","DOI":"10.1214\/18-EJS1468","article-title":"Online natural gradient as a Kalman filter","volume":"12","author":"Ollivier","year":"2018","journal-title":"Electron. J. Stat."},{"key":"ref_45","unstructured":"Ollivier, Y. (2019). The extended Kalman filter is a natural gradient descent in trajectory space. arXiv."},{"key":"ref_46","first-page":"482","article-title":"An overview of the simultaneous perturbation method for efficient optimization","volume":"19","author":"Spall","year":"1998","journal-title":"Johns Hopkins APL Tech. Dig."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Antal, C., Granichin, O., and Levi, S. (2010, January 15\u201317). Adaptive autonomous soaring of multiple UAVs using simultaneous perturbation stochastic approximation. Proceedings of the 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA.","DOI":"10.1109\/CDC.2010.5717903"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1109\/79.543975","article-title":"The expectation-maximization algorithm","volume":"13","author":"Moon","year":"1996","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_49","unstructured":"Beal, M.J. (2003). Variational Algorithms for Approximate Bayesian Inference. [Ph.D. Thesis, Cambridge University]."},{"key":"ref_50","unstructured":"Bishop, C.M. (2006). Pattern Pecognition and Machine Learning, Springer."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2136","DOI":"10.1109\/TAES.2019.2942706","article-title":"Joint target detection and tracking in multipath environment: A variational Bayesian approach","volume":"56","author":"Lan","year":"2020","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2008","DOI":"10.1109\/TPAMI.2018.2889774","article-title":"Advances in variational inference","volume":"41","author":"Zhang","year":"2019","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cja.2021.08.033","article-title":"Variational Bayesian Kalman filter using natural gradient","volume":"35","author":"Hu","year":"2021","journal-title":"Chin. J. Aeronaut."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Absil, P.A., Mahony, R., and Sepulchre, R. (2009). Optimization Algorithms on Matrix Manifolds, Princeton University Press.","DOI":"10.1515\/9781400830244"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1386","DOI":"10.1109\/78.668800","article-title":"Posterior Cram\u00e9r-Rao bounds for discrete-time nonlinear filtering","volume":"46","author":"Tichavsky","year":"1998","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Richards, M.A., Scheer, J., Holm, W.A., and Melvin, W.L. (2010). Principles of Modern Radar, Citeseer.","DOI":"10.1049\/SBRA021E"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3084","DOI":"10.1109\/TAES.2012.6324679","article-title":"On information resolution of radar systems","volume":"48","author":"Cheng","year":"2012","journal-title":"IEEE Trans. Aerosp. Electron. Syst."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1235\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:37:17Z","timestamp":1760128637000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1235"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,18]]},"references-count":57,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["e25081235"],"URL":"https:\/\/doi.org\/10.3390\/e25081235","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,18]]}}}