{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:31:52Z","timestamp":1760236312179,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"22","license":[{"start":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T00:00:00Z","timestamp":1637193600000},"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":["U20A20163"],"award-info":[{"award-number":["U20A20163"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific Research Project of Beijing Municipal Education Commission","award":["KZ202111232049","KM202011232021"],"award-info":[{"award-number":["KZ202111232049","KM202011232021"]}]},{"name":"Qin Xin Talents Cultivation Program","award":["QXTCP A201902"],"award-info":[{"award-number":["QXTCP A201902"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Aimed at the problems in which the performance of filters derived from a hypothetical model will decline or the cost of time of the filters derived from a posterior model will increase when prior knowledge and second-order statistics of noise are uncertain, a new filter is proposed. In this paper, a Bayesian robust Kalman filter based on posterior noise statistics (KFPNS) is derived, and the recursive equations of this filter are very similar to that of the classical algorithm. Note that the posterior noise distributions are approximated by overdispersed black-box variational inference (O-BBVI). More precisely, we introduce an overdispersed distribution to push more probability density to the tails of variational distribution and incorporated the idea of importance sampling into two strategies of control variates and Rao\u2013Blackwellization in order to reduce the variance of estimators. As a result, the convergence process will speed up. From the simulations, we can observe that the proposed filter has good performance for the model with uncertain noise. Moreover, we verify the proposed algorithm by using a practical multiple-input multiple-output (MIMO) radar system.<\/jats:p>","DOI":"10.3390\/s21227673","type":"journal-article","created":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T02:43:09Z","timestamp":1637289789000},"page":"7673","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Overdispersed Black-Box Variational Bayesian\u2013Kalman Filter with Inaccurate Noise Second-Order Statistics"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0875-1549","authenticated-orcid":false,"given":"Lin","family":"Cao","sequence":"first","affiliation":[{"name":"Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing 100101, China"},{"name":"School of Information and Communication Engineering, Beijing Information Science and Technology University, Beijing 100101, China"}]},{"given":"Chuyuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing 100101, China"},{"name":"School of Information and Communication Engineering, Beijing Information Science and Technology University, Beijing 100101, China"}]},{"given":"Zongmin","family":"Zhao","sequence":"additional","affiliation":[{"name":"Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing 100101, China"},{"name":"School of Information and Communication Engineering, Beijing Information Science and Technology University, Beijing 100101, China"}]},{"given":"Dongfeng","family":"Wang","sequence":"additional","affiliation":[{"name":"Beijing TransMicrowave Technology Company, Beijing 100080, China"}]},{"given":"Kangning","family":"Du","sequence":"additional","affiliation":[{"name":"Key Laboratory of the Ministry of Education for Optoelectronic Measurement Technology and Instrument, Beijing Information Science and Technology University, Beijing 100101, China"},{"name":"School of Information and Communication Engineering, Beijing Information Science and Technology University, Beijing 100101, China"}]},{"given":"Chong","family":"Fu","sequence":"additional","affiliation":[{"name":"Department of Communication and Electronics Engineering, School of Computer Science and Engineering, Northeastern University, Shenyang 110169, China"}]},{"given":"Jianfeng","family":"Gu","sequence":"additional","affiliation":[{"name":"Moonshot Health, 3700 St-Patrick Street, Suite 102, Montreal, QC H4E 1A2, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,18]]},"reference":[{"key":"ref_1","first-page":"58","article-title":"Stable detection when the signal and spectrum of normal noise are inaccurately known","volume":"3031","author":"Kuznetsov","year":"1976","journal-title":"Telecommun. Radio Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/0016-0032(77)90011-4","article-title":"Robust Wiener filters","volume":"304","author":"Kassam","year":"1977","journal-title":"J. Frankl. Inst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1115\/1.3662552","article-title":"A new approach to linear filtering prediction problems","volume":"82","author":"Kalman","year":"1960","journal-title":"J. Basic Eng."},{"key":"ref_4","first-page":"86","article-title":"Cooperative Localization for Mobile Agents: A Recursive Decentralized Algorithm Based on Kalman-Filter Decoupling","volume":"36","author":"Kia","year":"2016","journal-title":"IEEE ControlSyst. Mag."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1016\/j.asoc.2017.04.007","article-title":"Fuzzy extended Kalman filter for dynamic mobile localization in urban area using wireless network","volume":"57","author":"Bouzera","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"32368","DOI":"10.1109\/ACCESS.2019.2903219","article-title":"Adaptive Cooperative Localization Using Relative Position Estimation for Networked Systems with Minimum Number of Communication Links","volume":"7","author":"Safaei","year":"2019","journal-title":"IEEE Access"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1982","DOI":"10.1109\/JSEN.2020.3020273","article-title":"Variational Bayesian-Based Maximum Correntropy Cubature Kalman Filter with Both Adaptivity and Robustness","volume":"21","author":"He","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1109\/JSEN.2019.2941273","article-title":"MEMS-Based IMU Drift Minimization: Sage Husa Adaptive Robust Kalman Filtering","volume":"20","author":"Narasimhappa","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2021.3063191","article-title":"Kalman Filtering with Adaptive Step Size Using a Covariance-Based Criterion","volume":"70","author":"Or","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1109\/TAC.2008.2008348","article-title":"Recursive noise adaptive Kalman filtering by variational Bayesian approximations","volume":"54","author":"Sarkka","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_11","first-page":"1","article-title":"Predicting the Noise Covariance with a Multitask Learning Model for Kalman Filter-Based GNSS\/INS Integrated Navigation","volume":"70","author":"Wu","year":"2021","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4793","DOI":"10.1109\/TAC.2019.2959998","article-title":"An Adaptive Gaussian Sum Kalman Filter Based on a Partial Variational Bayesian Method","volume":"65","author":"Xu","year":"2020","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1768","DOI":"10.1109\/TAC.2017.2757908","article-title":"Worst-Case Prediction Performance Analysis of the Kalman Filter","volume":"63","author":"Yasini","year":"2018","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2902","DOI":"10.1109\/TAC.2016.2601879","article-title":"Robust Kalman Filtering Under Model Perturbations","volume":"62","author":"Zorzi","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1109\/TIT.1976.1055611","article-title":"The Kalman filter: A robust estimator for some classes of linear quadratic problems","volume":"22","author":"Morris","year":"1976","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2465","DOI":"10.1109\/TSP.2011.2129516","article-title":"An iterative Kalman-like algorithm ignoring noise and initial conditions","volume":"59","author":"Shmaliy","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3447","DOI":"10.1109\/TSP.2018.2833811","article-title":"Comparing Robustness of the Kalman, H\u221e, and UFIR Filters","volume":"66","author":"Shmaliy","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3086","DOI":"10.1109\/TSP.2010.2045422","article-title":"Linear optimal FIR estimation of discrete time-invariant state-space models","volume":"58","author":"Shmaliy","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2284","DOI":"10.1109\/TSP.2016.2516960","article-title":"Fast Kalman-Like Optimal Unbiased FIR Filtering with Applications","volume":"64","author":"Zhao","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2531","DOI":"10.1109\/TSP.2017.2656845","article-title":"Intrinsically Bayesian robust Kalman filter: An innovation process approach","volume":"65","author":"Dehghannasiri","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1982","DOI":"10.1109\/TSP.2017.2788419","article-title":"Optimal Bayesian Kalman Filtering with Prior Update","volume":"66","author":"Dehghannasiri","year":"2018","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"13212","DOI":"10.3390\/s121013212","article-title":"An Adaptive Altitude Information Fusion Method for Autonomous Landing Processes of Small Unmanned Aerial Rotorcraft","volume":"12","author":"Lei","year":"2012","journal-title":"Sensors"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"13682","DOI":"10.1109\/JSEN.2020.3004621","article-title":"A Computationally Efficient Variational Adaptive Kalman Filter for Transfer Alignment","volume":"20","author":"Xu","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Shan, C., Zhou, W., Yang, Y., and Jiang, Z. (2021). Multi-Fading Factor and Updated Monitoring Strategy Adaptive Kalman Filter-Based Variational Bayesian. Sensors, 21.","DOI":"10.3390\/s21010198"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1109\/TAC.2017.2730480","article-title":"A Novel Adaptive Kalman Filter with Inaccurate Process and Measurement Noise Covariance Matrices","volume":"63","author":"Huang","year":"2018","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"74569","DOI":"10.1109\/ACCESS.2018.2883040","article-title":"A Novel Adaptive Kalman Filter with Colored Measurement Noise","volume":"6","author":"Zhang","year":"2018","journal-title":"IEEE Access"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1109\/TAC.1968.1099025","article-title":"An innovations approach to least-squares estimation\u2014Part I: Linear filtering in additive white noise","volume":"AC-13","author":"Kailath","year":"1968","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1063\/1.1699725","article-title":"An extension of Wiener\u2019s theory of prediction","volume":"21","author":"Zadeh","year":"1950","journal-title":"J. Appl. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1109\/JRPROC.1950.231821","article-title":"A simplified derivation of linear least square smoothing and prediction theory","volume":"38","author":"Bode","year":"1950","journal-title":"Proc. IRE"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1109\/TAC.1980.1102349","article-title":"On robust Wiener filtering","volume":"25","author":"Poor","year":"1980","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1109\/TIT.1984.1056875","article-title":"Robust Wiener-Kolmogorov theory","volume":"30","author":"Vastola","year":"1984","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1109\/TIT.1983.1056734","article-title":"Robust matched filters","volume":"29","author":"Poor","year":"1983","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_33","unstructured":"Ruiz, F.J., Titsias, M., and Blei, D.M. (2016, January 25\u201329). Overdispersed Black-Box Variational Inference. Proceedings of the Thirty-Second Conference on Uncertainty in Artificial Intelligence, Jersey City, NJ, USA."},{"key":"ref_34","first-page":"2121","article-title":"Adaptive subgradient methods for online learning and stochastic optimization","volume":"12","author":"Duchi","year":"2011","journal-title":"J. Mach. Learn. Res."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/22\/7673\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:32:19Z","timestamp":1760167939000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/22\/7673"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,18]]},"references-count":34,"journal-issue":{"issue":"22","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["s21227673"],"URL":"https:\/\/doi.org\/10.3390\/s21227673","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2021,11,18]]}}}