{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T15:56:38Z","timestamp":1758124598635,"version":"3.37.3"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,9,6]],"date-time":"2018-09-06T00:00:00Z","timestamp":1536192000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2018,9,6]],"date-time":"2018-09-06T00:00:00Z","timestamp":1536192000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1553281"],"award-info":[{"award-number":["CCF-1553281"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["EURASIP J. Adv. Signal Process."],"published-print":{"date-parts":[[2018,12]]},"DOI":"10.1186\/s13634-018-0577-1","type":"journal-article","created":{"date-parts":[[2018,9,6]],"date-time":"2018-09-06T13:30:31Z","timestamp":1536240631000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Bayesian robust Kalman smoothing framework for state-space models with uncertain noise statistics"],"prefix":"10.1186","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7413-3437","authenticated-orcid":false,"given":"Roozbeh","family":"Dehghannasiri","sequence":"first","affiliation":[]},{"given":"Xiaoning","family":"Qian","sequence":"additional","affiliation":[]},{"given":"Edward R.","family":"Dougherty","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,9,6]]},"reference":[{"issue":"1","key":"577_CR1","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1115\/1.3662552","volume":"82","author":"R.E. Kalman","year":"1960","unstructured":"R.E. Kalman, A new approach to linear filtering and prediction problems. J. Basic Eng.82(1), 35\u201345 (1960).","journal-title":"J. Basic Eng."},{"key":"577_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-47612-4","volume-title":"Kalman filtering","author":"C.K. Chui","year":"2017","unstructured":"C.K. Chui, G. Chen, et al., Kalman filtering (Springer, Cham, 2017)."},{"issue":"1","key":"577_CR3","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1080\/10618600.2012.730083","volume":"23","author":"C.C. Drovandi","year":"2014","unstructured":"C.C. Drovandi, J.M. McGree, A.N. Pettitt, A sequential Monte Carlo algorithm to incorporate model uncertainty in Bayesian sequential design. J. Comput. Graph. Stat.23(1), 3\u201324 (2014).","journal-title":"J. Comput. Graph. Stat."},{"issue":"3","key":"577_CR4","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1111\/j.1467-9868.2012.01046.x","volume":"75","author":"N. Chopin","year":"2013","unstructured":"N. Chopin, P.E. Jacob, O. Papaspiliopoulos, SMC2: an efficient algorithm for sequential analysis of state space models. J. R. Stat. Soc. Ser. B Stat Methodol.75(3), 397\u2013426 (2013).","journal-title":"J. R. Stat. Soc. Ser. B Stat Methodol."},{"issue":"4A","key":"577_CR5","doi-asserted-by":"publisher","first-page":"3039","DOI":"10.3150\/17-BEJ954","volume":"24","author":"D. Crisan","year":"2018","unstructured":"D. Crisan, J. Miguez, et al., Nested particle filters for online parameter estimation in discrete-time state-space Markov models. Bernoulli. 24(4A), 3039\u20133086 (2018).","journal-title":"Bernoulli"},{"key":"577_CR6","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1016\/j.dsp.2016.09.011","volume":"60","author":"L. Martino","year":"2017","unstructured":"L. Martino, J. Read, V. Elvira, F. Louzada, Cooperative parallel particle filters for online model selection and applications to urban mobility. Digit. Signal Proc.60:, 172\u2013185 (2017).","journal-title":"Digit. Signal Proc."},{"key":"577_CR7","doi-asserted-by":"crossref","unstructured":"I. Urteaga, M.F. Bugallo, P.M. Djuri\u0107, in Statistical Signal Processing Workshop (SSP), 2016 IEEE. Sequential Monte Carlo methods under model uncertainty (IEEE, 2016), pp. 1\u20135.","DOI":"10.1109\/SSP.2016.7551747"},{"issue":"4","key":"577_CR8","doi-asserted-by":"publisher","first-page":"520","DOI":"10.1109\/TAC.1976.1101260","volume":"21","author":"K.A. Myers","year":"1976","unstructured":"K.A. Myers, B.D. Tapley, Adaptive sequential estimation with unknown noise statistics. IEEE Trans. Autom. Control. 21(4), 520\u2013523 (1976).","journal-title":"IEEE Trans. Autom. Control"},{"issue":"2","key":"577_CR9","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1109\/TAC.1970.1099422","volume":"15","author":"R.K. Mehra","year":"1970","unstructured":"R.K. Mehra, On the identification of variances and adaptive Kalman filtering. IEEE Trans. Autom. Control. 15(2), 175\u2013184 (1970).","journal-title":"IEEE Trans. Autom. Control"},{"issue":"3","key":"577_CR10","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1109\/TAC.1980.1102349","volume":"25","author":"H.V. Poor","year":"1980","unstructured":"H.V. Poor, On robust Wiener filtering. IEEE Trans. Autom. Control. 25(3), 531\u2013536 (1980).","journal-title":"IEEE Trans. Autom. Control"},{"issue":"2","key":"577_CR11","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1109\/TIT.1984.1056876","volume":"30","author":"S. Verdu","year":"1984","unstructured":"S. Verdu, H. Poor, On minimax robustness: a general approach and applications. IEEE Trans. Infor. Theory. 30(2), 328\u2013340 (1984).","journal-title":"IEEE Trans. Infor. Theory"},{"issue":"4","key":"577_CR12","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1109\/TAC.1981.1102756","volume":"26","author":"V. Poor","year":"1981","unstructured":"V. Poor, D.P. Looze, Minimax state estimation for linear stochastic systems with noise uncertainty. IEEE Trans. Autom. Control. 26(4), 902\u2013906 (1981).","journal-title":"IEEE Trans. Autom. Control"},{"issue":"3","key":"577_CR13","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1023\/A:1008356620614","volume":"11","author":"A.M. Grigoryan","year":"1999","unstructured":"A.M. Grigoryan, E.R. Dougherty, Design and analysis of robust binary filters in the context of a prior distribution for the states of nature. Math. Imag. Vision. 11(3), 239\u2013254 (1999).","journal-title":"Math. Imag. Vision"},{"issue":"12","key":"577_CR14","doi-asserted-by":"publisher","first-page":"2503","DOI":"10.1016\/S0165-1684(01)00144-X","volume":"81","author":"A.M. Grigoryan","year":"2001","unstructured":"A.M. Grigoryan, E.R. Dougherty, Bayesian robust optimal linear filters. Signal Process.81(12), 2503\u20132521 (2001).","journal-title":"Signal Process."},{"issue":"3","key":"577_CR15","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1109\/TSP.2013.2291213","volume":"62","author":"L. Dalton","year":"2014","unstructured":"L. Dalton, E. Dougherty, Intrinsically optimal Bayesian robust filtering. IEEE Trans. Signal Process.62(3), 657\u2013670 (2014).","journal-title":"IEEE Trans. Signal Process."},{"issue":"10","key":"577_CR16","doi-asserted-by":"publisher","first-page":"2531","DOI":"10.1109\/TSP.2017.2656845","volume":"65","author":"R. Dehghannasiri","year":"2017","unstructured":"R. Dehghannasiri, M.S. Esfahani, E.R. Dougherty, Intrinsically Bayesian robust Kalman filter: an innovation process approach. IEEE Trans. Signal Process.65(10), 2531\u20132546 (2017).","journal-title":"IEEE Trans. Signal Process."},{"issue":"23","key":"577_CR17","doi-asserted-by":"publisher","first-page":"6243","DOI":"10.1109\/TSP.2016.2605072","volume":"64","author":"X. Qian","year":"2016","unstructured":"X. Qian, E. Dougherty, Bayesian regression with network prior: optimal Bayesian filtering perspective. IEEE Trans. Signal Process.64(23), 6243 (2016).","journal-title":"IEEE Trans. Signal Process."},{"issue":"8","key":"577_CR18","doi-asserted-by":"publisher","first-page":"1982","DOI":"10.1109\/TSP.2017.2788419","volume":"66","author":"R. Dehghannasiri","year":"2018","unstructured":"R. Dehghannasiri, M.S. Esfahani, X. Qian, E.R. Dougherty, Optimal Bayesian Kalman filtering with prior update. IEEE Trans. Signal Process.66(8), 1982\u20131996 (2018).","journal-title":"IEEE Trans. Signal Process."},{"issue":"5","key":"577_CR19","doi-asserted-by":"publisher","first-page":"1301","DOI":"10.1016\/j.patcog.2012.10.018","volume":"46","author":"L.A. Dalton","year":"2013","unstructured":"L.A. Dalton, E.R. Dougherty, Optimal classifiers with minimum expected error within a Bayesian framework\u2013part I: discrete and Gaussian models. Pattern Recogn.46(5), 1301\u20131314 (2013).","journal-title":"Pattern Recogn."},{"issue":"Supplement C","key":"577_CR20","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1016\/j.sigpro.2017.10.016","volume":"144","author":"R. Dehghannasiri","year":"2018","unstructured":"R. Dehghannasiri, X. Qian, E.R. Dougherty, Intrinsically Bayesian robust Karhunen-Lo\u00e8ve compression. Signal Process.144(Supplement C), 311\u2013322 (2018).","journal-title":"Signal Process."},{"issue":"6","key":"577_CR21","doi-asserted-by":"publisher","first-page":"646","DOI":"10.1109\/TAC.1968.1099025","volume":"13","author":"T. Kailath","year":"1968","unstructured":"T. Kailath, An innovations approach to least-squares estimation\u2013part I: linear filtering in additive white noise. IEEE Trans. Autom. Control. 13(6), 646\u2013655 (1968).","journal-title":"IEEE Trans. Autom. Control"},{"key":"577_CR22","first-page":"353","volume-title":"Proceedings of the Optimum System Synthesis Conference","author":"A. Bryson","year":"1963","unstructured":"A. Bryson, M. Frazier, in Proceedings of the Optimum System Synthesis Conference. Smoothing for linear and nonlinear dynamic systems (DTIC DocumentOhio, 1963), pp. 353\u2013364."},{"issue":"4","key":"577_CR23","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1109\/TAC.1963.1105600","volume":"8","author":"H. Rauch","year":"1963","unstructured":"H. Rauch, Solutions to the linear smoothing problem. IEEE Trans. Autom. Control. 8(4), 371\u2013372 (1963).","journal-title":"IEEE Trans. Autom. Control"},{"issue":"8","key":"577_CR24","doi-asserted-by":"publisher","first-page":"1445","DOI":"10.2514\/3.3166","volume":"3","author":"H.E. Rauch","year":"1965","unstructured":"H.E. Rauch, C. Striebel, F. Tung, Maximum likelihood estimates of linear dynamic systems. AIAA J.3(8), 1445\u20131450 (1965).","journal-title":"AIAA J."},{"issue":"1","key":"577_CR25","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1137\/0305006","volume":"5","author":"J.S. Meditch","year":"1967","unstructured":"J.S. Meditch, Orthogonal projection and discrete optimal linear smoothing. SIAM J. Control. 5(1), 74\u201389 (1967).","journal-title":"SIAM J. Control"},{"issue":"2","key":"577_CR26","doi-asserted-by":"publisher","first-page":"316","DOI":"10.1109\/TSP.2012.2223696","volume":"61","author":"H. Zhao","year":"2013","unstructured":"H. Zhao, P. Cui, W. Wang, D. Yang, h \u221e fixed-interval smoothing estimation for time-delay systems. IEEE Trans. Signal Process.61(2), 316\u2013326 (2013).","journal-title":"IEEE Trans. Signal Process."},{"issue":"10","key":"577_CR27","doi-asserted-by":"publisher","first-page":"2437","DOI":"10.1109\/TAC.2008.2007858","volume":"53","author":"B. Ait-El-Fquih","year":"2008","unstructured":"B. Ait-El-Fquih, F. Desbouvries, On Bayesian fixed-interval smoothing algorithms. IEEE Trans. Autom. Control. 53(10), 2437\u20132442 (2008).","journal-title":"IEEE Trans. Autom. Control"},{"issue":"4","key":"577_CR28","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1109\/TAC.1969.1099196","volume":"14","author":"D. Fraser","year":"1969","unstructured":"D. Fraser, J. Potter, The optimum linear smoother as a combination of two optimum linear filters. IEEE Trans. Autom. Control. 14(4), 387\u2013390 (1969).","journal-title":"IEEE Trans. Autom. Control"},{"issue":"1","key":"577_CR29","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s10463-009-0236-2","volume":"62","author":"M. Briers","year":"2010","unstructured":"M. Briers, A. Doucet, S. Maskell, Smoothing algorithms for state\u2013space models. Ann. Inst. Stat. Math.62(1), 61\u201389 (2010).","journal-title":"Ann. Inst. Stat. Math."},{"issue":"6","key":"577_CR30","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1016\/S0019-9958(67)91040-6","volume":"10","author":"J.S. Meditch","year":"1967","unstructured":"J.S. Meditch, On optimal linear smoothing theory. J. Inf. Control.10(6), 598\u2013615 (1967).","journal-title":"J. Inf. Control."},{"issue":"6","key":"577_CR31","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1109\/TAC.1968.1099019","volume":"13","author":"T. Kailath","year":"1968","unstructured":"T. Kailath, P. Frost, An innovations approach to least-squares estimation\u2013part II: linear smoothing in additive white noise. IEEE Trans. Autom. Control. 13(6), 655\u2013660 (1968).","journal-title":"IEEE Trans. Autom. Control"},{"issue":"5","key":"577_CR32","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1016\/j.apm.2005.05.004","volume":"30","author":"S. Nakamori","year":"2006","unstructured":"S. Nakamori, A. Hermoso-Carazo, J. Linares-P\u00e9rez, Design of a fixed-interval smoother using covariance information based on the innovations approach in linear discrete-time stochastic systems. Appl. Math. Model.30(5), 406\u2013417 (2006).","journal-title":"Appl. Math. Model."},{"issue":"1","key":"577_CR33","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/S0096-3003(03)00707-0","volume":"154","author":"S. Nakamori","year":"2004","unstructured":"S. Nakamori, A. Hermoso-Carazo, J. Linares-P\u00e9rez, M. S\u00e1nchez-Rodr\u0131guez, Fixed-interval smoothing problem from uncertain observations with correlated signal and noise. Appl. Math. Comput.154(1), 239\u2013255 (2004).","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"577_CR34","doi-asserted-by":"publisher","first-page":"498","DOI":"10.1109\/18.910572","volume":"47","author":"F.R. Kschischang","year":"2001","unstructured":"F.R. Kschischang, B. J. Frey, A. H-Loeliger, Factor graphs and the sum-product algorithm. IEEE Trans. Inf. Theory. 47(2), 498\u2013519 (2001).","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"10","key":"577_CR35","doi-asserted-by":"publisher","first-page":"1890","DOI":"10.2514\/3.5477","volume":"7","author":"R.E. Griffin","year":"1969","unstructured":"R.E. Griffin, A. P. Sage, Sensitivity analysis of discrete filtering and smoothing algorithms. AIAA J.7(10), 1890\u20131897 (1969).","journal-title":"AIAA J."},{"key":"577_CR36","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511975837","volume-title":"Fundamentals of object tracking","author":"S. Challa","year":"2011","unstructured":"S. Challa, M.R. Morelande, D. Musicki, R.J. Evans, Fundamentals of object tracking (Cambridge University Press, Cambridge, 2011)."},{"issue":"3","key":"577_CR37","doi-asserted-by":"publisher","first-page":"1664","DOI":"10.1109\/TAES.2015.130550","volume":"51","author":"J.L. Williams","year":"2015","unstructured":"J.L. Williams, Marginal multi-Bernoulli filters: RFS derivation of MHT, JIPDA, and association-based MeMBer. IEEE Trans. Aerosp. Electron. Syst.51(3), 1664\u20131687 (2015).","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"577_CR38","doi-asserted-by":"crossref","unstructured":"H. Liu, S. Zhou, H. Liu, H. Wang, in Radar Conference (Radar), 2014 International. Radar detection during tracking with constant track false alarm rate (IEEE, 2014), pp. 1\u20135.","DOI":"10.1109\/RADAR.2014.7060437"},{"issue":"7","key":"577_CR39","doi-asserted-by":"publisher","first-page":"3327","DOI":"10.1109\/TSP.2008.919638","volume":"56","author":"L. Qian","year":"2008","unstructured":"L. Qian, H. Wang, E.R. Dougherty, Inference of noisy nonlinear differential equation models for gene regulatory networks using genetic programming and Kalman filtering. IEEE Trans. Signal Process.56(7), 3327\u20133339 (2008).","journal-title":"IEEE Trans. Signal Process."},{"issue":"1","key":"577_CR40","doi-asserted-by":"publisher","first-page":"202","DOI":"10.1109\/TCBB.2013.143","volume":"11","author":"M.S. Esfahani","year":"2014","unstructured":"M.S. Esfahani, E.R. Dougherty, Incorporation of biological pathway knowledge in the construction of priors for optimal Bayesian classification. IEEE\/ACM Trans. Comput. Biol. Bioinf.11(1), 202\u2013218 (2014).","journal-title":"IEEE\/ACM Trans. Comput. Biol. Bioinf."}],"container-title":["EURASIP Journal on Advances in Signal Processing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1186\/s13634-018-0577-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1186\/s13634-018-0577-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1186\/s13634-018-0577-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T21:05:07Z","timestamp":1661979907000},"score":1,"resource":{"primary":{"URL":"https:\/\/asp-eurasipjournals.springeropen.com\/articles\/10.1186\/s13634-018-0577-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,6]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,12]]}},"alternative-id":["577"],"URL":"https:\/\/doi.org\/10.1186\/s13634-018-0577-1","relation":{},"ISSN":["1687-6180"],"issn-type":[{"type":"electronic","value":"1687-6180"}],"subject":[],"published":{"date-parts":[[2018,9,6]]},"assertion":[{"value":"21 January 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 August 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 September 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Publisher\u2019s Note"}}],"article-number":"55"}}