Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

  • Tian-cheng Li
  • Jin-ya Su
  • Wei Liu
  • Juan M. Corchado


Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity.

Key words

Kalman filter Gaussian filter Time series estimation Bayesian filtering Nonlinear filtering Constrained filtering Gaussian mixture Maneuver Unknown inputs 

CLC number




T. Li would like to acknowledge Prof. Yu-chi (Larry) Ho with Harvard University for his high patience and generous encouragement shown in repeated discussion and comments on the topics involved in Sections 3.2 and 7.2 of this paper in the last several years since 2013.


  1. Adurthi, N., Singla, P., Singh, T., 2017. Conjugate unscented transformation: applications to estimation and control. J. Dyn. Syst. Meas. Contr., 140(3):030907. Scholar
  2. Afshari, H., Gadsden, S., Habibi, S., 2017. Gaussian filters for parameter and state estimation: a general review of theory and recent trends. Signal Process., 135:218–238. Scholar
  3. Agamennoni, G., Nebot, E.M., 2014. Robust estimation in non-linear state-space models with state-dependent noise. IEEE Trans. Signal Process., 62(8):2165–2175. Scholar
  4. Ali-Loytty, S.S., 2010. Box Gaussian mixture filter. IEEE Trans. Autom. Contr., 55(9):2165–2169. Scholar
  5. Arasaratnam, I., Haykin, S., 2008. Square-root quadrature Kalman filtering. IEEE Trans. Signal Process., 56(6):2589–2593. Scholar
  6. Arasaratnam, I., Haykin, S., 2009. Cubature Kalman filters. IEEE Trans. Autom. Contr., 54(6):1254–1269. Scholar
  7. Aravkin, A., Burke, J.V., Pillonetto, G., 2012. Robust and trend-following Kalman smoothers using Student’s t. IFAC Proc. Vol., 45(16):1215–1220. Scholar
  8. Ardeshiri, T., Granström, K., Ozkan, E., et al., 2015. Greedy reduction algorithms for mixtures of exponential family. IEEE Signal Process. Lett., 22(6):676–680. Scholar
  9. Arulampalam, M.S., Maskell, S., Gordon, N., et al., 2002. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process., 50(2):174–188. Scholar
  10. Azam, S.E., Chatzi, E., Papadimitriou, C., 2015. A dual Kalman filter approach for state estimation via outputonly acceleration measurements. Mech. Syst. Signal Process., 60–61:866–886. Scholar
  11. Bavdekar, V.A., Deshpande, A.P., Patwardhan, S.C., 2011. Identification of process and measurement noise covariance for state and parameter estimation using extended Kalman filter. J. Process Contr., 21(4):585–601. Scholar
  12. Bell, B.M., Cathey, F.W., 1993. The iterated Kalman filter update as a Gauss–Newton method. IEEE Trans. Autom. Contr., 38(2):294–297. Scholar
  13. Bielecki, T.R., Jakubowski, J., Niewegłowski, M., 2017. Conditional Markov chains: properties, construction and structured dependence. Stoch. Process. Their Appl., 127(4):1125–1170. Scholar
  14. Bilik, I., Tabrikian, J., 2010. MMSE-based filtering in presence of non-Gaussian system and measurement noise. IEEE Trans. Aerosp. Electron. Syst., 46(3):1153–1170. Scholar
  15. Bilmes, J.A., 1999. Buried Markov models for speech recognition. Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, p.713–716. Scholar
  16. Bogler, P.L., 1987. Tracking a maneuvering target using input estimation. IEEE Trans. Aerosp. Electron. Syst., 23(3):298–310. Scholar
  17. Bordonaro, S., Willett, P., Bar-Shalom, Y., 2014. Decorrelated unbiased converted measurement Kalman filter. IEEE Trans. Aerosp. Electron. Syst., 50(2):1431–1444. Scholar
  18. Bordonaro, S., Willett, P., Bar-Shalom, Y., 2017. Consistent linear tracker with converted range, bearing and range rate measurements. IEEE Trans. Aerosp. Electron. Syst., 53(6):3135–3149. Scholar
  19. Bugallo, M.F., Elvira, V., Martino, L., et al., 2017. Adaptive importance sampling: the past, the present, and the future. IEEE Signal Process. Mag., 34(4):60–79. Scholar
  20. Cappé, O., Godsill, S.J., Moulines, E., 2007. An overview of existing methods and recent advances in sequential Monte Carlo. Proc. IEEE, 95(5):899–924. Scholar
  21. Carrassi, A., Bocquet, M., Bertino, L., et al., 2017. Data assimilation in the geosciences—an overview on methods, issues and perspectives. arXiv:1709.02798. Scholar
  22. Chang, L., Hu, B., Li, A., et al., 2013. Transformed unscented Kalman filter. IEEE Trans. Autom. Contr., 58(1):252–257. Scholar
  23. Chen, B., Principe, J.C., 2012. Maximum correntropy estimation is a smoothed MAP estimation. IEEE Signal Process. Lett., 19(8):491–494. Scholar
  24. Chen, B., Liu, X., Zhao, H., et al., 2017. Maximum correntropy Kalman filter. Automatica, 76:70–77. Scholar
  25. Chen, F.C., Hsieh, C.S., 2000. Optimal multistage Kalman estimators. IEEE Trans. Autom. Contr., 45(11):2182–2188. Scholar
  26. Chen, H.D., Chang, K.C., Smith, C., 2010. Constraint optimized weight adaptation for Gaussian mixture reduction. SPIE, 7697:76970N. Scholar
  27. Chen, R., Liu, J.S., 2000. Mixture Kalman filters. J. R. Stat. Soc. Ser. B, 62(3):493–508. Scholar
  28. Cheng, Y., Ye, H., Wang, Y., et al., 2009. Unbiased minimum-variance state estimation for linear systems with unknown input. Automatica, 45(2):485–491. Scholar
  29. Clark, J.M.C., Vinter, R.B., Yaqoob, M.M., 2007. Shifted Rayleigh filter: a new algorithm for bearings-only tracking. IEEE Trans. Aerosp. Electron. Syst., 43(4):1373–1384. Scholar
  30. Crassidis, J.L., Markley, F.L., Cheng, Y., 2007. Survey of nonlinear attitude estimation methods. J. Guid. Contr. Dyn., 30(1):12–28. Scholar
  31. Crouse, D.F., Willett, P., Pattipati, K., et al., 2011. A look at Gaussian mixture reduction algorithms. 14th Int. Conf. on Information Fusion, p.1–8.Google Scholar
  32. Darouach, M., Zasadzinski, M., 1997. Unbiased minimum variance estimation for systems with unknown exogenous inputs. Automatica, 33(4):717–719. Scholar
  33. Daum, F., Huang, J., 2010. Generalized particle flow for nonlinear filters. SPIE, 7698:76980I. Scholar
  34. Deisenroth, M.P., Turner, R.D., Huber, M.F., et al., 2012. Robust filtering and smoothing with Gaussian processes. IEEE Trans. Autom. Contr., 57(7):1865–1871. Scholar
  35. del Moral, P., Arnaud, D., 2014. Particle methods: an introduction with applications. Proc. ESAIM, 44:1–46. Scholar
  36. DeMars, K.J., Bishop, R.H., Jah, M.K., 2013. Entropybased approach for uncertainty propagation of nonlinear dynamical systems. J. Guid. Contr. Dyn., 36(4):1047–1057. Scholar
  37. Djurić, P.M., Miguez, J., 2002. Sequential particle filtering in the presence of additive Gaussian noise with unknown parameters. Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, p.1621–1624. Scholar
  38. Dong, H., Wang, Z., Gao, H., 2010. Robust H filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts. IEEE Trans. Signal Process., 58(4):1957–1966. Scholar
  39. Duan, Z., Li, X.R., 2013. The role of pseudo measurements in equality-constrained state estimation. IEEE Trans. Aerosp. Electron. Syst., 49(3):1654–1666. Scholar
  40. Duan, Z., Li, X.R., 2015. Analysis, design, and estimation of linear equality-constrained dynamic systems. IEEE Trans. Aerosp. Electron. Syst., 51(4):2732–2746. Scholar
  41. Duník, J., Šimandl, M., Straka, O., 2010. Multiple-model filtering with multiple constraints. Proc. American Control Conf., p.6858–6863. Scholar
  42. Duník, J., Straka, O., Šimandl, M., 2013. Stochastic integration filter. IEEE Trans. Autom. Contr., 58(6):1561–1566. Scholar
  43. Duník, J., Straka, O., Šimandl, M., et al., 2015. Randompoint-based filters: analysis and comparison in target tracking. IEEE Trans. Aerosp. Electron. Syst., 51(2):1403–1421. Scholar
  44. Duník, J., Straka, O., Mallick, M., et al., 2016. Survey of nonlinearity and non-Gaussianity measures for state estimation. 19th Int. Conf. on Information Fusion, p.1845–1852.Google Scholar
  45. Duník, J., Straka, O., Ajgl, J., et al., 2017a. From competitive to cooperative filter design. Proc. 20th Int. Conf. on Information Fusion, p.235–243. Scholar
  46. Duník, J., Straka, O., Kost, O., et al., 2017b. Noise covariance matrices in state-space models: a survey and comparison of estimation methods—part I. Int. J. Adapt. Contr. Signal Process., 31(11):1505–1543. Scholar
  47. Eldar, Y.C., 2008. Rethinking biased estimation: improving maximum likelihood and the Cramér-Rao bound. Found. Trends Signal Process., 1(4):305–449. Scholar
  48. Evensen, G., 2003. The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn., 53(4):343–367. Scholar
  49. Fan, H., Zhu, Y., Fu, Q., 2011. Impact of mode decision delay on estimation error for maneuvering target interception. IEEE Trans. Aerosp. Electron. Syst., 47(1):702–711. Scholar
  50. Fang, H., de Callafon, R.A., 2012. On the asymptotic stability of minimum-variance unbiased input and state estimation. Automatica, 48(12):3183–3186. Scholar
  51. Faubel, F., McDonough, J., Klakow, D., 2009. The split and merge unscented Gaussian mixture filter. IEEE Signal Process. Lett., 16(9):786–789. Scholar
  52. Friedland, B., 1969. Treatment of bias in recursive filtering. IEEE Trans. Autom. Contr., 14(4):359–367. Scholar
  53. Frigola-Alcade, R., 2015. Bayesian Time Series Learning with Gaussian Pocesses. PhD Thesis, University of Cambridge, Cambridge, UK.Google Scholar
  54. Fritsche, C., Orguner, U., Gustafsson, F., 2016. On parametric lower bounds for discrete-time filtering. Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, p.4338–4342. Scholar
  55. García-Fernández, A.F., Svensson, L., 2015. Gaussian map filtering using Kalman optimization. IEEE Trans. Autom. Contr., 60(5):1336–1349. Scholar
  56. García-Fernández, A.F., Morelande, M.R., Grajal, J., et al., 2015a. Adaptive unscented Gaussian likelihood approximation filter. Automatica, 54:166–175. Scholar
  57. García-Fernández, A.F., Svensson, L., Morelande, M.R., et al., 2015b. Posterior linearization filter: principles and implementation using sigma points. IEEE Trans. Signal Process., 63(20):5561–5573. Scholar
  58. Geeter, J.D., Brussel, H.V., Schutter, J.D., et al., 1997. A smoothly constrained Kalman filter. IEEE Trans. Patt. Anal. Mach. Intell., 19(10):1171–1177. Scholar
  59. Gerstner, T., Griebel, M., 1998. Numerical integration using sparse grids. Numer. Algor., 18(3):209–232. Scholar
  60. Ghahremani, E., Kamwa, I., 2011. Dynamic state estimation in power system by applying the extended Kalman filter with unknown inputs to phasor measurements. IEEE Trans. Power Syst., 26(4):2556–2566. Scholar
  61. Ghoreyshi, A., Sanger, T.D., 2015. A nonlinear stochastic filter for continuous-time state estimation. IEEE Trans. Autom. Contr., 60(8):2161–2165. Scholar
  62. Gigerenzer, G., Brighton, H., 2009. Homo heuristicus: why biased minds make better inferences. Top. Cogn. Sci., 1(1):107–143. Scholar
  63. Gillijns, S., Moor, B.D., 2007. Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 43(5):934–937. Scholar
  64. Girón, F.J., Rojano, J.C., 1994. Bayesian Kalman filtering with elliptically contoured errors. Biometrika, 81(2):390–395.MathSciNetMATHCrossRefGoogle Scholar
  65. Godsill, S., Clapp, T., 2001. Improvement strategies for Monte Carlo particle filters. In: Doucet, A., de Freitas, N., Gordon, N. (Eds.), Sequential Monte Carlo Methods in Practice. Springer, New York, USA. Scholar
  66. Gordon, N., Percival, J., Robinson, M., 2003. The Kalman-Lévy filter and heavy-tailed models for tracking manoeuvring targets. Proc. 6th Int. Conf. on Information Fusion, p.1024–1031. Scholar
  67. Gorman, J.D., Hero, A.O., 1990. Lower bounds for parametric estimation with constraints. IEEE Trans. Inform. Theory, 36(6):1285–1301. Scholar
  68. Granström, K., Willett, P., Bar-Shalom, Y., 2015. Systematic approach to IMM mixing for unequal dimension states. IEEE Trans. Aerosp. Electron. Syst., 51(4):2975–2986. Scholar
  69. Grewal, M.S., Andrews, A.P., 2014. Kalman Filtering: Theory and Practice with MATLAB. Wiley-IEEE Press, New York, USA.CrossRefMATHGoogle Scholar
  70. Guo, Y., Fan, K., Peng, D., et al., 2015. A modified variable rate particle filter for maneuvering target tracking. Front. Inform. Technol. Electron. Eng., 16(11):985–994. Scholar
  71. Guo, Y., Tharmarasa, R., Rajan, S., et al., 2016. Passive tracking in heavy clutter with sensor location uncertainty. IEEE Trans. Aerosp. Electron. Syst., 52(4):1536–1554. Scholar
  72. Hanebeck, U.D., Briechle, K., Rauh, A., 2003. Progressive Bayes: a new framework for nonlinear state estimation. SPIE, 5099:256–267. Scholar
  73. Hendeby, G., 2008. Performance and Implementation Aspects of Nonlinear Filtering. PhD Thesis, Linköping University, Linköping, Sweden.Google Scholar
  74. Hewett, R.J., Heath, M.T., Butala, M.D., et al., 2010. A robust null space method for linear equality constrained state estimation. IEEE Trans. Signal Process., 58(8):3961–3971. Scholar
  75. Ho, Y., Lee, R., 1964. A Bayesian approach to problems in stochastic estimation and control. IEEE Trans. Autom. Contr., 9(4):333–339. Scholar
  76. Hsieh, C.S., 2009. Extension of unbiased minimum-variance input and state estimation for systems with unknown inputs. Automatica, 45(9):2149–2153. Scholar
  77. Hsieh, C.S., 2000. Robust two-stage Kalman filters for systems with unknown inputs. IEEE Trans. Autom. Contr., 45(12):2374–2378. Scholar
  78. Hu, X., Bao, M., Zhang, X.P., et al., 2015. Generalized iterated Kalman filter and its performance evaluation. IEEE Trans. Signal Process., 63(12):3204–3217. Scholar
  79. Huang, Y., Zhang, Y., Wang, X., et al., 2015. Gaussian filter for nonlinear systems with correlated noises at the same epoch. Automatica, 60:122–126. Scholar
  80. Huang, Y., Zhang, Y., Li, N., et al., 2016a. Design of Gaussian approximate filter and smoother for nonlinear systems with correlated noises at one epoch apart. Circ. Syst. Signal Process., 35(11):3981–4008. Scholar
  81. Huang, Y., Zhang, Y., Li, N., et al., 2016b. Design of Sigma-point Kalman filter with recursive updated measurement. Circ. Syst. Signal Process., 35(5):1767–1782. Scholar
  82. Huang, Y., Zhang, Y., Li, N., et al., 2017. A novel robust Student’s t-based Kalman filter. IEEE Trans. Aerosp. Electron. Syst., 53(3):1545–1554. Scholar
  83. Huber, M.F., 2015. Nonlinear Gaussian Filtering: Theory, Algorithms, and Applications. KIT Scientific Publishing, Karlsruhe, Germany.Google Scholar
  84. Huber, M.F., Hanebeck, U.D., 2008. Progressive Gaussian mixture reduction. 11th Int. Conf. on Information Fusion, p.1–8.Google Scholar
  85. Ishihara, S., Yamakita, M., 2009. Constrained state estimation for nonlinear systems with non-Gaussian noise. 48th IEEE Conf. on Decision Control, p.1279–1284. Scholar
  86. Ito, K., Xiong, K., 2000. Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Contr., 45(5):910–927. Scholar
  87. Jazwinski, A.H., 1970. Stochastic Processes and Filtering Theory. Academic Press, New York, USA, p.349–351.MATHGoogle Scholar
  88. Jia, B., Xin, M., Cheng, Y., 2012. Sparse-grid quadrature nonlinear filtering. Automatica, 48(2):327–341. Scholar
  89. Jia, B., Xin, M., Cheng, Y., 2013. High-degree cubature Kalman filter. Automatica, 49(2):510–518. Scholar
  90. Judd, K., 2015. Tracking an object with unknown accelerations using a shadowing filter. arXiv:1502.07743. Scholar
  91. Judd, K., Stemler, T., 2009. Failures of sequential Bayesian filters and the successes of shadowing filters in tracking of nonlinear deterministic and stochastic systems. Phys. Rev. E, 79(6):066206. Scholar
  92. Julier, S.J., LaViola, J.J., 2007. On Kalman filtering with nonlinear equality constraints. IEEE Trans. Signal Process., 55(6):2774–2784. Scholar
  93. Julier, S.J., Uhlmann, J.K., 2004. Unscented filtering and nonlinear estimation. Proc. IEEE, 92(3):401–422. Scholar
  94. Kalman, R., 1960. A new approach to linear filtering and prediction problems. J. Basic Eng., 82(1):35–45. Scholar
  95. Kalogerias, D.S., Petropulu, A.P., 2016. Grid based nonlinear filtering revisited: recursive estimation asymptotic optimality. IEEE Trans. Signal Process., 64(16):4244–4259. Scholar
  96. Kandepu, R., Foss, B., Imsland, L., 2008. Applying the unscented Kalman filter for nonlinear state estimation. J. Process Contr., 18(7–8): 753–768. Scholar
  97. Kim, K.S., Rew, K.H., 2013. Reduced order disturbance observer for discrete-time linear systems. Automatica, 49(4):968–975. Scholar
  98. Kim, K., Shevlyakov, G., 2008. Why Gaussianity? IEEE Signal Process. Mag., 25(2):102–113. Scholar
  99. Kitanidis, P.K., 1987. Unbiased minimum-variance linear state estimation. Automatica, 23(6):775–778. Scholar
  100. Ko, J., Fox, D., 2009. GP-Bayes filters: Bayesian filtering using Gaussian process prediction and observation models. Auton. Robots, 27(1):75–90. Scholar
  101. Ko, S., Bitmead, R.R., 2007. State estimation for linear systems with state equality constraints. Automatica, 43(8):1363–1368. Scholar
  102. Koch, K.R., Yang, Y., 1998. Robust Kalman filter for rank deficient observation models. J. Geod., 72(7–8): 436–441. Scholar
  103. Kostelich, E., Schreiber, T., 1993. Noise-reduction in chaotic time-series data: a survey of common methods. Phys. Rev. E, 48(3):1752–1763. Scholar
  104. Kotecha, J.H., Djurić, P.M., 2003a. Gaussian particle filtering. IEEE Trans. Signal Process., 51(10):2592–2601. Scholar
  105. Kotecha, J.H., Djurić, P.M., 2003b. Gaussian sum particle filtering. IEEE Trans. Signal Process., 51(10):2602–2612. Scholar
  106. Kurz, G., Gilitschenski, I., Hanebeck, U.D., 2016. Recursive Bayesian filtering in circular state spaces. IEEE Aerosp. Electron. Syst. Mag., 31(3):70–87. Scholar
  107. Kwon, W.H., Kim, P.S., Park, P., 1999. A receding horizon Kalman FIR filter for discrete time-invariant systems. IEEE Trans. Autom. Contr., 44(9):1787–1791. Scholar
  108. Lan, H., Liang, Y., Yang, F., et al., 2013. Joint estimation and identification for stochastic systems with unknown inputs. IET Contr. Theory Appl., 7(10):1377–1386. Scholar
  109. Lan, J., Li, X.R., 2015. Nonlinear estimation by LMMSEbased estimation with optimized uncorrelated augmentation. IEEE Trans. Signal Process., 63(16):4270–4283. Scholar
  110. Lan, J., Li, X.R., 2017. Multiple conversions of measurements for nonlinear estimation. IEEE Trans. Signal Process., 65(18):4956–4970. Scholar
  111. Lan, J., Li, X.R., Jilkov, V.P., et al., 2013. Second-order Markov chain based multiple-model algorithm for maneuvering target tracking. IEEE Trans. Aerosp. Electron. Syst., 49(1):3–19. Scholar
  112. Lerro, D., Bar-Shalom, Y., 1993. Tracking with debiased consistent converted measurements versus EKF. IEEE Trans. Aerosp. Electron. Syst., 29(3):1015–1022. Scholar
  113. Li, B., 2013. State estimation with partially observed inputs: a unified Kalman filtering approach. Automatica, 49(3):816–820. Scholar
  114. Li, T., Bolić, M., Djurić, P.M., 2015a. Resampling methods for particle filtering: classification, implementation, and strategies. IEEE Signal Process. Mag., 32(3):70–86. Scholar
  115. Li, T., Villarrubia, G., Sun, S., et al., 2015b. Resampling methods for particle filtering: identical distribution, a new method, and comparable study. Front. Inform. Technol. Electron. Eng., 16(11):969–984. Scholar
  116. Li, T., Corchado, J.M., Bajo, J., et al., 2016a. Effectiveness of Bayesian filters: an information fusion perspective. Inform. Sci., 329:670–689. Scholar
  117. Li, T., Prieto, J., Corchado, J.M., 2016b. Fitting for smoothing: a methodology for continuous-time target track estimation. Int. Conf. on Indoor Positioning and Indoor Navigation, p.1–8. Scholar
  118. Li, T., Corchado, J.M., Sun, S., et al., 2017a. Clustering for filtering: multi-object detection and estimation using multiple/massive sensors. Inform. Sci., 388–389:172–190. Scholar
  119. Li, T., Corchado, J., Prieto, J., 2017b. Convergence of distributed flooding and its application for distributed Bayesian filtering. IEEE Trans. Signal Inform. Process. Netw., 3(3):580–591. Scholar
  120. Li, T., Chen, H., Sun, S., et al., 2017c. Joint smoothing, tracking, and forecasting based on continuous-time target trajectory fitting. arXiv:1708.02196. Scholar
  121. Li, T., Corchado, J., Chen, H., et al., 2017d. Track a smoothly maneuvering target based on trajectory estimation. Proc. 20th Int. Conf. on Information Fusion, p.800–807. Scholar
  122. Li, T., la Prieta Pintado, F.D., Corchado, J.M., et al., 2018a. Multi-source homogeneous data clustering for multitarget detection from cluttered background with misdetection. Appl. Soft Comput., 60:436–446. Scholar
  123. Li, T., Corchado, J., Sun, S., et al., 2018b. Partial consensus and conservative fusion of Gaussian mixtures for distributed PHD fusion. arXiv:1711.10783. Scholar
  124. Li, X.R., Bar-Shalom, Y., 1996. Multiple-model estimation with variable structure. IEEE Trans. Autom. Contr., 41(4):478–493. Scholar
  125. Li, X.R., Jilkov, V.P., 2002. Survey of maneuvering target tracking: decision-based methods. SPIE, 4728:511–534. Scholar
  126. Li, X.R., Jilkov, V.P., 2005. Survey of maneuvering target tracking. Part V. Multiple-model methods. IEEE Trans. Aerosp. Electron. Syst., 41(4):1255–1321. Scholar
  127. Li, X.R., Jilkov, V.P., 2012. A survey of maneuvering target tracking, Part VIc: approximate nonlinear density filtering in discrete time. SPIE, 8393:83930V. Scholar
  128. Liang, Y., An, D.X., Zhou, D.H., et al., 2004. A finitehorizon adaptive Kalman filter for linear systems with unknown disturbances. Signal Process., 84(11):2175–2194. Scholar
  129. Liang, Y., Zhou, D.H., Zhang, L., et al., 2008. Adaptive filtering for stochastic systems with generalized disturbance inputs. IEEE Signal Process. Lett., 15:645–648. Scholar
  130. Lindley, D.V., Smith, A.F.M., 1972. Bayes estimates for the linear model. J. R. Stat. Soc. Ser. B, 34(1):1–41.MathSciNetMATHGoogle Scholar
  131. Liu, W., 2015. Optimal estimation for discrete-time linear systems in the presence of multiplicative and timecorrelated additive measurement noises. IEEE Trans. Signal Process., 63(17):4583–4593. Scholar
  132. Liu, W., Pokharel, P.P., Principe, J.C., 2007. Correntropy: properties and applications in non-Gaussian signal processing. IEEE Trans. Signal Process., 55(11):5286–5298. Scholar
  133. Liu, Y., Li, X.R., 2015. Measure of nonlinearity for estimation. IEEE Trans. Signal Process., 63(9):2377–2388. Scholar
  134. Liu, Y., Li, X.R., Chen, H., 2013. Generalized linear minimum mean-square error estimation with application to space-object tracking. Asilomar Conf. on Signals, Systems, and Computers, p.2133–2137. Scholar
  135. Loxam, J., Drummond, T., 2008. Student-t mixture filter for robust, real-time visual tracking. European Conf. on Computer Vision, p.372–385. Scholar
  136. Ma, R., Coleman, T.P., 2011. Generalizing the posterior matching scheme to higher dimensions via optimal transportation. 49th Annual Allerton Conf. on Communication, Control, and Computing, p.96–102. Scholar
  137. Mahler, R., 2014. Advances in Statistical Multisource-Multitarget Information Fusion. Artech House, Norwood, USA.MATHGoogle Scholar
  138. Martino, L., Read, J., Elvira, V., et al., 2017. Cooperative parallel particle filters for online model selection and applications to urban mobility. Dig. Signal Process., 60:172–185. Scholar
  139. Mayne, D.Q., 1963. Optimal non-stationary estimation of the parameters of a linear system with Gaussian inputs. J. Electron. Contr., 14(1):101–112. Scholar
  140. Michalska, H., Mayne, D.Q., 1995. Moving horizon observers and observer-based control. IEEE Trans. Autom. Contr., 40(6):995–1006. Scholar
  141. Mitter, S.K., Newton, N.J., 2003. A variational approach to nonlinear estimation. SIAM J. Contr. Optim., 42(5):1813–1833. Scholar
  142. Mohammaddadi, G., Pariz, N., Karimpour, A., 2017. Modal Kalman filter. Asian J. Contr., 19(2):728–738. Scholar
  143. Morelande, M.R., García-Fernández, A.F., 2013. Analysis of Kalman filter approximations for nonlinear measurements. IEEE Trans. Signal Process., 61(22):5477–5484. Scholar
  144. Morrison, N., 2012. Tracking Filter Engineering: the Gauss–Newton and Polynomial Filters. IET, London, UK. Scholar
  145. Murphy, K.P., 2007. Conjugate Bayesian Analysis of the Gaussian Distribution. Technical Report, University of British Columbia, Vancouver, Canada.Google Scholar
  146. Nadjiasngar, R., Inggs, M., 2013. Gauss–Newton filtering incorporating Levenberg–Marquardt methods for tracking. Dig. Signal Process., 23(5):1662–1667. Scholar
  147. Nørgaard, M., Poulsen, N.K., Ravn, O., 2000. New developments in state estimation for nonlinear systems. Automatica, 36(11):1627–1638. Scholar
  148. Nurminen, H., Ardeshiri, T., Piché, R., et al., 2015. Robust inference for state-space models with skewed measurement noise. IEEE Signal Process. Lett., 22(11):1898–1902. Scholar
  149. Nurminen, H., Piché, R., Godsill, S., 2017. Gaussian flow sigma point filter for nonlinear Gaussian state-space models. Proc. 20th Int. Conf. on Information Fusion, p.1–8. Scholar
  150. Ostendorf, M., Digalakis, V.V., Kimball, O.A., 1996. From HMM’s to segment models: a unified view of stochastic modeling for speech recognition. IEEE Trans. Speech Audio Process., 4(5):360–378. Scholar
  151. Oudjane, N., Musso, C., 2000. Progressive correction for regularized particle filters. 3rd Int. Conf. on Information Fusion: THB2/10. Scholar
  152. Park, S., Serpedin, E., Qaraqe, K., 2013. Gaussian assumption: the least favorable but the most useful [lecture notes]. IEEE Signal Process. Mag., 30(3):183–186. Scholar
  153. Patwardhan, S.C., Narasimhan, S., Jagadeesan, P., et al., 2012. Nonlinear Bayesian state estimation: a review of recent developments. Contr. Eng. Pract., 20(10):933–953. Scholar
  154. Petrucci, D.J., 2005. Gaussian Mixture Reduction for Bayesian Target Tracking in Clutter. BiblioScholar, Sydney, Australia.Google Scholar
  155. Piché, R., 2016. Cramér-Rao lower bound for linear filtering with t-distributed measurement noise. 19th Int. Conf. on Information Fusion, p.536–540.Google Scholar
  156. Piché, R., Särkkä, S., Hartikainen, J., 2012. Recursive outlier-robust filtering and smoothing for nonlinear systems using the multivariate student-t distribution. IEEE Int. Workshop on Machine Learning for Signal Processing, p.1–6. Scholar
  157. Pishdad, L., Labeau, F., 2015. Analytic MMSE bounds in linear dynamic systems with Gaussian mixture noise statistics. arXiv:1505.01765. Scholar
  158. Qi, W., Zhang, P., Deng, Z., 2014. Robust weighted fusion Kalman filters for multisensor time-varying systems with uncertain noise variances. Signal Process., 99:185–200. Scholar
  159. Raitoharju, M., Svensson, L., García-Fernández, Á.F., et al., 2017. Damped posterior linearization filter. arXiv:1704.01113. Scholar
  160. Rasmussen, C.E., Williams, C.K.I., 2005. Gaussian Processes for Machine Learning. MIT Press, Cambridge, USA.MATHGoogle Scholar
  161. Reece, S., Roberts, S., 2010. Generalised covariance union: a unified approach to hypothesis merging in tracking. IEEE Trans. Aerosp. Electron. Syst., 46(1):207–221. Scholar
  162. Ristic, B., Wang, X., Arulampalam, S., 2017. Target motion analysis with unknown measurement noise variance. Proc. 20th Int. Conf. on Information Fusion, p.1663–1670. Scholar
  163. Rosti, A.V.I., Gales, M.J.F., 2003. Switching linear dynamical systems for speech recognition. UK Speech Meeting.Google Scholar
  164. Roth, M., Özkan, E., Gustafsson, F., 2013. A Student’s t filter for heavy tailed process and measurement noise. Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, p.5770–5774. Scholar
  165. Roth, M., Hendeby, G., Gustafsson, F., 2016. Nonlinear Kalman filters explained: a tutorial on moment computations and sigma point methods. J. Adv. Inform. Fus., 11(1):47–70.Google Scholar
  166. Roth, M., Ardeshiri, T., Özkan, E., et al., 2017a. Robust Bayesian filtering and smoothing using student’s t distribution. arXiv:1703.02428. Scholar
  167. Roth, M., Hendeby, G., Fritsche, C., et al., 2017b. The ensemble Kalman filter: a signal processing perspective. arXiv:1702.08061. Scholar
  168. Ru, J., Jilkov, V.P., Li, X.R., et al., 2009. Detection of target maneuver onset. IEEE Trans. Aerosp. Electron. Syst., 45(2):536–554. Scholar
  169. Runnalls, A.R., 2007. Kullback-Leibler approach to Gaussian mixture reduction. IEEE Trans. Aerosp. Electron. Syst., 43(3):989–999. Scholar
  170. Salmond, D.J., 1990. Mixture reduction algorithms for target tracking in clutter. SPIE, 1305:434–445. Scholar
  171. Särkkä, S., Hartikainen, J., Svensson, L., et al., 2016. On the relation between Gaussian process quadratures and sigma-point methods. J. Adv. Inform. Fus., 11(1):31–46. Scholar
  172. Sarmavuori, J., Särkkä, S., 2012. Fourier-Hermite Kalman filter. IEEE Trans. Autom. Contr., 57(6):1511–1515. Scholar
  173. Saul, L.K., Jordan, M.I., 1999. Mixed memory Markov models: decomposing complex stochastic processes as mixtures of simpler ones. Mach. Learn., 37(1):75–87. Scholar
  174. Scardua, L.A., da Cruz, J.J., 2017. Complete offline tuning of the unscented Kalman filter. Automatica, 80:54–61. Scholar
  175. Schieferdecker, D., Huber, M.F., 2009. Gaussian mixture reduction via clustering. 12th Int. Conf. on Information Fusion, p.1536–1543.Google Scholar
  176. Šimandl, M., Duník, J., 2009. Derivative-free estimation methods: new results and performance analysis. Automatica, 45(7):1749–1757. Scholar
  177. Šimandl, M., Královec, J., Söderström, T., 2006. Advanced point-mass method for nonlinear state estimation. Automatica, 42(7):1133–1145. Scholar
  178. Simon, D., 2006. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. John Wiley & Sons, New York, USA.CrossRefGoogle Scholar
  179. Simon, D., 2010. Kalman filtering with state constraints: a survey of linear and nonlinear algorithms. IET Contr. Theory Appl., 4(8):1303–1318. Scholar
  180. Singpurwalla, N.D., Polson, N.G., Soyer, R., 2017. From least squares to signal processing and particle filtering. Technometrics, 2017:1–15 Scholar
  181. Smith, L.A., Cuellar, M.C., Du, H., et al., 2010. Exploiting dynamical coherence: a geometric approach to parameter estimation in nonlinear models. Phys. Lett. A, 374(26):2618–2623. Scholar
  182. Snidaro, L., García, J., Llinas, J., 2015. Context-based information fusion: a survey and discussion. Inform. Fus., 25(Supplement C):16–31. Scholar
  183. Song, P., 2000. Monte Carlo Kalman filter and smoothing for multivariate discrete state space models. Can. J. Statist., 28(3):641–652. Scholar
  184. Sorenson, H.W., 1970. Least-squares estimation: from Gauss to Kalman. IEEE Spectr., 7(7):63–68. Scholar
  185. Sorenson, H., Alspach, D., 1971. Recursive Bayesian estimation using Gaussian sums. Automatica, 7(4):465–479. Scholar
  186. Sornette, D., Ide, K., 2001. The Kalman–Lévy filter. Phys. D, 151(2-4):142174. Scholar
  187. Spinello, D., Stilwell, D.J., 2010. Nonlinear estimation with state-dependent Gaussian observation noise. IEEE Trans. Autom. Contr., 55(6):1358–1366. Scholar
  188. Stano, P., Lendek, Z., Braaksma, J., et al., 2013. Parametric Bayesian filters for nonlinear stochastic dynamical systems: a survey. IEEE Trans. Cybern., 43(6):1607–1624. Scholar
  189. Steinbring, J., Hanebeck, U.D., 2014. Progressive Gaussian filtering using explicit likelihoods. 17th Int. Conf. on Information Fusion, p.1–8.Google Scholar
  190. Stoica, P., Babu, P., 2011. The Gaussian data assumption leads to the largest Cramér-Rao bound [lecture notes]. IEEE Signal Process. Mag., 28(3):132–133. Scholar
  191. Stoica, P., Moses, R.L., 1990. On biased estimators and the unbiased Cramér-Rao lower bound. Signal Process., 21(4):349350. Scholar
  192. Straka, O., Duník, J., Šimandl, M., 2012. Truncation nonlinear filters for state estimation with nonlinear inequality constraints. Automatica, 48(2):273–286. Scholar
  193. Straka, O., Duník, J., Šimandl, M., 2014. Unscented Kalman filter with advanced adaptation of scaling parameter. Automatica, 50(10):2657–2664. Scholar
  194. Su, J., Chen, W.H., 2017. Model-based fault diagnosis system verification using reachability analysis. IEEE Trans. Syst. Man Cybern. Syst., 99:1–10. Scholar
  195. Su, J., Li, B., Chen, W.H., 2015a. On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs. Automatica, 53:149–154. Scholar
  196. Su, J., Li, B., Chen, W.H., 2015b. Simultaneous state and input estimation with partial information on the inputs. Syst. Sci. Contr. Eng., 3(1):445–452. Scholar
  197. Su, J., Chen, W.H., Yang, J., 2016. On relationship between time-domain and frequency-domain disturbance observers and its applications. J. Dyn. Syst. Meas. Contr., 138(9):091013. Scholar
  198. Svensson, A., Schön, T.B., Lindsten, F., 2017. Learning of state-space models with highly informative observations: a tempered sequential Monte Carlo solution. arXiv:1702.01618. Scholar
  199. Tahk, M., Speyer, J.L., 1990. Target tracking problems subject to kinematic constraints. IEEE Trans. Autom. Contr., 35(3):324–326. Scholar
  200. Teixeira, B.O., Tôrres, L.A., Aguirre, L.A., et al., 2010. On unscented Kalman filtering with state interval constraints. J. Process Contr., 20(1):45–57. Scholar
  201. Terejanu, G., Singla, P., Singh, T., et al., 2011. Adaptive Gaussian sum filter for nonlinear Bayesian estimation. IEEE Trans. Autom. Contr., 56(9):2151–2156. Scholar
  202. Tichavsky, P., Muravchik, C.H., Nehorai, A., 1998. Posterior Cramér-Rao bounds for discrete-time nonlinear filtering. IEEE Trans. Signal Process., 46(5):1386–1396. Scholar
  203. Tipping, M.E., Lawrence, N.D., 2005. Variational inference for Student-t models: robust Bayesian interpolation and generalised component analysis. Neurocomputing, 69(1–3): 123–141. Scholar
  204. van der Merwe, R., Doucet, A., de Freitas, N., et al., 2000. The unscented particle filter. Proc. NIPS, p.563–569.Google Scholar
  205. van Trees, H.L., 1968. Detection, Estimation and Modulation Theory. Wiley, New York, USA.MATHGoogle Scholar
  206. van Trees, H.L., Bell, K.L., 2007. Bayesian bounds for parameter estimation and nonlinear filtering/tracking. IET Radar Sonar Navig., 3(3):285–286. Scholar
  207. Vo, B.N., Ma, W.K., 2006. The Gaussian mixture probability hypothesis density filter. IEEE Trans. Signal Process., 54(11):4091–4104. Scholar
  208. Wang, J.M., Fleet, D.J., Hertzmann, A., 2008. Gaussian process dynamical models for human motion. IEEE Trans. Patt. Anal. Mach. Intell., 30(2):283–298. Scholar
  209. Wang, X., Fu, M., Zhang, H., 2012. Target tracking in wireless sensor networks based on the combination of KF and MLE using distance measurements. IEEE Trans. Mob. Comput., 11(4):567–576. Scholar
  210. Wang, X., Liang, Y., Pan, Q., et al., 2014. Design and implementation of Gaussian filter for nonlinear system with randomly delayed measurements and correlated noises. Appl. Math. Comput., 232:1011–1024. Scholar
  211. Wang, X., Liang, Y., Pan, Q., et al., 2015. Nonlinear Gaussian smoothers with colored measurement noise. IEEE Trans. Autom. Contr., 60(3):870–876. Scholar
  212. Wang, X., Song, B., Liang, Y., et al., 2017. EM-based adaptive divided difference filter for nonlinear system with multiplicative parameter. Int. J. Robust Nonl. Contr., 27(13):2167–2197. Scholar
  213. Wen, W., Durrant-Whyte, H.F., 1992. Model-based multisensor data fusion. Proc. IEEE Int. Conf. on Robotics and Automation, p.1720–1726. Scholar
  214. Williams, J.L., Maybeck, P.S., 2006. Cost-function-based hypothesis control techniques for multiple hypothesis tracking. Math. Comput. Model., 43(9–10): 976–989. Scholar
  215. Wu, Y., Hu, D., Wu, M., et al., 2006. A numerical-integration perspective on Gaussian filters. IEEE Trans. Signal Process., 54(8):2910–2921. Scholar
  216. Wu, Z., Shi, J., Zhang, X., et al., 2015. Kernel recursive maximum correntropy. Signal Process., 117:11–16. Scholar
  217. Xu, L., Li, X.R., Duan, Z., et al., 2013. Modeling and state estimation for dynamic systems with linear equality constraints. IEEE Trans. Signal Process., 61(11):2927–2939. Scholar
  218. Xu, L., Li, X.R., Duan, Z., 2016. Hybrid grid multiple-model estimation with application to maneuvering target tracking. IEEE Trans. Aerosp. Electron. Syst., 52(1):122–136. Scholar
  219. Yang, C., Blasch, E., 2009. Kalman filtering with nonlinear state constraints. IEEE Trans. Aerosp. Electron. Syst., 45(1):70–84. Scholar
  220. Yang, T., Laugesen, R.S., Mehta, P.G., et al., 2016. Multivariable feedback particle filter. Automatica, 71:10–23. Scholar
  221. Yang, Y., He, H., Xu, G., 2001. Adaptively robust filtering for kinematic geodetic positioning. J. Geod., 75(2–3): 109–116. Scholar
  222. Yi, D., Su, J., Liu, C., et al., 2016. Data-driven situation awareness algorithm for vehicle lane change. 19th IEEE Int. Conf. on Intelligent Transportation Systems, p.998–1003. Scholar
  223. Yong, S.Z., Zhu, M., Frazzoli, E., 2016. A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems. Automatica, 63:321–329. Scholar
  224. Zanetti, R., 2012. Recursive update filtering for nonlinear estimation. IEEE Trans. Autom. Contr., 57(6):1481–1490. Scholar
  225. Zen, H., Tokuda, K., Kitamura, T., 2007. Reformulating the HMM as a trajectory model by imposing explicit relationships between static and dynamic feature vector sequences. Comput. Speech Lang., 21(1):153–173. Scholar
  226. Zhan, R., Wan, J., 2007. Iterated unscented Kalman filter for passive target tracking. IEEE Trans. Aerosp. Electron. Syst., 43(3):1155–1163. Scholar
  227. Zhang, C., Zhi, R., Li, T., et al., 2016. Adaptive Mestimation for robust cubature Kalman filtering. Sensor Signal Processing for Defence, p.114–118. Scholar
  228. Zhang, Y., Huang, Y., Li, N., et al., 2015. Embedded cubature Kalman filter with adaptive setting of free parameter. Signal Process., 114:112–116. Scholar
  229. Zhao, S., Shmaliy, Y.S., Liu, F., 2016a. Fast Kalman-like optimal unbiased FIR filtering with applications. IEEE Trans. Signal Process., 64(9):2284–2297. Scholar
  230. Zhao, S., Shmaliy, Y.S., Liu, F., et al., 2016b. Unbiased, optimal, and in-betweens: the trade-off in discrete finite impulse response filtering. IET Signal Process., 10(4):325–334. Scholar
  231. Zheng, Y., Ozdemir, O., Niu, R., et al., 2012. New conditional posterior Cramér-Rao lower bounds for nonlinear sequential Bayesian estimation. IEEE Trans. Signal Process., 60(10):5549–5556. Scholar
  232. Zhou, D.H., Frank, P.M., 1996. Strong tracking filtering of nonlinear time-varying stochastic systems with coloured noise: application to parameter estimation and empirical robustness analysis. Int. J. Contr., 65(2):295–307. Scholar
  233. Zuo, L., Niu, R., Varshney, P.K., 2011. Conditional posterior Cramér-Rao lower bounds for nonlinear sequential Bayesian estimation. IEEE Trans. Signal Process., 59(1):1–14. Scholar

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© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of SciencesUniversity of SalamancaSalamancaSpain
  2. 2.Department of Aeronautical and Automotive EngineeringLoughborough UniversityLoughboroughUK
  3. 3.Department of Electronic and Electrical EngineeringUniversity of SheffieldSheffieldUK

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