Parametric bayesian filters for nonlinear stochastic dynamical systems: A survey

IEEE Transactions on Cybernetics

Volumn 43, Issue 6, 2013, Pages 1607-1624

  Stano, Pawe ^a   Lendek, Zsófia ^a,b   Braaksma, Jelmer ^c   Babuška, Robert ^a   De Keizer, Cees ^c   Den Dekker, Arnold J ^a  

^a DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

^b TECHNICAL UNIVERSITY OF CLUJ NAPOCA   (Romania)

^c IHC Systems   (Netherlands)

Author keywords

Analysis of variance; Bayesian methods; Nonlinear filters; Parametric methods

Indexed keywords

BAYESIAN METHODS; CENTRAL DIFFERENCE FILTERS; GAUSSIAN SUM APPROXIMATION; ITERATED EXTENDED KALMAN FILTER; MINIMUM MEAN SQUARED ERROR; PARAMETRIC METHOD; STATISTICAL APPROXIMATION; STOCHASTIC DYNAMICAL SYSTEM;

ANALYSIS OF VARIANCE (ANOVA); BAYESIAN NETWORKS; NONLINEAR FILTERING;

ALGORITHMS;

BAYES THEOREM; COMPUTER SIMULATION; NONLINEAR SYSTEM; REVIEW; SIGNAL PROCESSING; STATISTICAL MODEL; STATISTICS;

BAYES THEOREM; COMPUTER SIMULATION; MODELS, STATISTICAL; NONLINEAR DYNAMICS; SIGNAL PROCESSING, COMPUTER-ASSISTED; STOCHASTIC PROCESSES;

EID: 84885931401     PISSN: 21682267     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSMCC.2012.2230254     Document Type: Article

References (124)

1. S. Haykin, Adaptive Filter Theory. Englewood Cliffs,NJ: Prentice-Hall, 1991.
2. J. Weare, "Particle filtering with path sampling and an application to a bimodal ocean current model," J. Comput. Phys., vol. 228, pp. 4312-4331, 2009.
3. M. Murshed, B. Huang, and K. Nandakumar, "Estimation and control of solid oxide fuel cell system," Comput. Chem. Eng., vol. 34, pp. 96-111, 2010.
4. R. Barbieri, L. M. Frank, D. P. Nguyen, M. C. Quirk, V. Solo, M. A. Wilson, and E. N. Brown, "Dynamic analyses of information encoding in neural ensembles," Neural Comput., vol. 16, pp. 277-307, 2004. (Pubitemid 38085366)
5. J. Danielsson, "Stochastic volatility in asset prices estimation with simulated maximum likelihood," J. Econometr., vol. 64, pp. 375-400, 1994.
6. J. Rodríguez-Millán and C. González, "Three mathematica supported proposals for the discretization of nonlinear dynamical control systems," Nonlinear Anal., vol. 63, pp. 617-628, 2005.
7. D. Talay, "Simulation of stochastic differential systems," Lect. Notes Phys.: Probabil. Methods Appl. Phys., vol. 451, pp. 54-96, 1995.
8. B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracking Application. Norwood, MA: Artech House, 2004.
9. S. Arulampalam, S.Maskell, N. Gordon, and T. Clapp, "A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking," IEEE Trans. Signal Process., vol. 50, no. 2, pp. 174-188, Feb. 2002. (Pubitemid 34291500)
10. T. R. Bewley and A. S. Sharma, "Efficient grid-based Bayesian estimation of nonlinear low-dimensional systems with sparse non-Gaussian PDFs," Automatica, vol. 48, pp. 1286-1290, 2012.
11. S. Kramer and H. Sorenson, "Recursive Bayesian estimation using piecewise constant approximations," Automatica, vol. 24, pp. 789-801, 1988.
12. M. Šimandl, J. Královec, and T. Š'oderstř'om, "Advanced point-mass method for nonlinear state estimation," Automatica, vol. 42, pp. 1133-1145, 2006.
13. M. Šimandl, J. Královec, and T. Ś'oderstŕ'om, "Anticipative grid design in point-mass approach to nonlinear state estimation," IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 699-702, Apr. 2002.
14. V. E. Beneš, "Exact finite-dimensional filters for certain diffusions with nonlinear drift," Stochastics, vol. 5, pp. 65-92, 1981. (Pubitemid 12424272)
15. F. E. Daum, "Exact finite-dimensional nonlinear filters," IEEE Trans. Autom. Control, vol. 31, no. 7, pp. 616-622, Jul. 1986.
16. F. E. Daum, "Nonlinear filters: Beyond the Kalman filter," IEEE Aerosp. Electron. Syst. Mag., vol. 20, no. 8, pp. 57-69, Aug. 2005.
17. K. Ito and K. Xiong, "Gaussian filters for nonlinear filtering problems," IEEE Trans. Autom. Control, vol. 45, no. 5, pp. 910-927, May 2000.
18. M. Nørgaard, N. Poulsen, and O. Ravn, "New developments in state estimation for nonlinear systems," Automatica, vol. 36, pp. 1627-1638, 2000.
19. S. Julier and J. Uhlmann, "Unscented filtering and nonlinear estimation," Proc. IEEE, vol. 92, no. 3, pp. 401-422, Mar. 2004. (Pubitemid 40890750)
20. S. Julier and J. Uhlmann, "Corrections to unscented filtering and nonlinear estimation," Proc. IEEE, vol. 92, no. 12, p. 1958, Dec. 2004. (Pubitemid 40890668)
21. A. Doucet, S. Godsill, and C. Andrieu, "On sequential Monte Carlo samplingmethods for Bayesian filtering," J. Statist. Comput., vol. 10, pp. 197-208, 2000.
22. D. Cristian and A. Doucet, "A survey of convergence results on particle filtering methods for practitioners," IEEE Trans. Signal Process., vol. 50, no. 3, pp. 736-746, Mar. 2002. (Pubitemid 34295113)
23. H. A. Blom and E. A. Bloem, "Optimal decomposed particle filtering of two closely spaced Gaussian targets," in Proc. IEEE Conf. Decis. Control Eur. Control Conf., Orlando, FL, Dec. 2011, pp. 7895-7901.
24. F. L. Gland, F. Monbet, and V. D. Tran, "Large sample asymptotics for the ensemble Kalman filter," in The Oxford Handbook of Nonlinear Filtering, D. Crisan and B. Rozovskii, Eds. Oxford, U.K.: Oxford Univ. Press, 2011, ch. 22, pp. 598-631.
25. G. Burgers, P. J. van Leeuwen, and G. Evensen, "Analysis scheme in the ensemble Kalman filter," Month. Weather Rev., vol. 126, pp. 1719-1724, 1998. (Pubitemid 128597660)
26. G. Evensen, "The ensemble Kalman filter: Theoretical formulation and practical implementation," Ocean Dynam., vol. 53, pp. 343-367, 2003.
27. G. Evensen, Data Assimilation: The Ensemble Kalman Filter. Berlin, Germany: Springer-Verlag, 2006.
28. S. Särkkä, "Bayesian estimation of time-varying systems: Discrete-time systems," Written material for the course held in Spring 2012.
29. I. Arasaratnam and S. Haykin, "Cubature Kalman filters," IEEE Trans. Autom. Control, vol. 54, no. 6, pp. 1254-1269, Jun. 2009.
30. T. Yang, P. Mehta, and S. Meyn, "A mean-field control-oriented approach to particle filtering," in Proc. Amer. Control Conf., San Francisco, CA, 2011, pp. 2037-2043.
31. S. Roweis and Z. Ghahramani, "Learning nonlinear dynamical systems using the expectation-maximization algorithms," in Kalman Filtering of Neural Networks, S.Haykin, Ed. NewYork:Wiley, 2001, ch. 6, pp. 175-220.
32. O. Cappé, E. Moulines, and T. Rydén, Inference in HiddenMarkovModels. New York: Springer-Verlag, 2009.
33. O. Cappé and E. Moulines, "On-line expectation-maximization algorithm for latent data models," J. Royal Statist. Soc. Ser. B, vol. 71, pp. 593-613, 2009.
34. D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. New York: Wiley-Interscience, 2006.
35. Y. Bar-Shalom, X.-R. Li, and T. Kirubarajan, Estimation With Applications to Tracking and Navigation. New York: Wiley-Interscience, 2001.
36. G. Kitagawa, "Non-Gaussian state-space modeling of nonstationary time series," J. Amer. Statist. Assoc., vol. 82, pp. 1032-1041, 1987.
37. M. Šimandl and J. Duník, "Derivative-free estimation methods: New results and performance analysis," Automatica, vol. 45, pp. 1749-1757, 2009.
38. S. J. Godsill, A. Doucet, and M. West, "Monte Carlo smoothing for nonlinear time series," J. Amer. Statist. Assoc., vol. 99, no. 465, pp. 156-168, 2004. (Pubitemid 38545203)
39. H. E. Rauch, F. Tung, and C. T. Striebel, "Maximum likelihood estimates of linear dynamic systems," AIAA J., vol. 3, no. 8, pp. 1445-1450, 1965.
40. M. Šimandl, J. Královec, and P. Tichavsky, "Filtering, predictive, and smoothing Cramér-Rao bounds for discrete-time nonlinear dynamic systems," Automatica, vol. 37, pp. 1703-1716, 2001. (Pubitemid 32833041)
41. S. Särkkä and J. Hartikainen, "On Gaussian optimal smoothing of nonlinear state space models," IEEE Trans. Autom. Control, vol. 55, no. 8, pp. 1938-1941, Aug. 2010.
42. H. A. Blom and E. A. Bloem, "Decomposed particle filtering and track swap estimation in tracking two closely spaced targets," in Proc. 14th Int. Conf. Inf. Fusion, Chicago, IL, 2011, pp. 1-8.
43. E. H. Aoki, A. Bagchi, P. Mandal, and Y. Boers, "A theoretical analysis of Bayes-optimal multi-target tracking and labelling," University of Twente, Twente, The Netherlands, Tech. Rep. ISSN 1874-4850, 2011.
44. S. Rangan, A. Fletcher, and V. Goyal, "Asymptotic analysis of map estimation via the replica method and compressed sensing," in Proc. 23rd Annu. Conf. Neur. Inf. Process. Syst., Vancouver, BC, Canada, 2009, pp. 1545-1553.
45. J.-L. Gauvain and C.-H. Lee, "Maximum a posteriori estimation for multivariate Gaussian mixture observations of Markov chains," IEEE Trans. Speech Audio Process., vol. 2, no. 2, pp. 291-298, Apr. 1994.
46. L. Ljung and S. Gunnarsson, "Adaptation and tracking in system identification - a survey," Automatica, vol. 26, pp. 7-21, 1990.
47. L. H. Matthies, R. Szeliski, and T. Kanade, "Kalman filter-based algorithms for estimating depth from image sequences," Int. J. Comput. Vis., vol. 3, pp. 209-236, 1989.
48. G. Welch and G. Bishop, "An introduction to the Kalman filter," Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, Tech. Rep. 95-041, 1995
49. B. Øksendal, "The filtering problem," in Stochastic Differential Equations: An Introduction With Applications. New York: Springer Verlag, 2003, pp. 81-108.
50. Y. C. Ho and R. C. Lee, "Bayesian approach to problems in stochastic estimation and control," IEEE Trans. Autom. Control, vol. 9, no. 4, pp. 333-339, Oct. 1964.
51. R. Kalman, "A new approach to linear filtering and prediction problems," Trans. ASME - J. Basic Eng., vol. 82, pp. 35-45, 1960.
52. B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979.
53. R. Mehra, "On the identification of variances and adaptive Kalman filtering," IEEE Trans. Autom. Control, vol. 15, no. 2, pp. 175-184, Apr. 1970.
54. D. Alspach, "Comments on the identification of variances and adaptive Kalman filtering," IEEE Trans. Autom. Control, vol. 17, no. 6, pp. 843-845, Dec. 1972.
55. S. Sangsuk-Iam and T. Bullock, "Analysis of discrete-time Kalman filtering under incorrect noise covariances," IEEE Trans. Autom. Control, vol. 35, no. 12, pp. 1304-1308, Dec. 1990.
56. B. M. Akesson, J. B. Jorgensen, N. K. Poulsen, and S. B. Jorgensen, "A generalized autocovariance least-squares method for Kalman filter tuning," J. Process Control, vol. 18, pp. 769-779, 2008.
57. M. Rajamani and J. Rawlings, "Estimation of the disturbance structure from data using semidefinite programming and optimal weighting," Automatica, vol. 45, pp. 142-148, 2009.
58. R. Reynolds, "Robust estimation of covariance matrices," IEEE Trans. Autom. Control, vol. 35, no. 9, pp. 1047-1051, Sep. 1990.
59. J. M. Mendel, "State estimation: Filtering examples," in Lessons in Estimation Theory for Signal Processing, Communications, and Control. Englewood Cliffs, NJ: Prentice-Hall, 1995.
60. V. Pugachev and I. Sinitsyn, Stochastic Systems: Theory and Applications. Singapore: World Scientific, 2001.
61. H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time (Applications of Mathematics). New York: Springer-Verlag, 2001.
62. H. van Trees, Detection, Estimation, and Modulation Theory: Part I. New York: Wiley, 1968.
63. H. Cramer, Mathematical Methods of Statistics. Princeton, NJ: Princeton Univ. Press, 1946.
64. C. R. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc., vol. 37, pp. 81-91, 1945.
65. A. Farina, B. Ristic, and D. Benvenuti, "Tracking a ballistic target: Comparison of several nonlinear filters," IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 3, pp. 854-867, Jul. 2002. (Pubitemid 35377500)
66. T. Kerr, "Status of CR-like lower bounds for nonlinear filtering," IEEE Trans. Aerosp. Electron. Syst., vol. 25, no. 5, pp. 590-600, Sep. 1989.
67. P. Tichavsky, C. Muravchik, and A. Nehorai, "Posterior Cramer-Rao bounds for discrete-time nonlinear filtering," IEEE Trans. Signal Process., vol. 46, no. 5, pp. 1386-1396, May 1998. (Pubitemid 128743876)
68. M. Lei, B. van Wyk, and Y. Qi, "Online estimation of the approximate posterior Cramer-Rao lower bound for discrete-time nonlinear filtering," IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 1, pp. 37-57, Jan. 2011.
69. L. Zuo, R. Niu, and P. K. Varshney, "Conditional posterior Cramer-Rao lower bounds for nonlinear sequential Bayesian estimation," IEEE Trans. Signal Process., vol. 59, no. 1, pp. 1-14, Jan. 2011.
70. M. Hernandez, B. Ristic, A. Farina, and L. Timmoneri, "A comparison of two Cramer-Rao bounds for nonlinear filtering with pd < 1," IEEE Trans. Signal Process., vol. 52, no. 9, pp. 2361-2370, Sep. 2004.
71. R. Niu, P. Willett, and Y. Bar-Shalom, "Matrix CRLB scaling due to measurements of uncertain origin," IEEE Trans. Signal Process., vol. 49, no. 7, pp. 1325-1335, Jul. 2001. (Pubitemid 32615797)
72. X. Zhang, P. Willett, and Y. Bar-Shalom, "Dynamic Cramer-Rao bound for target tracking in clutter," IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 4, pp. 1154-1167, Oct. 2005. (Pubitemid 43033910)
73. D. Crisan and B. Rozovskii, The Oxford Handbook of Nonlinear Filtering. London, U.K.: Oxford Univ. Press, 2011.
74. R. van Handel, "Nonlinear filtering and systems theory," in Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, Hungary, 2010.
75. B. Ristic, S. Arulampalam, and N. Gordon, "Suboptimal nonlinear filters," in Beyond the Kalman Filter: Particle Filters for Tracking Application, Norwood, MA, Artech House, 2004, pp. 19-34.
76. H. Tanizaki, "Approximations of nonlinear filters," in Nonlinear Filters: Estimation and Applications. Berlin, Germany: Springer-Verlag, 1996, pp. 175-204.
77. M. Verlaan and A. W. Heemink, "Nonlinearity in data assimilation applications: A practical method for analysis," Month. Weather Rev., vol. 129, pp. 1578-1589, 2001. (Pubitemid 32680051)
78. S. Ali-Löytty, "EfficientGaussian mixture filter for hybrid positioning," in Proc. IEEE/ION Posit. Locat. Navigat. Symp., Monterey, CA, May 2008, pp. 60-66.
79. H. Tanizaki and R. S. Mariano, "Nonlinear filters based on Taylor series expansions," Commun. Statist.: Theor. Methods, vol. 25, pp. 1261-1282, 1996. (Pubitemid 126711035)
80. K. Xiong, H. Zhang, and L. Liu, "Adaptive robust extended Kalman filter for nonlinear stochastic systems," IET Control Theor. Appl., vol. 3, pp. 239-250, 2008. (Pubitemid 351357562)
81. S. Julier and J. Uhlmann, "A general method for approximating nonlinear transformations of probability distributions," Robot, Res. Group, Dept. Eng. Sci., Univ. Oxford, Oxford, U.K., Tech. Rep., 1996.
82. S. Bolognani, L. Tubiana, and M. Zigliotto, "Extended Kalman filter tuning in sensorless PMSM drives," IEEE Trans. Ind. Appl., vol. 39, no. 6, pp. 1741-1747, Nov.-Dec. 2003.
83. C.-B. Chang and J. Tabaczynski, "Application of state estimation to target tracking," IEEE Trans. Autom. Control, vol. 29, no. 2, pp. 98-109, Feb. 1984. (Pubitemid 14570748)
84. H. Tanizaki, Nonlinear Filters: Estimation and Applications. Berlin, Germany: Springer-Verlag, 1996.
85. A. Jazwinski, Stochastic Processing and Filtering Theory. NewYork: Academic, 1970.
86. B. M. Bell and F. W. Cathey, "The iterated Kalman filter update as a Gauss-Newton method," IEEE Trans. Autom. Control, vol. 38, no. 2, pp. 294-297, Feb. 1993.
87. T. Lefebvre, H. Bruyninckx, and J. Schutter, "Kalman filters for nonlinear systems: A comparison of performance," Int. J. Control, vol. 77, pp. 639-653, 2004.
88. K. Spingarn, "Passive position location estimation using the extended Kalman filter," IEEE Trans. Aerosp. Electron. Syst., vol. 23, no. 4, pp. 558-567, Jul. 1987. (Pubitemid 18537391)
89. R. Wishner, J. Tabaczynski, and M. Athans, "A comparison of three nonlinear filters," Automatica, vol. 5, pp. 487-496, 1969.
90. S. J. Gosh, D. Roy, and C. S. Manohar, "New forms of extended Kalman filter via transversal linearization and applications to structural system identification," Comput.Meth. Appl.Mech. Eng., vol. 196, pp. 5063-5083, 2007. (Pubitemid 47423964)
91. N. Saha and D. Roy, "Extended Kalman filters using explicit and derivative-free local linearizations," Appl. Math. Modell., vol. 33, pp. 2545-2563, 2009.
92. X. Song, P. Willett, and S. Zhou, "Posterior Cramer-Rao bounds for Doppler biased multistatic range-only tracking," in Proc. 14th Int. Conf. Inf. Fusion, Chicago, IL, 2011, pp. 1-8.
93. B. Ripley, Stochastic Simulation. New York: Wiley, 1987.
94. S. Julier, J. Uhlmann, and H. F. Durrant-Whyte, "A new method for the nonlinear transformation of means and covariances in filters and estimators," IEEE Trans. Autom. Control, vol. 45, no. 3, pp. 477-482, Mar. 2000.
95. E. A.Wan and R. van der Merwe, "The unscented Kalman filter for nonlinear estimation," in Proc. IEEE 2000 Adapt. Syst. Signal Process. Commun. Control Symp., Lake Louise, AB, Canada, 2000, pp. 153-158.
96. E. A.Wan and R. V. D. Merwe, "The unscented Kalman filter," in Kalman Filtering and Neural Networks, S. Haykin, Ed. New York:Wiley, 2001, ch. 7, pp. 221-280.
97. I. Arasaratnam, S. Haykin, and T. R. Hurd, "Cubature Kalman filtering for continuous-discrete systems: Theory and simulations," IEEE Trans. Signal Process., vol. 58, no. 10, pp. 4977-4993, Oct. 2010.
98. J. Duník, M. Šimandl, and O. Straka, "Adaptive choice of scaling parameter in derivative-free local filters," in Proc. 13th Int. Conf. Inf. Fusion, Edinburgh, U.K., 2010, pp. 1-8.
99. S. Julier, "The scaled unscented transform," in Proc. Amer. Control Conf., Anchorage, AK, 2002, vol. 6, pp. 4555-4559.
100. D. Tenne and T. Singh, "The higher order unscented filter," in Proc. Amer. Control Conf., Denver, CO, Jun. 4-6, 2003, pp. 2441-2446.
101. Y. Wu, M. Wu, D. Hu, and X. Hu, "An improvement to unscented transformation," in Proc. 17th Aust. Joint Conf. Artif. Intell., Cairns, Australia, 2004, vol. 3339, pp. 1024-1029. (Pubitemid 41055474)
102. I. Arasaratnam, S. Haykin, and R. Elliott, "Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature," Proc. IEEE, vol. 95, no. 5, pp. 953-977, May 2007.
103. S. Julier and J.Uhlmann, "Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations," in Proc. Amer. Control Conf., Anchorage, AK, 2002, vol. 2, pp. 887-892.
104. A. Honkela, "Approximating nonlinear transformations of probability distributions for nonlinear independent component analysis," in Proc. Int. Joint Conf. Neural Netw., Budapest, Hungary, 2004, vol. 3, pp. 2169-2174. (Pubitemid 40006594)
105. G. H. Golub and J. H. Welsch, "Calculation of gauss quadrature rules," Math. Comput., vol. 23, pp. 221-230, 1969.
106. K. Pomorski, "Gauss-Hermite approximation formula," Comput. Phys. Commun., vol. 174, pp. 181-186, 2006. (Pubitemid 43133097)
107. T. S. Schei, "A finite-difference method for linearization in nonlinear estimation algorithms," Automatica, vol. 33, pp. 2053-2058, 1997. (Pubitemid 127397989)
108. I. Bilik and J. Tabrikian, "Target tracking in glint noise environment using nonlinear non-Gaussian Kalman filter," in Proc. IEEE Int. Radar Conf., Verona, NY, Apr. 2006, pp. 282-286.
109. N. M.Kwok, Q. M. Ha, S.Huang, G. Dissanayake, and G. Fang, "Mobile robot localization and mapping using a Gaussian sum filter," Int. J. Control Autom. Syst., vol. 5, pp. 251-268, 2007. (Pubitemid 46878376)
110. D. L. Alspach and H. W. Sorenson, "Nonlinear Bayesian estimation using Gaussian sum approximations," IEEE Trans. Autom. Control, vol. 17, no. 4, pp. 439-448, Aug. 1972.
111. D. Alspach and H. Sorenson, "Recursive Bayesian estimation using Gaussian sums," Automatica, vol. 7, pp. 465-479, 1971.
112. J. J. Verbeek, J. R. Nunnik, and N. Vlassis, "Accelerated EM-based clustering of large data sets," Data Min. Knowl. Disc., vol. 13, pp. 291-307, 2006. (Pubitemid 44455022)
113. J. Vermaak, S. Maskell, and M. Briers, "Online sensor registration," in Proc. IEEE Aerosp. Conf., Big Sky, MT, 2005, pp. 2117-2125.
114. W. I. Tam, K. N. Plataniotis, and D. Hatzinakos, "An adaptive Gaussian sum algorithm for radar tracking," Signal Process., vol. 77, pp. 85-104, 1999.
115. J. T. Horwood and A. B. Poore, "Adaptive Gaussian sum filters for space surveillance," IEEE Trans. Autom. Control, vol. 56, no. 8, pp. 1777-1790, Aug. 2011.
116. G. Terejanu, P. Singla, T. Singh, and P. D. Scott, "A novel Gaussian sum filter method for accurate solution to the nonlinear filtering problem," in Proc. 11th Int. Conf. Inf. Fusion, Cologne, Germany, 2008, pp. 1-8.
117. S. Ali-Löytty, "On the convergence of the Gaussian mixture filter," Dept. Math., Tampere Univ. Technol., Tampere, Finland, Tech. Rep. 89, 2008.
118. J. L. Williams. (2003). Gaussian mixture reduction for tracking multiple maneuvering targets in clutter, M.S. thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH [Online]. Available: http://ssg.mit. edu/jlwil
119. P. Zarchan, Tactical and Strategic Missile Guidanc (Progress in Astronautics and Aeronautics). Reston, VA: AIAA, 2002
120. X. R. Li and V. P. Jilkov, "Survey of maneuvering target tracking - I: Dynamic models," IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 4, pp. 1333-1364, Oct. 2003.
121. Mathworks [Online]. Available: www.mathworks.com/matlabcentral/ fileexchange/18189
122. Mathworks [Online]. Available: www.mathworks.com/matlabcentral/ fileexchange/18217
123. G. Welch and G. Bishop. Dept. Comput. Sci., Univ. Northern Carolina at Chapel Hill, Chapel Hill [Online]. Available: www.cs.unc.edu/welch/ kalman
124. Identification and Decision Making Research Group, University ofWest Bohemia, Pilsen, Czech Republic [Online]. Available: http://nft.kky.zcu. cz/tools