220
C. Greco and M. Vasile
Gaussian filters, which approximate the posterior as Gaussian via moment matching
approximations, was sketched. Lastly, the sampling-based particle filter was derived
as a general practical method to compute the filtering solution when the assumptions
of the previous methods are too restrictive, e.g. when distributions other than
Gaussian are involved.
References
1. H. Akashi, H. Kumamoto, Random sampling approach to state estimation in switching
environments. Automatica 13(4), 429–434 (1977)
2. I. Arasaratnam, S. Haykin, Cubature Kalman filters. Trans. Autom. Control. 54(6), 1254–1269
(2009)
3. M.S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for online
nonlinear/non-gaussian bayesian tracking. IEEE Trans. Signal Proc. 50(2), 174–188 (2002)
4. R.H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised edn.
(American Institute of Aeronautics and Astronautics, Reston, 1999)
5. A. Beskos, D. Crisan, A. Jasra, K. Kamatani, Y. Zhou, A stable particle filter in highdimensions (2014). Preprint arXiv:1412.3501
6. AT. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications
(McGraw-Hill, New York, 1960)
7. G.J. Bierman, Factorization Methods for Discrete Sequential Estimation (Dover, Illinois, 2006)
8. S. Brooks, A. Gelman, G.L. Jones, X.L. Meng, Handbook of Markov Chain Monte Carlo
(Chapman & Hall/CRC, Boca Raton, 2011)
9. R. Bucy, P. Joseph, Filtering for Stochastic Processes (Wiley, Hoboken, 1968)
10. J.V. Candy, Bayesian Signal Processing: Classical, Modern and Particle Filtering Methods,
2nd edn. (Wiley, Hoboken, 2016). https://doi.org/10.1002/9781119125495
11. C. Casella, C.P. Rober, Rao-Blackwellisation of sampling schemes. Biometrika 83(1), 81–94
(1996)
12. S. Chakravorty, M. Kumar, P. Singla, A quasi-Gaussian Kalman filter, in American Control
Conference, Minneapolis (2006). https://doi.org/10.1109/ACC.2006.1655484
13. S. Challa, Y. Bar-Shalom, Nonlinear filter design using Fokker-Planck-Kolmogorov probability
density evolutions. IEEE Trans. Aerosp. Electron. Syst. 36(1), 309–315 (2000). https://doi.org/
10.1109/7.826335
14. S. Challa, Y. Bar-Shalom, V. Krishnamurthy, Nonlinear filtering via generalized Edgeworth
series and Gauss-Hermite quadrature. IEEE Trans. Signal Proc. 48(6), 1816–1820 (2000).
https://doi.org/10.1109/78.845944
15. S. Challa, F.A. Faruqi, Application of Chebechev’s inequality theorem in the design of optimal
non-linear filters, in Proceedings of the IEEE International Conference on Acoustics, Speech
and Signal Processing, vol. 3 (1998). https://doi.org/10.1109/ICASSP.1998.681678
16. H. Cox, On the Estimation of state variables and parameters for noisy dynamic systems. IEEE
Trans. Autom. Control 9(1), 5–12 (1964). https://doi.org/10.1109/TAC.1964.1105635
17. W.F. Denham, S. Pines, Sequential estimation when measurement function nonlinearity is
comparable to measurement error. AIAA J. 4(6), 1071–1076 (1966). https://doi.org/10.2514/3.
3606
18. A. Doucet, W. Xiaodong, Monte Carlo methods for signal processing: a review in the statistical
signal processing context. IEEE Signal Proc. Mag. 22(6), 152–170 (2005). https://doi.org/10.
1109/MSP.2005.1550195
19. A. Doucet, S. Godsill, C. Andrieu, On sequential Monte Carlo sampling methods for bayesian
filtering. Stat. Comput. 10(3), 197–208 (2000). https://doi.org/10.1023/A:1008935410038
C. Greco and M. Vasile
Gaussian filters, which approximate the posterior as Gaussian via moment matching
approximations, was sketched. Lastly, the sampling-based particle filter was derived
as a general practical method to compute the filtering solution when the assumptions
of the previous methods are too restrictive, e.g. when distributions other than
Gaussian are involved.
References
1. H. Akashi, H. Kumamoto, Random sampling approach to state estimation in switching
environments. Automatica 13(4), 429–434 (1977)
2. I. Arasaratnam, S. Haykin, Cubature Kalman filters. Trans. Autom. Control. 54(6), 1254–1269
(2009)
3. M.S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for online
nonlinear/non-gaussian bayesian tracking. IEEE Trans. Signal Proc. 50(2), 174–188 (2002)
4. R.H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised edn.
(American Institute of Aeronautics and Astronautics, Reston, 1999)
5. A. Beskos, D. Crisan, A. Jasra, K. Kamatani, Y. Zhou, A stable particle filter in highdimensions (2014). Preprint arXiv:1412.3501
6. AT. Bharucha-Reid, Elements of the Theory of Markov Processes and Their Applications
(McGraw-Hill, New York, 1960)
7. G.J. Bierman, Factorization Methods for Discrete Sequential Estimation (Dover, Illinois, 2006)
8. S. Brooks, A. Gelman, G.L. Jones, X.L. Meng, Handbook of Markov Chain Monte Carlo
(Chapman & Hall/CRC, Boca Raton, 2011)
9. R. Bucy, P. Joseph, Filtering for Stochastic Processes (Wiley, Hoboken, 1968)
10. J.V. Candy, Bayesian Signal Processing: Classical, Modern and Particle Filtering Methods,
2nd edn. (Wiley, Hoboken, 2016). https://doi.org/10.1002/9781119125495
11. C. Casella, C.P. Rober, Rao-Blackwellisation of sampling schemes. Biometrika 83(1), 81–94
(1996)
12. S. Chakravorty, M. Kumar, P. Singla, A quasi-Gaussian Kalman filter, in American Control
Conference, Minneapolis (2006). https://doi.org/10.1109/ACC.2006.1655484
13. S. Challa, Y. Bar-Shalom, Nonlinear filter design using Fokker-Planck-Kolmogorov probability
density evolutions. IEEE Trans. Aerosp. Electron. Syst. 36(1), 309–315 (2000). https://doi.org/
10.1109/7.826335
14. S. Challa, Y. Bar-Shalom, V. Krishnamurthy, Nonlinear filtering via generalized Edgeworth
series and Gauss-Hermite quadrature. IEEE Trans. Signal Proc. 48(6), 1816–1820 (2000).
https://doi.org/10.1109/78.845944
15. S. Challa, F.A. Faruqi, Application of Chebechev’s inequality theorem in the design of optimal
non-linear filters, in Proceedings of the IEEE International Conference on Acoustics, Speech
and Signal Processing, vol. 3 (1998). https://doi.org/10.1109/ICASSP.1998.681678
16. H. Cox, On the Estimation of state variables and parameters for noisy dynamic systems. IEEE
Trans. Autom. Control 9(1), 5–12 (1964). https://doi.org/10.1109/TAC.1964.1105635
17. W.F. Denham, S. Pines, Sequential estimation when measurement function nonlinearity is
comparable to measurement error. AIAA J. 4(6), 1071–1076 (1966). https://doi.org/10.2514/3.
3606
18. A. Doucet, W. Xiaodong, Monte Carlo methods for signal processing: a review in the statistical
signal processing context. IEEE Signal Proc. Mag. 22(6), 152–170 (2005). https://doi.org/10.
1109/MSP.2005.1550195
19. A. Doucet, S. Godsill, C. Andrieu, On sequential Monte Carlo sampling methods for bayesian
filtering. Stat. Comput. 10(3), 197–208 (2000). https://doi.org/10.1023/A:1008935410038