A recursive linear MMSE filter for dynamic systems with unknown state vector means

Original Paper

Abstract

In this contribution we extend Kalman-filter theory by introducing a new recursive linear minimum mean squared error (MMSE) filter for dynamic systems with unknown state-vector means. The recursive filter enables the joint MMSE prediction and estimation of the random state vectors and their unknown means, respectively. We show how the new filter reduces to the Kalman-filter in case the state-vector means are known and we discuss the fundamentally different roles played by the intitialization of the two filters.

Keywords

Minimum mean squared error (MMSE) Best linear unbiased estimation (BLUE) Best linear unbiased prediction (BLUP) Kalman filter BLUE-BLUP recursion 

Mathematics Subject Classification

60G25 60G35 93E11 

References

  1. Acharya, R., Roy, B., Sivaraman, M., Dasgupta, A.: Estimation of equatorial electrojet from total electron content at geomagnetic equator using Kalman filter. Adv. Space Res. 47(6), 938–944 (2011)CrossRefGoogle Scholar
  2. Anderson, B.D.O., Moore, J.B.: Optimal Filtering, vol. 11. Prentice-hall Englewood Cliffs, New Jersey (1979)MATHGoogle Scholar
  3. Bar-Shalom, Y., Li, X.: Estimation and Tracking-Principles, Techniques, and Software. Artech House Inc, Norwood (1993)MATHGoogle Scholar
  4. Bertino, L., Evensen, G., Wackernagel, H.: Combining geostatistics and Kalman filtering for data assimilation in an estuarine system. Inverse Probl. 18(1), 1 (2002)CrossRefMATHGoogle Scholar
  5. Brammer, K., Siffling, G.: Kalman-Bucy Filters. Artech Hous, Norwood (1989)Google Scholar
  6. Candy, J.: Signal Processing: Model Based Approach. McGraw-Hill Inc, New York (1986)Google Scholar
  7. Cao, Y., Chen, Y., Li, P.: Wet refractivity tomography with an improved Kalman-filter method. Adv. Atmos. Sci. 23, 693–699 (2006)CrossRefGoogle Scholar
  8. Christensen R (2001) Advanced Linear Modeling: Multivariate, Time Series, and Spatial Data; Nonparametric Regression and Response Surface Maximization, 2nd edn. Springer, HeidelbergGoogle Scholar
  9. Ferraresi, M., Todini, E., Vignoli, R.: A solution to the inverse problem in groundwater hydrology based on Kalman filtering. J. Hydrol. 175(1), 567–581 (1996)CrossRefGoogle Scholar
  10. Gelb, A.: Applied Optimal Estimation. MIT Press, Cambridge (1974)Google Scholar
  11. Gibbs, B.: Advanced Kalman Filtering, Least-squares and Modeling. A Practical Handbook. Wiley, New York (2011)CrossRefGoogle Scholar
  12. Goldberger, A.: Best linear unbiased prediction in the generalized linear regression model. J. Am. Stat. Assoc. 57(298), 369–375 (1962)CrossRefMATHMathSciNetGoogle Scholar
  13. Grafarend, E.W.: Geodetic applications of stochastic processes. Phys. Earth Planet. Inter. 12(2), 151–179 (1976)CrossRefMathSciNetGoogle Scholar
  14. Grafarend, E.W., Rapp, R.H.: Advances in Geodesy: selected papers from reviews of Geophysics and Space Physics. American Geophysical Union 1, Washington DC (1984)Google Scholar
  15. Grewal, M.S., Andrews, A.P.: Kalman Filtering; Theory and Practice Using MATLAB, 3rd edn. Wiley, New York (2008)CrossRefGoogle Scholar
  16. Gross, R.S., Eubanks, T.M., Steppe, J.A., Freedman, A.P., Dickey, J.O., Runge, T.F.: A Kalman-filter-based approach to combining independent Earth-orientation series. J. Geod. 72(4), 215–235 (1998)CrossRefMATHGoogle Scholar
  17. Herring, T.A., Davis, J.L., Shapiro, I.I.: Geodesy by radio interferometry: the application of Kalman filtering to the analysis of very long baseline interferometry data. J. Geophys. Res (1978–2012) 95(B8), 12561–12581 (1990)CrossRefGoogle Scholar
  18. Ince, C.D., Sahin, M.: Real-time deformation monitoring with GPS and Kalman Filter. Earth Planets Space 52(10), 837–840 (2000)Google Scholar
  19. Jazwinski, A.: Stochastic Processes and Filtering Theory. Dover Publications, New York (1991)Google Scholar
  20. Kailath, T.: A view of three decades of linear filtering theory. IEEE Trans. Inf. Theory 20(2), 146–181 (1974)CrossRefMATHMathSciNetGoogle Scholar
  21. Kailath, T.: Lectures on Wiener and Kalman Filtering, p. 140. Springer, Heidelberg (1981)CrossRefMATHGoogle Scholar
  22. Kailath, T., Sayed, A.H., Hassibi, B.: Linear Estimation. Prentice-Hall, New Jersey (2000)Google Scholar
  23. Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35–45 (1960)CrossRefGoogle Scholar
  24. Marx, B.A., Potthast, R.W.E.: On instabilities in data assimilation algorithms. GEM-Int. J. Geomath. 3(2), 253–278 (2012)CrossRefMATHMathSciNetGoogle Scholar
  25. Maybeck, P.: Stochastic Models, Estimation, and Control, vol. 1, Academic Press, Massachusetts (1979)Google Scholar
  26. Sanso, F.: The minimum mean square estimation error principle in physical geodesy (stochastic and non-stochastic interpretation). Boll. Geod. Sci. Affi. 39(2), 112–129 (1980)Google Scholar
  27. Sanso, F.: Statistical methods in physical geodesy. In: Sunkel, H. (ed.) Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, vol. 7, pp. 49–155. Springer, Berlin (1986)CrossRefGoogle Scholar
  28. Simon, D.: Optimal state estimation: Kalman, H (infinity) and Nonlinear Approaches. Wiley, Hoboken (2006)CrossRefGoogle Scholar
  29. Sorenson, H.W.: Kalman filtering techniques. In: Leondes, C.T., (ed.) Advances in Control Systems Theory and Applications, 3, 219–292 (1966)Google Scholar
  30. Stark, H., Woods, J.: Probability, Random Processes, and Estimation Theory for Engineers. Prentice-Hall Englewood Cliffs, New Jersey (1986)Google Scholar
  31. Teunissen, P.J.G.: Best prediction in linear models with mixed integer/real unknowns: theory and application. J. Geod. 81(12), 759–780 (2007)CrossRefMATHGoogle Scholar
  32. Teunissen, P.J.G., Khodabandeh, A.: BLUE, BLUP and the Kalman filter: some new results. J. Geod. 87(5), 1–13 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Spatial Sciences, GNSS Research CentreCurtin University of TechnologyPerthAustralia
  2. 2.Department of Geoscience and Remote SensingDelft University of TechnologyDelftThe Netherlands

Personalised recommendations