Abstract
In this contribution, we extend ‘Kalman-filter’ theory by introducing a new BLUE–BLUP recursion of the partitioned measurement and dynamic models. Instead of working with known state-vector means, we relax the model and assume these means to be unknown. The recursive BLUP is derived from first principles, in which a prominent role is played by the model’s misclosures. As a consequence of the mean state-vector relaxing assumption, the recursion does away with the usual need of having to specify the initial state-vector variance matrix. Next to the recursive BLUP, we introduce, for the same model, the recursive BLUE. This extension is another consequence of assuming the state-vector means unknown. In the standard Kalman filter set-up with known state-vector means, such difference between estimation and prediction does not occur. It is shown how the two intertwined recursions can be combined into one general BLUE–BLUP recursion, the outputs of which produce for every epoch, in parallel, the BLUP for the random state-vector and the BLUE for the mean of the state-vector.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson BDO, Moore JB (1979) Optimal filtering, vol 11. Prentice-hall, Englewood Cliffs
Ansley CF, Kohn R (1985) Estimation, filtering, and smoothing in state space models with incompletely specified initial conditions. Ann Stat 13(4):1286–1316
Bar-Shalom Y, Li X (1993) Estimation and tracking—principles, techniques, and software. Artech House, Inc., Norwood
Bode H, Shannon C (1950) A simplified derivation of linear least square smoothing and prediction theory. Proc IRE 38(4):417–425
Brammer K, Siffling G (1989) Kalman-Bucy filters. Artech House, Norwood
Candy J (1986) Signal processing: model based approach. McGraw-Hill, Inc, New York
Gelb A (1974) Applied optimal estimation. MIT Press, Cambridge
Gibbs B (2011) Advanced Kalman filtering, least-squares and modeling: a practical handbook. Wiley, New York
Goldberger A (1962) Best linear unbiased prediction in the generalized linear regression model. J Am Stat Assoc 57(298):369–375
Grewal MS, Andrews AP (2008) Kalman filtering: theory and practice using MATLAB, 3rd edn. Wiley, New York
Harvey AC, Phillips GDA (1979) Maximum likelihood estimation of regression models with autoregressive-moving average disturbances. Biometrika 66(1):49–58
Jazwinski A (1991) Stochastic processes and filtering theory. Dover Publications, New York
de Jong P (1991) The diffuse Kalman filter. Ann Stat 19(2):1073–1083
Kailath T (1968) An innovations approach to least-squares estimation—part I: Linear filtering in additive white noise. IEEE Trans Autom Control 13(6):646–655
Kailath T (1981) Lectures on Wiener and Kalman filtering. Springer, Berlin
Kailath T, Sayed AH, Hassibi B (2000) Linear estimation. Prentice-Hall, Englewood Cliffs
Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82(1):35–45
Maybeck P (1979) Stochastic models, estimation, and control, vol 1. Academic Press, Waltham, republished 1994
Sanso F (1986) Statistical methods in physical geodesy. In: Sunkel H (ed) Mathematical and numerical techniques in physical geodesy. Lecture notes in earth sciences, vol 7. Springer, Berlin, pp 49–155
Simon D (2006) Optimal state estimation: Kalman, H [infinity] and nonlinear approaches. Wiley, New York
Sorenson HW (1966) Kalman filtering techniques. In: Leondes CT (ed) Advances in control systems: theory and applications, vol 3. pp 219–292
Stark H, Woods J (1986) Probability, random processes, and estimation theory for engineers. Prentice-Hall, Englewood Cliffs
Teunissen PJG (2000) Adjustment theory: an introduction. In: Series on Mathematical Geodesy and Positioning. Delft University Press
Teunissen PJG (2007) Best prediction in linear models with mixed integer/real unknowns: theory and application. J Geod 81(12):759–780
Teunissen PJG (2008) On a stronger-than-best property for best prediction. J Geod 82(3):167–175
Teunissen PJG, Simons DG, Tiberius CCJM (2005) Probability and observation theory. Delft University, Faculty of Aerospace Engineering, Delft University of Technology, lecture notes AE2-E01
Tienstra J (1956) Theory of the adjustment of normally distributed observation. Argus, Amsterdam
Zadeh LA, Ragazzini JR (1950) An extension of Wiener’s theory of prediction. J Appl Phys 21(7):645–655
Acknowledgments
P.J.G. Teunissen is the recipient of an Australian Research Council Federation Fellowship (project number FF0883188). The research of A. Khodabandeh was carried out whilst a Curtin International Research Scholar at Curtin’s GNSS Research Centre. All this support is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Teunissen, P.J.G., Khodabandeh, A. BLUE, BLUP and the Kalman filter: some new results. J Geod 87, 461–473 (2013). https://doi.org/10.1007/s00190-013-0623-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-013-0623-6