# BLUE, BLUP and the Kalman filter: some new results

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## 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.

## Keywords

Best linear unbiased estimation (BLUE) Best linear unbiased prediction (BLUP) Minimum mean squared error (MMSE) Misclosures Kalman filter BLUE–BLUP recursion## Notes

### 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.

## References

- Anderson BDO, Moore JB (1979) Optimal filtering, vol 11. Prentice-hall, Englewood CliffsGoogle Scholar
- Ansley CF, Kohn R (1985) Estimation, filtering, and smoothing in state space models with incompletely specified initial conditions. Ann Stat 13(4):1286–1316Google Scholar
- Bar-Shalom Y, Li X (1993) Estimation and tracking—principles, techniques, and software. Artech House, Inc., NorwoodGoogle Scholar
- Bode H, Shannon C (1950) A simplified derivation of linear least square smoothing and prediction theory. Proc IRE 38(4):417–425CrossRefGoogle Scholar
- Brammer K, Siffling G (1989) Kalman-Bucy filters. Artech House, NorwoodGoogle Scholar
- Candy J (1986) Signal processing: model based approach. McGraw-Hill, Inc, New YorkGoogle Scholar
- Gelb A (1974) Applied optimal estimation. MIT Press, CambridgeGoogle Scholar
- Gibbs B (2011) Advanced Kalman filtering, least-squares and modeling: a practical handbook. Wiley, New YorkCrossRefGoogle Scholar
- Goldberger A (1962) Best linear unbiased prediction in the generalized linear regression model. J Am Stat Assoc 57(298):369–375CrossRefGoogle Scholar
- Grewal MS, Andrews AP (2008) Kalman filtering: theory and practice using MATLAB, 3rd edn. Wiley, New YorkCrossRefGoogle Scholar
- Harvey AC, Phillips GDA (1979) Maximum likelihood estimation of regression models with autoregressive-moving average disturbances. Biometrika 66(1):49–58Google Scholar
- Jazwinski A (1991) Stochastic processes and filtering theory. Dover Publications, New YorkGoogle Scholar
- de Jong P (1991) The diffuse Kalman filter. Ann Stat 19(2):1073–1083Google Scholar
- Kailath T (1968) An innovations approach to least-squares estimation—part I: Linear filtering in additive white noise. IEEE Trans Autom Control 13(6):646–655CrossRefGoogle Scholar
- Kailath T (1981) Lectures on Wiener and Kalman filtering. Springer, BerlinGoogle Scholar
- Kailath T, Sayed AH, Hassibi B (2000) Linear estimation. Prentice-Hall, Englewood CliffsGoogle Scholar
- Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82(1):35–45CrossRefGoogle Scholar
- Maybeck P (1979) Stochastic models, estimation, and control, vol 1. Academic Press, Waltham, republished 1994Google Scholar
- 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 49–155CrossRefGoogle Scholar
- Simon D (2006) Optimal state estimation: Kalman, H [infinity] and nonlinear approaches. Wiley, New YorkCrossRefGoogle Scholar
- Sorenson HW (1966) Kalman filtering techniques. In: Leondes CT (ed) Advances in control systems: theory and applications, vol 3. pp 219–292 Google Scholar
- Stark H, Woods J (1986) Probability, random processes, and estimation theory for engineers. Prentice-Hall, Englewood CliffsGoogle Scholar
- Teunissen PJG (2000) Adjustment theory: an introduction. In: Series on Mathematical Geodesy and Positioning. Delft University PressGoogle Scholar
- Teunissen PJG (2007) Best prediction in linear models with mixed integer/real unknowns: theory and application. J Geod 81(12):759–780CrossRefGoogle Scholar
- Teunissen PJG (2008) On a stronger-than-best property for best prediction. J Geod 82(3):167–175CrossRefGoogle Scholar
- Teunissen PJG, Simons DG, Tiberius CCJM (2005) Probability and observation theory. Delft University, Faculty of Aerospace Engineering, Delft University of Technology, lecture notes AE2-E01Google Scholar
- Tienstra J (1956) Theory of the adjustment of normally distributed observation. Argus, AmsterdamGoogle Scholar
- Zadeh LA, Ragazzini JR (1950) An extension of Wiener’s theory of prediction. J Appl Phys 21(7):645–655CrossRefGoogle Scholar