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Block-Wise Weighted Least Square

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FundamentalsFundamentals
Title Block-Wise Weighted Least Square
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Advanced
Year of Publication 2011

Let's consider two linear equations systems, sharing the same unknown parameters vector :


where and are the covariance matrices of measurement vectors , .


Thence the two systems can be combined into a common system as:


where no correlation between the two measurement vectors and is assumed in matrix .


From (3) and (4) (see Best Linear Unbiased Minimum-Variance Estimator (BLUE))


it is easy to show that taking the corresponding augmented matrices and , the WLS solution of previous system (2) yields:


Comments:

  • Recursive computation: From previous approach, the following recursive computation of estimate can be written:


Note: If only the final estimate is desired, it is best not to process data sequentially using (7), but instead to apply (see Best Linear Unbiased Minimum-Variance Estimator (BLUE))
and (6), that accumulates the equations without solving until the end [Bierman, 1976] [1]. This could be especially useful in case of numerical instabilities, avoiding the propagation of the numerical inaccuracies along the recursive steps.


  • Constrains: A priory information can be added to the linear system (1) as constrain equations with a given weight . Indeed:


References

  1. ^ [Bierman, 1976] Bierman, G., 1976. Factorization Methods fro Discrete Sequential estimation. Academic Press, New York, New York, USA.