View source for Block-Wise Weighted Least Square

{{Article Infobox2
|Category=Fundamentals
|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
|Level=Advanced
|YearOfPublication=2011
|Title={{PAGENAME}}
}}
Let's consider two linear <math>[m_1\times n], [m_2\times n]</math> equations systems, sharing the same unknown parameters vector <math>{\mathbf X}</math>:

:<math>
\begin{array}{l}
{\mathbf Y_1}={\mathbf G_1}\,{\mathbf X};{\mathbf R_1}\\[0.3cm]
{\mathbf Y_2}={\mathbf G_2}\,{\mathbf X};{\mathbf R_2}\\
\end{array} \qquad \mbox{(1)}
</math>


where  <math>{\mathbf R_1}</math> and  <math>{\mathbf R_2}</math> are the covariance matrices of measurement vectors <math>{\mathbf Y_1}</math>, <math>{\mathbf Y_2}</math>.


Thence the two systems can be combined into a common <math>[(m_1+m_2)\times n]</math> system as:

:<math>
\left[
\begin{array}{c}
{\mathbf Y_1} \\
{\mathbf Y_2}
\end{array}
\right] =
\left[
\begin{array}{c}
{\mathbf G_1}\\[0.2cm]
{\mathbf G_2}
\end{array}
\right] {\mathbf X} ;\qquad
{\mathbf R}=\left[
\begin{array}{cc}
{\mathbf R_1} & {\mathbf 0}  \\[0.2cm]
{\mathbf 0} & {\mathbf R_2}
\end{array}
\right]
\qquad \mbox{(2)}
</math>


where no correlation between the two measurement vectors  <math>{\mathbf Y_1}</math> and  <math>{\mathbf Y_2}</math> is assumed in matrix <math>{\mathbf R}</math>.


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

:<math>
\hat{\mathbf X}=({\mathbf G}^T\,{\mathbf R}^{-1}\,{\mathbf G})^{-1}{\mathbf G}^T\,{\mathbf R^{-1}}\,{\mathbf Y} \qquad \mbox{(3)}
</math>

:<math>
{\mathbf P}=({\mathbf G}^T\,{\mathbf R}^{-1}\,{\mathbf G})^{-1} \qquad \mbox{(4)}
</math>


it is easy to show that taking the corresponding augmented matrices <math>{\mathbf Y}</math> and <math>{\mathbf G}</math>, the WLS solution of previous system (2) yields:

:<math>
 \hat{\mathbf X}=\left [{\mathbf G_1}^T\,{\mathbf R_1}^{-1}\,{\mathbf G_1} + {\mathbf G_2}^T\,{\mathbf R_2}^{-1}\,{\mathbf G_2} \right ]^{-1} \left [{\mathbf G_1}^T\,{\mathbf R_1^{-1}}\,{\mathbf Y_1} + {\mathbf G_2}^T\,{\mathbf R_2^{-1}}\,{\mathbf Y_2} \right ]
\qquad \mbox{(5)}
</math>

:<math>
{\mathbf P}=\left [{\mathbf G_1}^T\,{\mathbf R_1}^{-1}\,{\mathbf G_1} + {\mathbf G_2}^T\,{\mathbf R_2}^{-1}\,{\mathbf G_2} \right ]^{-1}
\qquad \mbox{(6)}
</math>


'''''Comments:'''''

*''' Recursive computation:''' From previous approach, the following recursive computation of estimate <math>{\mathbf X}</math> can be written:

:<math>
\begin{array}{rl}
{\mathbf P_1}=&\left [ {\mathbf G_1}^T\,{\mathbf R_1}^{-1}\,{\mathbf G_1} \right ]^{-1}\\[0.2cm]
\hat{\mathbf X}_{(1)}=&{\mathbf P_1} \cdot \left [{\mathbf G_1}^T\,{\mathbf R_1^{-1}}\,{\mathbf Y_1} \right ]\\[0.4cm]
{\mathbf P_2}=&\left [{\mathbf P_1}^{-1}+ {\mathbf G_2}^T\,{\mathbf R_2}^{-1}\,{\mathbf G_2} \right ]^{-1}\\[0.2cm]
 \hat{\mathbf X}_{(2)}=& {\mathbf P_2} \cdot \left [{\mathbf P_1^{-1}}\,{\mathbf X_{(1)}} + {\mathbf G_2}^T\,{\mathbf R_2^{-1}}\,{\mathbf Y_2} \right ]\\
\end{array}
\qquad \mbox{(7)}
</math>


:''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)]])

::<math>
\hat{\mathbf X}=({\mathbf G}^T\,{\mathbf R}^{-1}\,{\mathbf G})^{-1}{\mathbf G}^T\,{\mathbf R^{-1}}\,{\mathbf Y}
\qquad \mbox{(8)}
</math>

:and (6), that accumulates the equations without solving until the end [Bierman, 1976] <ref>[Bierman, 1976] Bierman, G., 1976. Factorization Methods fro Discrete Sequential estimation. Academic Press, New York, New York, USA. </ref>. 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 <math>{\mathbf \Lambda}={\mathbf A} {\mathbf X}</math>  with a given weight <math>{\mathbf W}={\mathbf R_\Lambda}^{-1}</math>. Indeed:

::<math>
\begin{array}{l}
{\mathbf Y}={\mathbf G}\,\,{\mathbf X}~;~{\mathbf R}\\[0.1cm]
{\mathbf  \Lambda}={\mathbf A}\,\,{\mathbf X}~;~{\mathbf R_\Lambda}
\end{array}
\qquad \mbox{(9)}
</math>


==References==
<references/>

[[Category:Fundamentals]]
[[Category:Solving Navigation Equations]]