Block-Wise Weighted Least Square - Revision history

Carlos.Lopez at 11:20, 23 February 2012

2012-02-23T11:20:51Z

← Older revision		Revision as of 11:20, 23 February 2012
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	\|Level=Advanced		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Jaume.Sanz at 12:35, 23 January 2012

2012-01-23T12:35:24Z

← Older revision		Revision as of 12:35, 23 January 2012
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	</math>		</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.		: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.

Jaume.Sanz at 12:31, 23 January 2012

2012-01-23T12:31:25Z

← Older revision		Revision as of 12:31, 23 January 2012
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	{{Article Infobox2		{{Article Infobox2
	\|Category=Fundamentals		\|Category=Fundamentals
	~~\|Title={{PAGENAME}}~~		\|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
	\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. ~~Hernandez~~-Pajares, University of ~~Catalunia~~, Spain.
	\|Level=Advanced		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	\|Logo=gAGE		\|Logo=gAGE
			\|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>:		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>:

Carlos.Lopez at 09:59, 28 October 2011

2011-10-28T09:59:03Z

← Older revision		Revision as of 09:59, 28 October 2011
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	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.		\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
	\|Level=~~Medium~~		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	\|Logo=gAGE		\|Logo=gAGE

Carlos.Lopez at 08:10, 9 August 2011

2011-08-09T08:10:50Z

← Older revision		Revision as of 08:10, 9 August 2011
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	[[Category:Fundamentals]]		[[Category:Fundamentals]]
			[[Category:Solving Navigation Equations]]

Carlos.Lopez at 13:30, 5 August 2011

2011-08-05T13:30:16Z

← Older revision		Revision as of 13:30, 5 August 2011
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	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 (~~\ref{eq:bbwlms}~~) yields:		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>		:<math>
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	* '''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:		* '''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>		::<math>
	\begin{array}{l}		\begin{array}{l}
	{\mathbf Y}={\mathbf G}\,\,{\mathbf X};{\mathbf R}\\[0.1cm]		{\mathbf Y}={\mathbf G}\,\,{\mathbf X}~;~{\mathbf R}\\[0.1cm]
	{\mathbf \Lambda}={\mathbf A}\,\,{\mathbf X};{\mathbf R_\Lambda}		{\mathbf \Lambda}={\mathbf A}\,\,{\mathbf X}~;~{\mathbf R_\Lambda}
	\end{array}		\end{array}
	\qquad \mbox{(9)}		\qquad \mbox{(9)}

Carlos.Lopez at 09:19, 25 July 2011

2011-07-25T09:19:08Z

← Older revision		Revision as of 09:19, 25 July 2011
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	\qquad \mbox{(8)}		\qquad \mbox{(8)}
	</math>		</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.		: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.

Carlos.Lopez at 09:18, 25 July 2011

2011-07-25T09:18:05Z

← Older revision		Revision as of 09:18, 25 July 2011
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	\left[		\left[
	\begin{array}{c}		\begin{array}{c}
	{\mathbf Y_1}		{\mathbf Y_1} \\
	{\mathbf Y_2}		{\mathbf Y_2}
	\end{array}		\end{array}
	\right] =		\right] =
	~~\begin{array}{c}~~
	\left[		\left[
	~~{\mathbf G_1}~~
	~~{\mathbf G_2}~~
	~~\end{array}~~
	~~\right]~~
	~~</math>~~
	~~<math>~~
	~~{\mathbf X}; \qquad~~
	~~{\mathbf R}= \left[~~
	~~\begin{array}{cc}~~
	~~{\mathbf R_1} & {\mathbf 0}~~
	~~{\mathbf 0} & {\mathbf R_2}~~
	~~\end{array}~~
	~~\right]~~
	~~</math>~~




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

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

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

	:<math>		:<math>
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	</math>		</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>		:''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}		\hat{\mathbf X}=({\mathbf G}^T\,{\mathbf R}^{-1}\,{\mathbf G})^{-1}{\mathbf G}^T\,{\mathbf R^{-1}}\,{\mathbf Y}
	\qquad \mbox{(8)}		\qquad \mbox{(8)}
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	* 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:
			* '''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>		:<math>
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	</math>		</math>





	~~==Notes==~~
	~~<references group="footnotes"/>~~

	==References==		==References==

Carlos.Lopez: Created page with "{{Article Infobox2 |Category=Fundamentals |Title={{PAGENAME}} |Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain. |Level=Medium..."

2011-07-25T08:58:53Z

Created page with "{{Article Infobox2 |Category=Fundamentals |Title={{PAGENAME}} |Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain. |Level=Medium..."

New page

{{Article Infobox2
|Category=Fundamentals
|Title={{PAGENAME}}
|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
|Level=Medium
|YearOfPublication=2011
|Logo=gAGE
}}

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] =
\begin{array}{c}
\left[
{\mathbf G_1}
{\mathbf G_2}
\end{array}
\right]
</math>
<math>
{\mathbf X}; \qquad
{\mathbf R}= \left[
\begin{array}{cc}
{\mathbf R_1} & {\mathbf 0}
{\mathbf 0} & {\mathbf R_2}
\end{array}
\right]
</math>

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

==Notes==
<references group="footnotes"/>

==References==
<references/>

[[Category:Fundamentals]]