Parameters adjustment - Revision history

Dsosa at 10:22, 26 February 2019

2019-02-26T10:22:34Z

← Older revision		Revision as of 10:22, 26 February 2019
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	Due to the error term <math>{\boldsymbol \varepsilon}</math>, in general <math>{\mathbf Y}={\mathbf G}\,{\mathbf X}</math> defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector <math>{\mathbf ~~\hat{~~X}}</math> that minimises the discrepancy in the equations system. That is, the vector <math>{\mathbf ~~\hat{~~X}}</math> providing the "best fit" of <math>{\mathbf Y} \simeq {\mathbf G}\,{\mathbf ~~\hat{~~X}}</math> in a given sense.		Due to the error term <math>{\boldsymbol \varepsilon}</math>, in general <math>{\mathbf Y}={\mathbf G}\,{\mathbf X}</math> defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector <math>\hat{\mathbf X}</math> that minimises the discrepancy in the equations system. That is, the vector <math>\hat{\mathbf X}</math> providing the "best fit" of <math>{\mathbf Y} \simeq {\mathbf G}\,\hat{\mathbf X}</math> in a given sense.


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	:<math>		:<math>
	\begin{array}{l}		\begin{array}{l}
	min \\|{\mathbf Y}-{\mathbf ~~\hat{~~Y}}\\|^2=min \left[ \sum_{i=1}^n{\left ( y_i-\hat{y}_i \right )^2 }\right ] \qquad where \qquad {\mathbf ~~\hat{~~Y}}={\mathbf G}\,{\mathbf ~~\hat{~~X}}		min \\|{\mathbf Y}-\hat{\mathbf Y}\\|^2=min \left[ \sum_{i=1}^n{\left ( y_i-\hat{y}_i \right )^2 }\right ] \qquad where \qquad \hat{\mathbf Y}={\mathbf G}\,\hat{\mathbf X}
	\qquad \mbox{(3)} \end{array}		\qquad \mbox{(3)} \end{array}
	</math>		</math>


	The discrepancy vector between the measurements <math>{\mathbf Y}</math> and the fitted model <math>{\mathbf ~~\hat{~~Y}}={\mathbf G}\,{\mathbf ~~\hat{~~X}}</math> is usually called the'' residual vector:''		The discrepancy vector between the measurements <math>{\mathbf Y}</math> and the fitted model <math>\hat{\mathbf Y }={\mathbf G}\,\hat{\mathbf X}</math> is usually called the'' residual vector:''

	:<math>		:<math>
	{\mathbf r}={\mathbf Y}-{\mathbf ~~\hat{~~Y}}={\mathbf Y}-{\mathbf G}\,{\mathbf ~~\hat{~~X}}		{\mathbf r}={\mathbf Y}-\hat{\mathbf Y}={\mathbf Y}-{\mathbf G}\,\hat{\mathbf X }
	\qquad \mbox{(4)} </math>		\qquad \mbox{(4)} </math>


	Thence, the Least-Squares estimator solution defined by equation (3), gives the vector <math>{\mathbf ~~\hat{~~X}}</math> that minimises <ref group="footnotes">The equation (3), where a quadratic sum is minimised, could be interpreted in physical terms as minimising the energy of the error fit. Thence the estimate <math>{\mathbf ~~\hat{~~X}}</math> can be seen as an equilibrium solution.</ref> the residuals quadratic norm <math>\|\|{\mathbf r}\|\|^2</math>.		Thence, the Least-Squares estimator solution defined by equation (3), gives the vector <math>\hat{\mathbf X}</math> that minimises <ref group="footnotes">The equation (3), where a quadratic sum is minimised, could be interpreted in physical terms as minimising the energy of the error fit. Thence the estimate <math>\hat{\mathbf X}</math> can be seen as an equilibrium solution.</ref> the residuals quadratic norm <math>\|\|{\mathbf r}\|\|^2</math>.


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	where <math>{\mathbf S}</math> is a symmetrical, idempotent Projection matrix		where <math>{\mathbf S}</math> is a symmetrical, idempotent Projection matrix

	:<math>{\mathbf S}={\mathbf I}-{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,;\;\;{\mathbf S}^T={\mathbf S}\,\;\;;\;\;{\mathbf S}^2={\mathbf S}\,;\;\; {\mathbf r}= {\mathbf S}\, {\mathbf Y} \, {\boldsymbol \perp} \,{\mathbf ~~\hat{~~Y}}		:<math>{\mathbf S}={\mathbf I}-{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,;\;\;{\mathbf S}^T={\mathbf S}\,\;\;;\;\;{\mathbf S}^2={\mathbf S}\,;\;\; {\mathbf r}= {\mathbf S}\, {\mathbf Y} \, {\boldsymbol \perp} \,\hat{\mathbf Y}
	\qquad \mbox{(6)}		\qquad \mbox{(6)}
	</math>		</math>
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	:<math>		:<math>
	{\mathbf{\Delta X}}={\mathbf ~~\hat{~~X}}-{\mathbf {X}}=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,{\boldsymbol \varepsilon}		{\mathbf{\Delta X}}=\hat{\mathbf X}-{\mathbf {X}}=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,{\boldsymbol \varepsilon}
	\qquad \mbox{(7)}		\qquad \mbox{(7)}
	</math>		</math>

Dsosa at 10:07, 26 February 2019

2019-02-26T10:07:34Z

← Older revision		Revision as of 10:07, 26 February 2019
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	Due to the error term <math>{\boldsymbol \varepsilon}</math>, in general <math>{\mathbf Y}={\mathbf G}\,{\mathbf X}</math> defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector <math>{\mathbf \hat{X}}</math> that minimises the discrepancy in the equations system. That is, the vector <math>{\mathbf~~} {~~\hat{X}}</math> providing the "best fit" of <math>{\mathbf Y} \simeq {\mathbf G}\,{\mathbf \hat{X}}</math> in a given sense.		Due to the error term <math>{\boldsymbol \varepsilon}</math>, in general <math>{\mathbf Y}={\mathbf G}\,{\mathbf X}</math> defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector <math>{\mathbf \hat{X}}</math> that minimises the discrepancy in the equations system. That is, the vector <math>{\mathbf \hat{X}}</math> providing the "best fit" of <math>{\mathbf Y} \simeq {\mathbf G}\,{\mathbf \hat{X}}</math> in a given sense.

Dsosa at 10:03, 26 February 2019

2019-02-26T10:03:46Z

← Older revision		Revision as of 10:03, 26 February 2019
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	Due to the error term <math>{\boldsymbol \varepsilon}</math>, in general <math>{\mathbf Y}={\mathbf G}\,{\mathbf X}</math> defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector <math>{\mathbf \hat{X}}</math> that minimises the discrepancy in the equations system. That is, the vector <math>{\mathbf \hat(X)}</math> providing the "best fit" of <math>{\mathbf Y} \simeq {\mathbf G}\,{\mathbf \hat{X}}</math> in a given sense.		Due to the error term <math>{\boldsymbol \varepsilon}</math>, in general <math>{\mathbf Y}={\mathbf G}\,{\mathbf X}</math> defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector <math>{\mathbf \hat{X}}</math> that minimises the discrepancy in the equations system. That is, the vector <math>{\mathbf} {\hat{X}}</math> providing the "best fit" of <math>{\mathbf Y} \simeq {\mathbf G}\,{\mathbf \hat{X}}</math> in a given sense.

Dsosa at 10:00, 26 February 2019

2019-02-26T10:00:42Z

← Older revision		Revision as of 10:00, 26 February 2019
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	Due to the error term <math>{\boldsymbol \varepsilon}</math>, in general <math>{\mathbf Y}={\mathbf G}\,{\mathbf X}</math> defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector <math>{\mathbf \hat{X}}</math> that minimises the discrepancy in the equations system. That is, the vector <math>{\mathbf \hat{X}}</math> providing the "best fit" of <math>{\mathbf Y} \simeq {\mathbf G}\,{\mathbf \hat{X}}</math> in a given sense.		Due to the error term <math>{\boldsymbol \varepsilon}</math>, in general <math>{\mathbf Y}={\mathbf G}\,{\mathbf X}</math> defines an incompatible system (i.e., there is not an "exact" solution fulfilling the system). In this context, the parameters' solution can be taken as the vector <math>{\mathbf \hat{X}}</math> that minimises the discrepancy in the equations system. That is, the vector <math>{\mathbf \hat(X)}</math> providing the "best fit" of <math>{\mathbf Y} \simeq {\mathbf G}\,{\mathbf \hat{X}}</math> in a given sense.

Jaume.Sanz at 19:49, 4 February 2013

2013-02-04T19:49:21Z

← Older revision		Revision as of 19:49, 4 February 2013
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	where <math>{\mathbf S}</math> is a symmetrical, idempotent Projection matrix		where <math>{\mathbf S}</math> is a symmetrical, idempotent Projection matrix

	:<math>{\mathbf S}={\mathbf I}-{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,;\;\;{\mathbf S}^T={\mathbf S}\,\;\;;\;\;{\mathbf S}^2={\mathbf S}\,;\;\; {\mathbf r}= {\mathbf S}\, {\mathbf Y} \, {\boldsymbol \perp} \,{\mathbf \hat{X}}		:<math>{\mathbf S}={\mathbf I}-{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,;\;\;{\mathbf S}^T={\mathbf S}\,\;\;;\;\;{\mathbf S}^2={\mathbf S}\,;\;\; {\mathbf r}= {\mathbf S}\, {\mathbf Y} \, {\boldsymbol \perp} \,{\mathbf \hat{Y}}
	\qquad \mbox{(6)}		\qquad \mbox{(6)}
	</math>		</math>

Jaume.Sanz at 12:30, 26 December 2012

2012-12-26T12:30:45Z

← Older revision		Revision as of 12:30, 26 December 2012
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	The ~~last two expressions~~ become simpler by assuming uncorrelated values with identical variance <math>\sigma^2</math>. That is, taking <math>{\mathbf R}=E[{\boldsymbol \varepsilon} \, {\boldsymbol \varepsilon}^T]=\sigma^2\,{\mathbf I}</math>, thence:		The expression of <math>{\mathbf P}</math> become simpler by assuming uncorrelated values with identical variance <math>\sigma^2</math>. That is, taking <math>{\mathbf R}=E[{\boldsymbol \varepsilon} \, {\boldsymbol \varepsilon}^T]=\sigma^2\,{\mathbf I}</math>, thence:

	:<math>		:<math>
	{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1}~~\;;\;\;\;~~		{\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1}
	~~\mbox{MSE}_{_{\mathbf \Delta X}}=n\,\sigma^2~~
	\qquad \mbox{(9)}		\qquad \mbox{(9)}
	</math>		</math>

Carlos.Lopez at 11:34, 23 February 2012

2012-02-23T11:34:52Z

← Older revision		Revision as of 11:34, 23 February 2012
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	\|Level=Advanced		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Jaume.Sanz at 12:13, 23 January 2012

2012-01-23T12:13:40Z

← Older revision		Revision as of 12:13, 23 January 2012
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	Substituting (5) and (2) in (4) the "post-fit"-residual vector is:		Substituting (5) and (2) in (4) the'' post-fit'' -residual vector is:

	:<math>		:<math>

Jaume.Sanz at 12:12, 23 January 2012

2012-01-23T12:12:55Z

← Older revision		Revision as of 12:12, 23 January 2012
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	Substituting (5) and (2) in (4) the ~~{\em~~ post-fit}-residual vector is:		Substituting (5) and (2) in (4) the "post-fit"-residual vector is:

	:<math>		:<math>

Jaume.Sanz at 11:51, 23 January 2012

2012-01-23T11:51:38Z

← Older revision		Revision as of 11:51, 23 January 2012
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	A common criterion used in GNSS is the Least-Squares ~~<math>{\mathbf Least-Squares}</math>~~ adjustment, which is defined by the condition:		A common criterion used in GNSS is the Least-Squares adjustment, which is defined by the condition:

	:<math>		:<math>