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Parameters adjustment

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

The equation (1) neglects the measurement noise and missmodelling

[math] R^j=\rho^j+c(\delta t-\delta t^j)+T^j+\hat{\alpha}\, I^j+TGD^j+\mathcal{M}^j+{\boldsymbol \varepsilon}^j \qquad \mbox{(1)} [/math]


If such errors ([math]{\boldsymbol \varepsilon}[/math]) are explicitly written, thence the lineal model is as follows:

[math] {\mathbf Y}={\mathbf G}\,{\mathbf X}+{\boldsymbol \varepsilon} \qquad \mbox{(2)} [/math]


where the error term [math]{\boldsymbol \varepsilon}[/math] is only known from some statistical properties, usually the mean [math]{\mathbf m}=E[{\boldsymbol \varepsilon}][/math] and covariance matrix [math]{\mathbf R}=E[{\boldsymbol \varepsilon} \, {\boldsymbol \varepsilon}^T][/math].


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.


A common criterion used in GNSS is the Least-Squares adjustment, which is defined by the condition:

[math] \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}} \qquad \mbox{(3)} \end{array} [/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:

[math] {\mathbf r}={\mathbf Y}-{\mathbf \hat{Y}}={\mathbf Y}-{\mathbf G}\,{\mathbf \hat{X}} \qquad \mbox{(4)} [/math]


Thence, the Least-Squares estimator solution defined by equation (3), gives the vector [math]{\mathbf \hat{X}}[/math] that minimises [footnotes 1] the residuals quadratic norm [math]||{\mathbf r}||^2[/math].


From basic results of linear algebra, it follows that the solution fulfilling the condition (3) is given by:

[math] \hat{\mathbf X}=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,{\mathbf Y} \qquad \mbox{(5)} [/math]


Substituting (5) and (2) in (4) the post-fit -residual vector is:

[math] {\mathbf r}=\left [\mathbf{I}-{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\right]\, {\mathbf Y}= {\mathbf S}\, {\mathbf Y}={\mathbf S}\,{\boldsymbol \varepsilon} [/math]


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}} \qquad \mbox{(6)} [/math]


From (5) and (2) the estimator error can be written as:

[math] {\mathbf{\Delta X}}={\mathbf \hat{X}}-{\mathbf {X}}=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,{\boldsymbol \varepsilon} \qquad \mbox{(7)} [/math]


Assuming that the measurements minus model (i.e., prefit-residuals) have mean zero errors ([math]E[{\boldsymbol \varepsilon}]=0[/math]) and covariance matrix [math]{\mathbf R}[/math], thence, the mean error, covariance matrix and Mean-Square Error (MSE) of the estimator are given by:

[math] \begin{array}{l} \begin{array}{rl} {\mathbf m}_{_{\mathbf \Delta X}} =&E[{\mathbf \Delta X}]=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T\,E[{\boldsymbol \varepsilon}]=0 \end{array}\\[0.3cm] \begin{array}{rl} {\mathbf P}_{_{\mathbf \Delta X}}= &E[{\mathbf \Delta X} \, {\mathbf \Delta X}^T]=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T \,E[{\boldsymbol \varepsilon}\, {\boldsymbol \varepsilon}^T]\,{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1}=({\mathbf G}^T\,{\mathbf G})^{-1}{\mathbf G}^T \,\,{\mathbf R}\,\,{\mathbf G}({\mathbf G}^T\,{\mathbf G})^{-1} \end{array} \\[0.3cm] \begin{array}{rl} \mbox{MSE}_{_{\mathbf \Delta X}}= &E[{\mathbf \Delta X}^T \, {\mathbf \Delta X}]= trace ({\mathbf P}_{_{\mathbf \Delta X}}) \end{array}~\\ \end{array} \qquad \mbox{(8)} [/math]


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] {\mathbf P}_{_{\mathbf{\Delta X}}}=\sigma^2\,({\mathbf G}^T\,{\mathbf G})^{-1} \qquad \mbox{(9)} [/math]


For more information, please go to the following articles:


Notes

  1. ^ 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.

References