Deformation (congruence) analysis - statistical tests

by Micha ⌂, Bad Vilbel, Friday, April 23, 2021, 21:45 (22 days ago) @ Krzysztof

Hello Krzysztof,

- In my opinion, please correct me if I am wrong, we may not test observations against possible blunders and points against possible displacements at the same stage.

Yes, I fully agree!

We should first test observations against blunders, based on the unconstrained model given by me in Eq. (2).

Yes, I agree. Estimating both epochs individually to check the data and to adapt the stochastic model is recommended.

Only then, we may test datum points against displacements, based, e.g., on the congruence/constrained model (implicit hypothesis method).

Yes, I agree. On this stage, we should test the datum points using explicit or implicit hypothesis tests or a similar technique.

In other words, we should perform these two testing procedures separately. Otherwise, possible blunder(s) and displacement(s) will mix and it will be difficult to identify and adapt them.

Yes, I fully agree. However, the test statistic

$T_{prio} = \frac{\mathbf{\nabla^T Q_{\nabla\nabla}^{-1} \nabla}}{m \sigma_0^2}$

used for detecting/identifying blunders is equivalent to the test statistic used for the point test. The only difference is the matrix $\mathbf{B}$, i.e., the (assumed) misspecified functional model of the null model. Misspecifying contains blunders, point shifts, missing parameters (e.g. zero point parameter of the EDM), a wrong functional equation etc. In my opinion, we only test against misspecifying - nothing more. Since powerful local tests are always performed in JAG3D, usually, I evaluate only these test statistics instead of the (weak) global test - in outlier detection as well as in deformation analysis.

- Please correct me if I am wrong, the global congruence test for the detection of datum point displacement(s) based on: vT*P*v - (viT*Pi*vi + vjT*Pj*vj) is more powerful, i.e., a given displacement will be detectable with a higher probability than by the test based on vT*P*v.

Yes, this should be correct. However, your suggested implicit test is not implemented in JAG3D, but powerful local tests are provided, which are specified as explicit hypothesis tests. The golden rule is: If a global test is rejected, we can conclude that something is wrong. Usually, we start with the localisation of the problem at this state. On the other hand, if the global test is not rejected, we cannot draw the conclusion that everything is fine; and again, we start with the (local) analysis. ;-)

Kind regards
Micha

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