Deformation (congruence) analysis - statistical tests

by Krzysztof, Wednesday, April 21, 2021, 11:05 (24 days ago)

Hi Micha,

I have two concerns about the congruence analysis package.
https://software.applied-geodesy.org/wiki/least-squares-adjustment/deformationanalysis

1. It seems to me that the global congruence test (detection test) is not performed for datum points in JAG3D and, immediately, single - datum - point displacements are tested individually (local tests) to identify possible unstable datum point(s). Am I right?

2. A posteriori test statistics for single datum points - T_post,j - are tested using F-test, where the degrees of freedom are: d1 = m (the degrees of freedom of the quadratic form in the nominator, in fact, the rank of the cofactor matrix in the nominator) and d2 = f (the redundancy of congruence model) - m. In other words, we obtain d2 = the redundancy of network model from epoch 0 + the redundancy of network model from epoch 1. And this does not raise my concerns.

2.1. However, it is not the case for the a posteriori test statistics for single object points - T_post,k. It means d2 = f in this case. While, it seems to me that it should be, as previously, d2 = f - m or, equivalently, d2 = the redundancy of network model from epoch 0 + the redundancy of network model from epoch 1. It is due to the fact that the estimate of variance factor (in the denominator of T_post,k) should be based on unconstrained model (i.e., the model of combined adjustment as, e.g., it is presented in Caspary's (2000) monography on p.121-123, and, not congruence model). Furthermore, Lehmann and Löesler (2017), Congruence analysis of geodetic networks – hypothesis tests versus model selection by information criteria, write on p.274: '... σ̂^2 is the estimate of σ2. Note that they are external estimates, i.e., they must be computed in the unconstrained model, in the sense of least squares.' The Caspary's experiments also seem to confirm this statement, e.g., d2 = 58 (the redundancy of network model from epoch 0 (f0 = 29) + the redundancy of network model from epoch 1 (f1 = 29)), in the second equation on p.153. What do you think about this?

Best regards,
Krzysztof


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