# Deformation (congruence) analysis - statistical tests

Hello Krzysztof,

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?

The congruence analysis in JAG3D is based on the joint adjustment of two epochs on the observation level. For that reason, there is no difference between a normal adjustment or a congruence analysis. This is some interpretation of the user not of the software. In general, there is no evident difference between outliers detection or detection of deformed points.

JAG3D performs a global test. The test result is given in the table of the variance components estimation, see the table here which shows a rejected global test.

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.

All quantiles used within the project are given in the table test statistic, see the table here.

And yes, you are right, because this is the constrained model but we need the quantile of the unconstrained model...

2.1. However, it is not the case for the a posteriori test statistics for single object points - T_post,k.

In contrast to datum points, object points are separated in JAG3D - one per epoch. Estimating two different positions (one per epoch) for an object point $P_{i}$, i.e., $P_{i,e1}$ and $P_{i,e2}$, is nothing else than to estimate a single position $P_{i}$ and the shift vector say $\nabla_{i}$, where $\nabla_{i} = P_{i,e2} - P_{i,e1}$. Since two different positions are estimated, the object point $P_{i}$ does not force the network (it is the unconstrained model). The vector $\nabla$ is already part of the parameters to be estimated.

The datum points are assumed to be stable in both epochs during the adjustment (only a single position is estimated - constrained model). The redundancy of the network does not contain the unknown $m$-vector $\nabla$ of extra parameters related to the tested datum point. If you estimate $\nabla$ within the adjustment, the number of parameters to be estimated increases by $m$ and, thus, the redundancy is reduced, i.e., $f = n_0 - u_0 - m + d$. Here $n$ is the number of observations (not changed), $u$ is the number of unknowns in the constrained model and $m$ denotes the additional parameters in $\nabla$. Thus, $u_0 + m$ represents the unknown in the unconstrained model.

(The equations used in JAG3D are also given in the textbook written by Jäger et al. (2005), p. 277, eg. Eq. 6.109.)

Kind regards
Micha

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