## 4 The Como Test
### Analysis of the Time Series between Como EPN and Rover
#### 4.2.2 Congruence Analysis  
The congruence analysis verifies how many estimated displacements between couples of sessions are congruent with the imposed ones. The approach explained in [18] is used here to define classification errors \(\varepsilon_{\omega}\) of estimated displacements. Let \(\omega_{k}\) define the class corresponding to imposed displacement \(\overline{\delta\mathbf{x}}_{k}=k\times 5\) mm, \(k=0,\ldots,N_{max}\). Let us suppose that the probability distribution of estimated \(\delta\mathbf{x}_{ij}\) conditioned to class \(\omega_{k}\) is normal with constant average \(\overline{\delta\mathbf{x}}_{k}\) and covariance \(\mathbf{C}\): a Bayesian approach is straightforward to classify each estimated \(\delta\mathbf{x}_{ij}\) in the most probable class \(\omega_{k}\).  
Step 1. The conditional probabilities \(P(\omega_{k}/\delta\mathbf{x}_{ij})\) are computed for each class \(\omega_{k}\);  
Step 2. \(\delta\mathbf{x}_{ij}\) is attributed to the class \(\omega_{k}\) with the maximum \(P(\omega_{k}/\delta\mathbf{x}_{ij})\); and  
Step 3. The class error is computed according to:  
\[\varepsilon_{\omega}=abs(\omega_{k}-\overline{\omega}_{ij}) \tag{6}\]  
where \(\overline{\omega}_{ij}\) is the true class corresponding to the imposed \(\overline{\delta\mathbf{x}}_{ij}\).  
Congruence analysis is performed on all the possible combinations of the 74 sessions: in total, 2775 differences, classifications, and their errors are available.  
East, north, vertical, and 2D horizontal displacements are tested. A class error equal to 0 means no classification error. A class error \(n\) means a classification error equal to:  
* \(\varepsilon=n\times 5/\sqrt{2}\) mm for east and north, and
* \(\varepsilon=n\times 5\) mm for 2D and up.  
Figure 6 provides the detail of the results. In Table 9 the cumulated percentages are shown. The congruence analysis in the east direction provides rather satisfactory results: 66% of the couples are in class 0-2, while the maximum class error is 8 for just one result; north results are still satisfactory, but slightly worse. The 2D analysis performs better than individual east and north results. Vertical results are worse, with a significant percentage of class errors (40%) exceeding class 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Movements between Couples** & **2D (\%)** & **U (\%)** & **3D (\%)** \\ \hline
0 & 78 & 85 & 80 \\
1 & 20 & 14 & 28 \\
2 & 29 & 17 & 46 \\
3 & 63 & 28 & 68 \\
4 & 88 & 31 & 90 \\
5 & 99 & 38 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Baseline between the primary reference and the rover. Significance analysis on estimated displacements and percentages of right answers to the test. Zero movement: right answer “no displacement”. Other movements: right answer “yes displacement”.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Class Error** & **E (\%)** & **N (\%)** & **2D (\%)** & **U (\%)** \\ \hline
0 & 16 & 14 & 18 & 7 \\
1 & 45 & 42 & 50 & 21 \\
2 & 71 & 66 & 76 & 35 \\
3 & 89 & 82 & 90 & 48 \\
4 & 97 & 92 & 98 & 60 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Baseline between the primary reference and the rover. Congruence analysis on estimated displacements. Cumulated percentages of classes of errors. E: east, N: north, 2D: horizontal, and U: up.