# Development and testing of in-context confidence regions for geodetic survey networks

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## Abstract

The objectives of the contract were to develop and numerically test in-context absolute and relative confidence regions for geodetic networks. In-context confidence regions are those that relate to many points simultaneously, rather than the conventional notion of speaking about the confidence region about only one point without regard to any others.
An out-of-context test is conducted on some piece of data without regard for the remaining data in the set. An in-context test is conducted on a quantity in the context of being a member of a larger set. Adjustment software, such as GHOST and GeoLab, that use the so-called Tau test of residuals are based on in-context testing. However, we are aware of no software that is capable of performing in-context testing on confidence regions for the estimated coordinate parameters.
Another issue needing clarification is the matter of local versus global testing.
Global testing is understood to be a single test involving the entire group of variates under examination. A global test statistic is typically a quadratic form which transforms the variates into a scalar quantity, containing all the information about the group. On the other hand, local testing is the process of testing individual variates in the group, either in-context or out-of-context. Since these tests can be conducted in either parameter or observation space, they should use a consistent approach in both spaces whenever possible.
The development of confidence regions corresponding to one solution is different from the statistical testing of the compatibility (or congruency) of one solution against another. In this report we focus on the development of confidence regions for the analysis of a single network solution, rather than the development of statistical tests for applications such as deformation analyses that require the comparison of two solutions.
The key issue of in-context testing is the formation of a mathematical link between the various statistical tests that may be conducted not only on the estimated parameters but also on the estimated residuals. The consequence of a mathematical link is compatibility of statistical tests throughout observation and parameter space.
Three approaches to the computation of in-context confidence regions were examined during this contract: the Bonferroni, Baarda and projection approaches. The Bonferroni approach equates the simultaneous probability of the individual in-context confidence regions to a selected global probability level. However, it neglects any correlations between the tested quantities, which can have serious consequences for parameter confidence regions. The Baarda (or Delft) approach uses the relation between Type I and II errors for both global and local testing, but arbitrarily assumes the probability and non-centrality parameters for both local and global Type II errors are the same. Finally, the projection approach simply uses the global confidence region or test and projects it to the individual subspaces for local confidence regions or tests. It uses the global expansion factor for all individual in-context confidence regions and tests, which results in unreasonably large confidence regions that can grow without bound. Strictly speaking this is not an in-context approach as defined above. It is effectively a global test on the individual quantities. That is, the failure of one individual local test also implies the failure of the global test.
To summarize, the projection method tests hypotheses that are different from what we want and its in-context expansion factors are unreasonably large and grow without bound for large networks. Baarda's approach gives relatively large in-context expansion factors which grow without bound (although much more slowly than in the projection approach). The Bonferroni approach yields the smallest and most reasonable expansion factors for in-context confidence regions and tests. The expansion factors are also bounded to reasonable values for even the largest of networks. However, this approach neglects the effects of correlations which can be very large between coordinate parameters in geodetic networks. Primarily because of the smaller expansion factors, we recommend to use the Bonferroni approach for in-context confidence regions and tests, in spite of the neglect of correlations. It is recommended to further investigate the effects of large correlations and possible ways of accounting for them.
The recommended approach for the in-context statistical analysis of the adjustment of a geodetic network is to first chose a global significance level (a) to be used as the basis for all global and local in-context tests and confidence regions. The specific significance levels to use for the various tests and confidence regions are:
• Global test on residuals (variance factor test). Use the global significance level (a).
• Local tests of individual residuals (outlier tests). Use the in-context significance level ao = a/n, where n is the degrees of freedom of the adjustment.
• Global confidence region. For a global confidence region for all points in the network, use the global significance level (a).
• Local absolute (point) confidence regions. For absolute in-context confidence regions at individual points in the network, use the in-context significance level ao = a/n, where n is the number of points being simultaneously assessed.
• Local relative confidence regions. For relative in-context confidence regions between pairs of points in the network, use the in-context significance level ao = a/m, where m is the number of linearly independent pairs of points to be simultaneously assessed.