Data Snooping for the Equality Constrained Nonlinear Gauss–Helmert Model Using Sensitivity Analysis

Abstract

To develop a universal-outliers processing algorithm under the conditions with equality constraints, the equality-constrained nonlinear Gauss–Helmert (GH) model, which contains the equality-constrained Gauss–Markov (GM) and errors-in-variables (EIV) models as special cases, is selected as the research object in this paper. The least squares solution for the nonlinear GH model with equality constraints is obtained using the Euler–Lagrange approach, and then, it is equivalently formulated as the standard constrained least squares (CLS) problem. To construct the test statistics for the outliers detection, a distinctive sensitivity analysis approach is introduced into this CLS problem. The local sensitivity of the weighted sum of squared residuals to the perturbations of observations in the CLS problem is discussed, and then, the local test statistics are constructed based on these sensitivity indicators. To verify the performance of the sensitivity-based test statistics, the proposed data-snooping algorithm for the equality-constrained nonlinear GH model is applied to a three-dimensional (3D) symmetric similarity transformation. The computational results of the simulated and real examples manifest that the proposed data-snooping algorithm using the sensitivity-based test statistics can effectually decrease the negative impact of the outliers and derive reliable parameters. It should be pointed out that the new algorithm is applicable in various kinds of equality-constrained least squares and total least squares problems.