Robust M estimation for 3D correlated vector observations based on modified bifactor weight reduction model

Abstract

This paper develops a robust M estimation approach applied for three-dimensional (3D) correlated vector observations. A modified bifactor reduction model is constructed, where the weight shrinking factor of the 3D vector observation is determined by a new test statistic that coincides with the estimated direction of the outlier vector and thus is more sensitive to vector-type outliers than the standardized residual used for most conventional robust M methods. With the proposed bifactor reduction model, the outlying vector observation is down-weighted directly along a specific direction, rather than separately at the three components. The new equivalent weight matrix derived from the proposed bifactor model still keeps symmetry, based on which the parameter estimation procedure is developed. A real 3D control network of GNSS vector observations is processed by simulating outliers with different types, sizes and locations. The results show the effectiveness of the proposed approach by comparing with other four conventional robust M method (IGGIII, Danish, Huber and Hampel).

Appendix

The Danish method was proposed by Krarup (Krarup et al. 1980) and is purely heuristic with no rigorous statistical theory. The methods work by performing IRLS with the following weight shrinking function (Caspary 1987; Knight and Wang 2009)

$$ \gamma_{ii} = \left\{ {\begin{array}{*{20}l} 1 & {\left| {\tilde{v}_{i} } \right| \le k_{0} } \\ {e^{{ - \frac{{\left| {\tilde{v}_{i} } \right|}}{{k_{0} }}}} } & {\left| {\tilde{v}_{i} } \right| > k_{0} } \\ \end{array} } \right. $$

(38)

The M estimators were first proposed by Huber (1964, 1981), and the weight reduction factor is given by

$$ \gamma_{ii} = \left\{ {\begin{array}{*{20}c} 1 & {\left| {\tilde{v}_{i} } \right| \le k_{0} } \\ {\frac{{k_{0} }}{{\left| {\tilde{v}_{i} } \right|}}} & {\left| {\tilde{v}_{i} } \right| > k_{0} } \\ \end{array} } \right. $$

(39)

The shrinking factor for Hampel method is expressed as (Hampel et al. 1986)

$$ \gamma_{ii} = \left\{ {\begin{array}{*{20}c} 1 & {\left| {\tilde{v}_{i} } \right| \le k_{0} } \\ {\frac{{k_{0} }}{{\left| {\tilde{v}_{i} } \right|}}} & {k_{0} < \left| {\tilde{v}_{i} } \right| \le k_{1} } \\ {\frac{{k_{0} }}{{\left| {\tilde{v}_{i} } \right|}}\frac{{k_{2} - \left| {\tilde{v}_{i} } \right|}}{{k_{2} - k_{1} }}} & {k_{1} < \left| {\tilde{v}_{i} } \right| \le k_{2} } \\ 0 & {\left| {\tilde{v}_{i} } \right| > k_{2} } \\ \end{array} } \right. $$

(40)

The above three weight reduction functions are used for Danish, Huber and Hampel methods, respectively, in Sect. 4, to replace Eq. (9) for IGGIII methods. To applying these functions into correlated observations, bifactor reduction scheme of Eqs. (6)–(8) is always used (Figs. 9, 10, 11, and 12).

Fig. 9

MPE results of the Danish, Huber and Hampel methods for case 1 when a 3D outlier vector is added into each baseline, respectively

Fig. 10

Cumulative distribution probability (CDF) of the test statistics $\left| {\tilde{v}_{i} } \right|$ for conventional methods (a) and $\left| {w_{il} } \right|$ for the proposed method (b) when an outlier vector with a size of σ, 2σ and 3σ is simulated at the baseline B2 to B13. (OF is for outlier-free case)

Fig. 11

Probability of test statistics $\left| {w_{il} } \right|, \left| {\tilde{v}_{i} } \right|$ smaller than the corresponding thresholds as the magnitude of the outlier at baseline B2 to B1 increases from 0.5σ to 6σ, respectively

Fig. 12

MPE results of the Danish, Huber and Hampel methods for case 2 when a scaled outlier is added into the vertical component of the baseline, respectively