Performance comparison of least squares, iterative and global L1 norm minimization and exhaustive search methods for outlier detection in leveling networks

Abstract

Different approaches have been proposed to determine the possible outliers existing in a dataset. The most widely used consists in the application of the data snooping test over the least squares adjustment results. This strategy is very likely to succeed for the case of zero or one outliers but, contrary to what is often assumed, the same is not valid for the multiple outlier case, even in its iterative application scheme. Robust estimation, computed by iteratively reweighted least squares or a global optimization method, is other alternative approach which often produces good results in the presence of outliers, as is the case of exhaustive search methods that explore elimination of every possible set of observations. General statements, having universal validity, about the best way to compute a geodetic network with multiple outliers are impossible to be given due to the many different factors involved (type of network, number and size of possible errors, available computational force, etc.). However, we see in this paper that some conclusions can be drawn for the case of a leveling network, which has a certain geometrical simplicity compared with planimetric or three-dimensional networks though a usually high number of unknowns and relatively low redundancy. Among other results, we experience the occasional failure in the iterative application of the data snooping test, the relatively successful results obtained by both methods computing the robust estimator, which perform equivalently in this case, and the successful application of the exhaustive search method, for different cases that become increasingly intractable as the number of outliers approaches half the number of degrees of freedom of the network.

Monte Carlo-Based Covariance Matrix of Residuals and Critical Values in Minimum L1-Norm

“…For all networks, the standard deviation of the observations was given by σ i � 1.0(mm) * �� K i , where K i (in km) is the length of the respective leveling line. In the ascending order of the observation index, the lengths (in km) of each leveling line were as follows: for network A, [42,38,27,22,23,33]; for network B, [37,28,33,26,40,32,39,29,34,41]; and for network C, [30,34,25,37,28,38,29,35,31,26,33,36,27,32,24]. erefore, for example, σ i of the 4 th observation of network A (which is also the lowest σ i of all networks) is σ 4(Network A) � 1.0(mm) * �� 22 √ � 4.69 mm.…”

“…In particular, Minimum L1-norm is likely to provide higher outlier identification success rate than IDS for low-redundancy networks [15,22]. e minimum L1-norm solution may not be unique [23], and its vector of residuals in geodetic networks tends to be sparse, with many residuals being equal to zero (see, e.g., [20,24]). is means that corresponding observations are accepted as "perfect" observations, without any measurement errors.…”

“…Another technique commonly employed to solve minimum L1-norm in geodetic networks is the iterative reweighted least squares (IRLS) (see, e.g., [24,[32][33][34]). It has also been applied in deformation analysis of geodetic networks [35][36][37].…”

“…Other techniques that were performed to compute minimum L1-norm in geodetic networks include simulated annealing [24]; genetic algorithm [39,40]; and linear programming by an interior point method [38]. Solutions of minimum L1-norm by these methods and by IRLS are outside the scope of this paper.…”

Application of Optimization Algorithms for Identification of Reference Points in a Monitoring Network

“…Some metaheuristic methods have also been used to solve some optimization problems in geodesy and surveying techniques, e.g., [ 37 , 38 , 39 ], but applications of SA in this field are rather limited. They include research on geodetic network design [ 40 , 41 ], the adjustment of geodetic measurements [ 42 , 43 ], LIDAR survey design [ 44 ] or coordinate transformation [ 34 ]. Deformation measurements which are the subject of this work can be included in the same area.…”