Two aspects on L1-norm adjustment of leveling networks

Abstract

L1-norm adjustment corresponds to the minimization of the sum of weighted absolute residuals. Unlike Least Squares, it is a robust estimator, i.e., insensitive to outliers. In geodetic networks, the main application of L1-norm refers to the identification of outliers. There is no general analytical expression for its solution. Linear programming is the usual strategy, but it demands decorrelated observations. In the context of Least Squares, it is well known that the application of Cholesky factorization decorrelates observations without changing the results of the adjustment. However, there is no mathematical proof that this is valid for L1-norm. Besides that, another aspect on L1-norm is that equal weights may guarantee maximum robustness in practice. Therefore, it is expected to also provide a better effectiveness in the identification of outliers. This work presents contributions on two aspects concerning L1-norm adjustment of leveling networks, being them: the validity of Cholesky factorization for decorrelation of observations and the effectiveness for identification of outliers of a stochastic model with equal weights for observations. Two experiments were conducted in leveling networks simulated by the Monte Carlo method. In the first one, results indicate that the application of the factorization as previously performed in the literature seems inappropriate and needs further investigation. In the second experiment, comparisons were made between L1 with equal weights and L1 with weights proportional to the inverse of the length of the leveling line. Results show that the first approach was more effective for the identification of outliers. Therefore, it is an interesting alternative for the stochastic model in L1-norm adjustment. Besides providing a better performance in the identification of outliers, the need for observation decorrelation becomes irrelevant if equal weights are adopted.

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.…”

“…is work focuses on the solution by the simplex method [28] of linear programming, the most widely used approach for solving minimum L1-norm [23]. In geodetic networks, it has already been applied by many authors (see, e.g., [19,20,[29][30][31]).…”

Avaliação da Variação Vertical Local Voltada à Integridade de Séries Temporais Maregráficas

“…Ao levar em consideração esta perspectiva, é importante ressaltar que dependendo da natureza ou comportamento dos erros aleatórios, o MMQ pode ou não ser o melhor estimador imparcial (KLEIN, 2012). Abordagens alternativas ao ajustamento pelo MMQ em redes altimétricas podem ser obtidas, por exemplo, em Suraci, Oliveira e Klein (2019) ou Suraci e Oliveira (2020).…”

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

“…In the present work we will work with the leveling network presented in Suraci et al (2019), Figure 2, and personal programming code developed under Matlab R2019b (Matlab, 2019) to examine the performance of the different approaches outlined above for outlier detection. As we can see the situation will become increasingly intractable as the number of outliers approaches half the number of degrees of freedom of the network.…”

“…The example network presented in Suraci et al (2019) was simulated with zero error in its measurements; therefore, the least squares adjustment yields only null residual for all its observations. To resemble more a real network we must consider the accidental errors which are inherent to any measuring process, so that we add to each It is worth recalling here that the data snooping test is based on the assumption of a possible single outlier only, although it is frequently used with the unjustified hope of a high efficiency for the multiple outlier case.…”