Estimation of parameters in a split functional model of geodetic observations (M split estimation)

Wiśniewski, Z.

doi:10.1007/s00190-008-0260-7

Cited by 15 publications

(34 citation statements)

References 25 publications

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“…Msplit(q) [17,18] assumes existence set of n observations l (l1,l2…ln) that contain q mixed random variables with expected values (1):…”

Section: M-estimationmentioning

confidence: 99%

The circle object detection with the use of Msplit estimation

Janowski

2018

E3S Web Conf.

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Abstract. The paper presents the use of Msplit(q) -estimation in the filtration and aggregation of point clouds containing a known number of elliptical shapes with preliminary unknown -locations and dimensions. These theoretical solutions may have practical relevance especially in the modelling of terrestrial laser scanning data of objects that have similar shape to circles. Mentioned shapes can be scanned of tree trunks, columns, gutters, etc., that are elliptical in the horizontal plane. The results are satisfied and encourage furthermore detailed research, particularly with the extension of 3D applications.

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“…Msplit(q) [17,18] assumes existence set of n observations l (l1,l2…ln) that contain q mixed random variables with expected values (1):…”

Section: M-estimationmentioning

confidence: 99%

The circle object detection with the use of Msplit estimation

Janowski

2018

E3S Web Conf.

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“…, n, (a i − ith row of the matrix A) have a fixed probability density function (PDF). Thus, each observation brings a piece of finformation I (y i ; X) = − ln f (y i ; X) = − ln f (v i ) (see, e.g., Jones and Jones 2000;Wiśniewski 2009). Considering the parameter X, all observations included within the vector y bring the global amount of information as follows…”

Section: Theoretical Foundationsmentioning

confidence: 99%

“…In such a case, the equivalent variance can be written as μ 2,i = μ 2,i /t (v i ). Probabilistic models are also applied in a development of M-estimation, namely M split estimation (Wiśniewski 2009(Wiśniewski , 2010Duchnowski and Wiśniewski 2011). In the basic variant of the method in question, it is assumed that P = {P X α , P X β : X α , X β ∈ R r } ⊂ W v .…”

Section: Theoretical Foundationsmentioning

confidence: 99%

M-estimation with probabilistic models of geodetic observations

Wiśniewski

2014

J Geod

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The paper concerns M-estimation with probabilistic models of geodetic observations that is called M P estimation. The special attention is paid to M P estimation that includes the asymmetry and the excess kurtosis, which are basic anomalies of empiric distributions of errors of geodetic or astrometric observations (in comparison to the Gaussian errors). It is assumed that the influence function of M P estimation is equal to the differential equation that defines the system of the Pearson distributions. The central moments μ k , k = 2, 3, 4, are the parameters of that system and thus, they are also the parameters of the chosen influence function. The M P estimation that includes the Pearson type IV and VII distributions (M PD(l) method) is analyzed in great detail from a theoretical point of view as well as by applying numerical tests. The chosen distributions are leptokurtic with asymmetry which refers to the general characteristic of empirical distributions. Considering M-estimation with probabilistic models, the Gram-Charlier series are also applied to approximate the models in question (M G−C method). The paper shows that M P estimation with the application of probabilistic models belongs to the class of robust estimations; M PD(l) method is especially effective in that case. It is suggested that even in the absence of significant anomalies the method in question should be regarded as robust against gross errors while its robustness is controlled by the pseudo-kurtosis.

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“…Thus, CDA is still developed or improved, for example, to cope with such more complicated cases or to increase its efficiency by applying a new alternative strategy that works on absolute deformation monitoring networks and that has been developed based on division into subnetworks which contain one objective point and reference points (see, for example, Hekimoglu et al 2010). On the other hand, there are also other less traditional methods that can be applied in deformation analyses, for example R-estimation (see, Duchnowski 2009Duchnowski , 2010 or M split estimation (see, Wiśniewski 2009). A special variant of the latter, which concerns the estimation of a shift of parameters, was proposed in Duchnowski and Wiśniewski (2012).…”

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confidence: 99%

“…There is no doubt that M-estimation is the most common robust method that is applied for solving problems in geodesy or surveying. Since the pioneering paper (Huber 1964) M-estimation was improved and developed (see, e.g., Andrews 1974;Huber 1981;Hampel et al 1986) and it also became an important method of robust estimation employed in geodetic applications (see, Xu 1989Xu , 2005Yang et al 2002;Wiśniewski 2009Wiśniewski , 2010Erenoglu and Hekimoglu 2010;Guo et al 2011). On the other hand, R-estimation which is a robust method often applied in other fields of science, is still almost unknown (or not applied) method in geodesy and surveying, The beginnings of the estimates that are derived from rank tests, namely R-estimates, are related to the paper of Hodges and Lehmann (1963), hence the alternative name for such type estimates, namely the Hodges-Lehmann estimates (HLE).…”

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confidence: 99%

Hodges–Lehmann estimates in deformation analyses

Duchnowski

2013

J Geod

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This paper presents new variants of the HodgesLehmann estimates, which belong to the class of R-estimates. The new approach to this method arises from the need of taking into account differences in accuracy of geodetic measurements, which is not possible while applying traditional R-estimates. The theoretical assumptions of the conventional Hodges-Lehmann estimates are supplemented with the information about the accuracy of observations and two new variants of the estimates in question are derived by applying the principles proposed by Hodges and Lehmann, hence they are called the Hodges-Lehmann weighted estimate. The main properties of the new estimates follow from such approach, and from the practical point of view, the most important seems to be their robustness against outliers. Since the first estimate proposed is a natural estimator of the shift between two samples, it can be applied in deformation analysis to estimate point displacements. The paper presents two numerical examples that show the properties as well as possible applications of the new estimates.

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Estimation of parameters in a split functional model of geodetic observations (M split estimation)

Cited by 15 publications

References 25 publications

The circle object detection with the use of Msplit estimation

The circle object detection with the use of Msplit estimation

M-estimation with probabilistic models of geodetic observations

Hodges–Lehmann estimates in deformation analyses

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