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M split estimation. Part I: Theoretical foundation

Zbigniew Wiśniewski
Geodesy and Cartography | 2009 | vol.58 | No 1 | 3-21
Keywords Geodetic adjustment M-estimation split potencial of an observation set
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Abstract

The present paper consists of two parts. The first part presents theoretical foundations of Msplit, estimation with reference to the previous author's paper (Wiśniewski, 2009). This time, some probabilistic assumptions are described in detail. A new quantity called f-information is also introduced to formulate the split potential in more general way. The main aim of this part of the paper is to generalize the target function of Msplit estimation that is the basis for a new formulation of the optimization problem. Such problem itself as well as its solution are presented in this part of the paper. The second part of the paper presents some special case of Mspli, estimation called squared Mspli, estimation (also with reference to the mentioned above paper of the author). That part presents a new solution and development in the theory of this version of M,plit estimation and some numerical examples that show properties of the method and its application scope.
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Authors and Affiliations

Zbigniew Wiśniewski

M split estimation. Part II: Squared M split estimation and numerical examples

Zbigniew Wiśniewski
Geodesy and Cartography | 2009 | vol.58 | No 1 | 23-48
Keywords Geodetic adjustment M estimation split potential of an observation set
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Abstract

This part of the paper presents particular case of Msplit estimation called a squared Msplit estimation whose target function is based on convex squared functions. One can find here theoretical foundations and algorithm of the squared Msplit estimation as well as some numerical examples.
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Authors and Affiliations

Zbigniew Wiśniewski

Absolute M split estimation as an alternative for robust M-estimation

Robert Duchnowski Patrycja Wyszkowska
Advances in Geodesy and Geoinformation | 2022 | vol. 71 | No 1 | e17 | DOI: 10.24425/gac.2022.141170
Keywords laser scanning robust estimation M-estimation absolute Msplit estimation
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Abstract

The problem of outlying observations is very well-known in the surveying data processing. Outliers might have several sources, different magnitudes, and shares within the whole observation set. It means that it is not possible to propose one universal method to deal with such observations. There are two general approaches in such a context: data cleaning or robust estimation. For example, the robust M-estimation has found many practical applications. However, there are other options, such as R-estimation or the absolute M split estimation. The latter method was created to be less sensitive to outliers than the squared M split estimation (the basic variant of Msplit estimation). From the theoretical point of view, the absolute M split estimation cannot be classified as a robust method; however, it was proved that it could be used in such a context under certain conditions. The paper presents the primary comparison between that method and a conventional robust M-estimation. The results show that the absolute M split estimation predominates over the classical methods, especially when the percentage of outliers is high. Thus, that method might be used to process LiDAR data, including mismeasured points. Processing synthetic data from terrestrial laser scanning or airborne laser scanning confirms that the absolute M split / estimation can deal with outliers sufficiently.
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Authors and Affiliations

Robert Duchnowski
1
ORCID: ORCID
Patrycja Wyszkowska
1
ORCID: ORCID
  1. University of Warmia and Mazury, Olsztyn, Poland

A Novel Influence Function M-Estimator-Based for Active Noise Control

Seyed Amir Hoseini Sabzevari Seyed Iman Hoseini Sabzevari
Archives of Acoustics | 2021 | vol. 46 | No 3 | 499-506 | DOI: 10.24425/aoa.2021.138142
Keywords active noise control adaptive FIR M-estimator noise model
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Abstract

M-estimators are widely used in active noise control (ANC) systems in order to update the adaptive FIR filter taps. ANC systems reduce the noise level by generating anti-noise signals. Up to now, the evaluation of M-estimators capabilities has shown that there exists a need for further improvements in this area. In this paper, a new improved M-estimator is proposed. The sensitivity of the proposed algorithm to the variations of its constant parameter is checked in feedforward control. The effectiveness of the algorithm in both types is proved by comparing it with previous studies. Simulation results show the steady performance and fast initial convergence of the proposed algorithm.
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Bibliography

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Authors and Affiliations

Seyed Amir Hoseini Sabzevari
1
Seyed Iman Hoseini Sabzevari
2
  1. Department of Mechanical Engineering, University of Gonabad, Gonabad, 9691957678, Iran
  2. Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, 9177948974, Iran

Theoretical geodesy

Andrzej Borkowski Wiesław Kosek
Geodesy and Cartography | 2015 | vol. 64 | No 2 | DOI: 10.1515/geocart-2015-0015
Keywords M-estimation robust estimation reliability time series polar motion
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Abstract

The paper presents a summary of research activities concerning theoretical geodesy performed in Poland in the period of 2011–2014. It contains the results of research on new methods of the parameter estimation, a study on robustness properties of the M-estimation, control network and deformation analysis, and geodetic time series analysis. The main achievements in the geodetic parameter estimation involve a new model of the M-estimation with probabilistic models of geodetic observations, a new Shift-M split estimation, which allows to estimate a vector of parameter differences and the Shift- M split (+) that is a generalisation of Shift- M split estimation if the design matrix A of a functional model has not a full column rank. The new algorithms of the coordinates conversion between the Cartesian and geodetic coordinates, both on the rotational and triaxial ellipsoid can be mentioned as a highlights of the research of the last four years. New parameter estimation models developed have been adopted and successfully applied to the control network and deformation analysis. New algorithms based on the wavelet, Fourier and Hilbert transforms were applied to find time-frequency characteristics of geodetic and geophysical time series as well as time-frequency relations between them. Statistical properties of these time series are also presented using different statistical tests as well as 2 nd , 3 rd and 4 th moments about the mean. The new forecasts methods are presented which enable prediction of the considered time series in different frequency bands.
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Authors and Affiliations

Andrzej Borkowski
Wiesław Kosek

Empirical tests of performance of some M – estimators

Marek Banaś Marcin Ligas
Geodesy and Cartography | 2014 | vol. 63 | No 2 | DOI: 10.2478/geocart-2014-0010
Keywords M – estimation iteratively reweighted least squares tuning constant outliers network adjustment
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Abstract

The paper presents an empirical comparison of performance of three well known M – estimators (i.e. Huber, Tukey and Hampel’s M – estimators) and also some new ones. The new M – estimators were motivated by weighting functions applied in orthogonal polynomials theory, kernel density estimation as well as one derived from Wigner semicircle probability distribution. M – estimators were used to detect outlying observations in contaminated datasets. Calculations were performed using iteratively reweighted least-squares (IRLS). Since the residual variance (used in covariance matrices construction) is not a robust measure of scale the tests employed also robust measures i.e. interquartile range and normalized median absolute deviation. The methods were tested on a simple leveling network in a large number of variants showing bad and good sides of M – estimation. The new M – estimators have been equipped with theoretical tuning constants to obtain 95% efficiency with respect to the standard normal distribution. The need for data – dependent tuning constants rather than those established theoretically is also pointed out.
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Authors and Affiliations

Marek Banaś
Marcin Ligas

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