L1 norm minimization in partial errors-in-variables model View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2017-09

AUTHORS

Jun Zhao, Qingming Gui, Feixiao Guo

ABSTRACT

The weighted total least-squares (WTLS) estimate is sensitive to outliers and will be strongly disturbed if there are outliers in the observations and coefficient matrix of the partial errors-in-variables (EIV) model. The L1 norm minimization method is a robust technique to resist the bad effect of outliers. Therefore, the computational formula of the L1 norm minimization for the partial EIV model is developed by employing the linear programming theory. However, the closed-form solution cannot be directly obtained since there are some unknown parameters in constrained condition equation of the presented optimization problem. The iterated procedure is recommended and the proper condition for stopping iteration is suggested. At the same time, by treating the partial EIV model as the special case of the non-linear Gauss–Helmert (G–H) model, another iterated method for the L1 norm minimization problem is also developed. At last, two simulated examples and a real data of 2D affine transformation are conducted. It is illustrated that the results derived by the proposed L1 norm minimization methods are more accurate than those by the WTLS method while the observations and elements of the coefficient matrix are contaminated with outliers. And the two methods for the L1 norm minimization problem are identical in the sense of robustness. By comparing with the data-snooping method, the L1 norm minimization method may be more reliable for detecting multiple outliers due to masking. But it leads to great computation burden. More... »

PAGES

389-406

References to SciGraph publications

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  • 2010-04. Robust estimation for correlated observations: two local sensitivity-based downweighting strategies in JOURNAL OF GEODESY
  • 2005-06. Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness in JOURNAL OF GEODESY
  • 2011-04. A Bayesian unmasking method for locating multiple gross errors based on posterior probabilities of classification variables in JOURNAL OF GEODESY
  • 2008-07. On weighted total least-squares adjustment for linear regression in JOURNAL OF GEODESY
  • 2013-06. Outlier separability analysis with a multiple alternative hypotheses test in JOURNAL OF GEODESY
  • 2002-07. Robust estimator for correlated observations based on bifactor equivalent weights in JOURNAL OF GEODESY
  • 2013-02. Robust estimation by expectation maximization algorithm in JOURNAL OF GEODESY
  • 2011-04. An iterative solution of weighted total least-squares adjustment in JOURNAL OF GEODESY
  • 2012-08. Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis in JOURNAL OF GEODESY
  • 2013-08. Weighted total least squares: necessary and sufficient conditions, fixed and random parameters in JOURNAL OF GEODESY
  • 2013-07. Data-snooping procedure applied to errors-in-variables models in STUDIA GEOPHYSICA ET GEODAETICA
  • 2007-10. A Bayesian approach to the detection of gross errors based on posterior probability in JOURNAL OF GEODESY
  • 1991-12. On the solution of the errors in variables problem using thel1 norm in BIT NUMERICAL MATHEMATICS
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    http://scigraph.springernature.com/pub.10.1007/s40328-016-0178-0

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    40 schema:description The weighted total least-squares (WTLS) estimate is sensitive to outliers and will be strongly disturbed if there are outliers in the observations and coefficient matrix of the partial errors-in-variables (EIV) model. The L1 norm minimization method is a robust technique to resist the bad effect of outliers. Therefore, the computational formula of the L1 norm minimization for the partial EIV model is developed by employing the linear programming theory. However, the closed-form solution cannot be directly obtained since there are some unknown parameters in constrained condition equation of the presented optimization problem. The iterated procedure is recommended and the proper condition for stopping iteration is suggested. At the same time, by treating the partial EIV model as the special case of the non-linear Gauss–Helmert (G–H) model, another iterated method for the L1 norm minimization problem is also developed. At last, two simulated examples and a real data of 2D affine transformation are conducted. It is illustrated that the results derived by the proposed L1 norm minimization methods are more accurate than those by the WTLS method while the observations and elements of the coefficient matrix are contaminated with outliers. And the two methods for the L1 norm minimization problem are identical in the sense of robustness. By comparing with the data-snooping method, the L1 norm minimization method may be more reliable for detecting multiple outliers due to masking. But it leads to great computation burden.
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