Three points method for searching the best least absolute deviations plane

https://doi.org/10.1016/j.amc.2009.06.023Get rights and content

Abstract

In this paper a new method for estimation of optimal parameters of a best least absolute deviations plane is proposed, which is based on the fact that there always exists a best least absolute deviations plane passing through at least three different data points. The proposed method leads to a solution in finitely many steps. Moreover, a modification of the aforementioned method is proposed that is especially adjusted to the case of a large number of data and the need to estimate parameters in real time. Both methods are illustrated by numerical examples on the basis of simulated data and by one practical example from the field of robotics.

Introduction

In this paper we consider the problem of a fast determination of a best least absolute deviations plane (LAD-plane) on the basis of the given set of points in space Λ={Ti=(xi,yi,zi)R3:iI}, where I={1,,m},m3, is a set of indices. It is thereby assumed that the number of data m can be great and that among the data a substantial amount of outliers (i.e. wild points) might appear. This problem could be naturally extended to determination of the best LAD hyperplane. Such problems are found very often in applied research (see e.g. [6], [8], [12], [15], [25], [27], [28]).

Searching for the best least absolute deviations hyperplane on the basis of a given set of points is well-known in the literature (see [1], [7], [11], [15], [19], [22], [30]). This principle is considered to have been proposed by the Croatian mathematician Bošković in the mid-eighteenth century (see [1], [8]).

The problem of determining optimal parameters in a mathematical model appears in all areas of applied research. Let us mention only applications in statistics, finance, engineering, bioinformatics, operations research, etc. The problem of estimating optimal parameters can be considered in several ways. Most frequently it is assumed that errors are normally distributed and that they can occur only in measured values of the dependent variable. In that case, it is generally taken that parameters are estimated by means of the Ordinary Least Squares method (see [7], [13]). If outliers can appear among the data, then the application of the l1-norm is much more acceptable (see [2], [4], [8], [15], [17], [20], [22], [27]). In literature this approach is known as the Least Absolute Deviations problem (LAD-problem).

If errors are assumed to occur in measured values of both (dependent and independent) variables, the problem in question is the errors-in-variables problem (see e.g. [24]). Especially, if thereby the square of the l2 norm is used, it is the Total Least Squares problem, and if the l1 or the l2 norm is used, then the problems in question are the orthogonal l1 and the orthogonal l2 problem, respectively (see e.g. [26], [29]).

In this paper a new method for estimating optimal parameters of a best LAD-plane is proposed, which is based on the fact that there always exists a best LAD-plane passing through at least three different data points. The proposed method leads to the solution of this LAD-problem in finitely many steps. Moreover, a modification of the aforementioned method is proposed that is especially adjusted to the case of a large number of data and the need to estimate parameters in real time.

First, some important facts necessary for consideration and analysis of the problem are given. After that, basic properties of a best LAD-plane are considered, on the basis of which it is possible to design and analyze the proposed methods. Methods are illustrated and tested on several numerical examples. Finally, an example of applying these methods in robotics is given.

Section snippets

A best LAD-plane

Let us now define the LAD-plane problem. Let I={1,,m},m3 be a set of indices, and Λ={Ti=(xi,yi,zi)R3:iI} a set of points in space with corresponding data weights wi>0. Thereby zi’s are the measured values of the unknown function g:R2R at points Pi=(xi,yi),i=1,,m among which outliers appear. A best LAD-plane should be determined, i.e. optimal parameters a,b,cR of the functionf(x,y;a,b,c)=ax+by+c,should be determined such thatG(a,b,c)=min(a,b,c)R3G(a,b,c),G(a,b,c)=i=1mwi|zi-axi-byi-

Methods of searching for a best LAD-plane

To search for optimal parameters of a best LAD-plane, general methods of minimization without using derivatives can be used (see [13], [16]), but their efficiency in the case of a large number of data is very low. Efficient methods for solving this problem are based on Linear Programming (see e.g. [1], [4], [19], [30]). There are also various specializations of the Gauss–Newton method (see e.g. [4], [7], [11], [19], [30]), special methods of combinatorial optimization (see [17], [31]), as well

Application in robotics

Detection of planar surfaces is an important problem in robot vision. Most mobile robots today are constructed to operate in indoor environment moving on a flat horizontal surface called herein the ground plane. Safe obstacle avoidance can be performed by classifying all environment points into two sets. The first represents all points lying sufficiently close to the ground plane so that the robot can safely roll over them. The second set contains all other points representing the obstacles

Acknowledgement

We would like to thank an anonymous referee for useful comments and remarks, which helped us to improve the paper significantly.

References (31)

  • E. Castillo et al.

    Dealing with the multiplicity of solutions of the l1 and l regression models

    European J. Oper. Res.

    (2008)
  • R. Scitovski et al.

    Approximating surface by moving total least squares method

    Appl. Math. Comput.

    (1998)
  • S. Se et al.

    Ground plane estimation error analysis and applications

    Robot. Auton. Syst.

    (2002)
  • P. Bloomfield et al.

    Least Absolute Deviations: Theory Applications and Algorithms

    (1983)
  • J. Brimberg et al.

    Properties of three-dimensional median line location models

    Ann. Operat. Res.

    (2003)
  • D. Burschka, S. Lee, G. Hager, Stereo-based obstacle avoidance in indoor environments with active sensor...
  • J. Corso, D. Burschka, G. Hager, Direct plane tracking in stereo images for mobile navigation, in: Proceedings of the...
  • R. Cupec, Scene reconstruction and free space representation for biped walking robots, Dr. Ing. Dissertation TU...
  • E.Z. Demidenko

    Optimization and Regression

    (1989)
  • Y. Dodge (Ed.), Statistical data analysis based on the L1-norm and related methods, in: Proceedings of The Third...
  • O. Faugeras

    Three-Dimensional Computer Vision: A Geometric Viewpoint

    (1993)
  • M.A. Fischler et al.

    Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography

    Graphics Image Process.

    (1981)
  • C. Gurwitz

    Weighted median algorithms for L1 approximation

    BIT

    (1990)
  • K.P. Hadeler et al.

    Least squares problems for Michaelis Menten kinetics

    Math. Methods Appl. Sci.

    (2007)
  • C.T. Kelley

    Iterative Methods for Optimization

    (1999)
  • Cited by (0)

    This work was supported by the Ministry of Science, Education and Sports, Republic of Croatia, through Research Grants 235-2352818-1034, 165-0361621-2000 and 036-0363078-3018.

    View full text