Alternatives to least squares

Details Alternatives to least squares Alternatives to least squares

Jun 1982

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1982aj.....87..928b&link_type=abstract

Astronomical Journal, vol. 87, June 1982, p. 928-937.

Mathematics

9

Data Reduction, Error Analysis, Least Squares Method, Sum Rules, Algorithms, Average, Chebyshev Approximation, Matrices (Mathematics), Optimization, Signal Processing, Simultaneous Equations

Scientific paper

The mathematical discussion of the results of observation almost invariably can be reduced to the form A.x=d, where A is the matrix of the equations of condition, d is the m vector of the observations, and x is the desired solution vector of dimension n. Because of the errors of the observations, the system is inconsistent and no one vector x will satisfy all of the equations. In practice, one makes the m vector r of the residuals, defined as r=A.x-d, as small as possible in some sense. The criterion is almost always that of the method of least squares involving the minimization of r(T).r. However, many modern authorities feel that the underlying assumptions upon which the method is based may not be valid. In an evaluation of three alternatives to traditional least squares it is found that the method of averages and the Chebyshev method are totally unsuitable for the analysis of observational data. The method of minimum sum, however, is very competitive with least squares and is deserving of greater attention by astronomers.

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