Chapter

Least-squares fitting of parameters

Peter Main

in Crystal Structure Analysis

Second edition

Published in print June 2009 | ISBN: 9780199219469
Published online September 2009 | e-ISBN: 9780191722516 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199219469.003.0012

Series: International Union of Crystallography Texts on Crystallography

Least-squares fitting of parameters

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In crystallography, numerical parameters for the structure are derived from experimental data. This chapter discusses how the data and parameters are related, and introduces data fitting procedures including unweighted and weighted means, and least-squares criteria for a ‘best fit’. The simple case of linear regression for two parameters of a straight line is treated in some detail in order to explain the least-squares tools of observational equations and matrix algebra, leading to variances and covariances. Restraints and constraints are applied, and their important distinction made clear. Non-linearity in the observational equations leads to further complications, with only parameter shifts rather than the parameters themselves obtainable through least-squares treatment. Ill-conditioning and matrix singularity are explained, with reference to crystallographic relevance. Computing aspects are considered, since least-squares refinement is particularly expensive computationally.

Keywords: least-squares refinement; data; parameters; weights; variance and covariance; linear least-squares; non-linear least-squares; restraints; constraints; ill-conditioning; singularity

Chapter.  5073 words. 

Subjects: Crystallography

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