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Crystal Structure AnalysisPrinciples and Practice$
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William Clegg, Alexander J Blake, Jacqueline M Cole, John S O Evans, Peter Main, Simon Parsons, and David J Watkin

Print publication date: 2009

Print ISBN-13: 9780199219469

Published to Oxford Scholarship Online: September 2009

DOI: 10.1093/acprof:oso/9780199219469.001.0001

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Least-squares fitting of parameters

Least-squares fitting of parameters

(p.155) 12 Least-squares fitting of parameters
Crystal Structure Analysis

Peter Main

Oxford University Press

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

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