A test of a robust/resistant refinement procedure on synthetic data sets

Abstract

The conventional crystallographic least-squares procedure has been compared with a robust/resistant modification in which the weight of each reflection is multiplied by a function of the ratio of its residual to a resistant measure of the width of the residual distribution on the previous cycle. Three synthetic data sets were created by adding random errors, according to various probability distributions, to the calculated structure factors for a known crystal structure. A set with a Gaussian error distribution was refined with two sets of weights: one assigned correctly in proportion to the reciprocals of the variances of the data points, the other using unit weights throughout. The second error distribution was Gaussian contaminated by 10% drawn from another Gaussian distribution with its variance nine times greater. The third distribution was a long-tailed distribution derived by dividing a random variable with a Gaussian distribution by an independent random variable with a uniform distribution. Each of the first three cases was refined to convergence using both conventional and robust/resistant procedures, with the modified procedure leading to a result at least as close to the known structure as the conventional procedure. In the fourth case, the conventional procedure gave a poor fit, but the robust/ resistant procedure converged to a reasonable approximation to the correct structure.