Statistical problems arising from crystal structure analysis

This thesis is concerned with the application of statistical techniques in the field of crystallography - a branch of science dealing with the structure, classification and properties of crystals - and an analysis of some of the associated statistical problems. We shall concentrate throughout on the estimation of atomic co-ordinates within the unit cells of crystals. The science of X-ray crystallography will be introduced and a review of some of the existing methodology given. We shall then consider how statistical ideas may be used to improve this methodology. We shall be particularly concerned with the area of sequential experimentation, in which the data collection process itself is modified as a result of analysing the data already collected. Sequential experimentation for improved efficiency in any particular crystallographic problem requires that decisions be made as to which additional data should be collected in order to achieve the desired objective. Ways of selecting suitable sampling strategies will be described, together with associated stopping rules. We will also describe methods for handling relevant prior information - e.g. structural information available in crystallographic data bases - and nuisance parameters, and procedures for dealing with the inherent non-linearity of the crystallographic model, matrix updating and the recursive addition of data. The central problem of X-ray crystallography - the 'phase problem' - will also be analysed from a statistical perspective. Practical application of some of our ideas will be given. Much emphasis is placed on non-linear parameter estimation problems such as those arising in crystallography. A review of relevant statistical work in this general field is undertaken, and geometry-based ideas of our own proposed. We concentrate on either seeking suitable re-parameterisations (in a sense which we define) or on seeking alternatives to the standard tangent plane approximation to the solution surface based on relevant curvature measures. The thesis ends with a few relevant concluding comments and some ideas for further related statistical work in the area of X-ray crystallography.