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Statistical analysis on high-dimensional spheres and shape spaces

We consider the statistical analysis of data on high-dimensional spheres and shape spaces. The work is of particular relevance to applications where high-dimensional data are available--a commonly encountered situation in many disciplines. First the uniform measure on the infinite-dimensional sphere is reviewed, together with connections with Wiener measure. We then discuss densities of Gaussian measures with respect to Wiener measure. Some nonuniform distributions on infinite-dimensional spheres and shape spaces are introduced, and special cases which have important practical consequences are considered. We focus on the high-dimensional real and complex Bingham, uniform, von Mises-Fisher, Fisher-Bingham and the real and complex Watson distributions. Asymptotic distributions in the cases where dimension and sample size are large are discussed. Approximations for practical maximum likelihood based inference are considered, and in particular we discuss an application to brain shape modeling.Comment: Published at http://dx.doi.org/10.1214/009053605000000264 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Exploration of Space

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Bayesian matching of unlabelled point sets using Procrustes and configuration models

The problem of matching unlabelled point sets using Bayesian inference is considered. Two recently proposed models for the likelihood are compared, based on the Procrustes size-and-shape and the full configuration. Bayesian inference is carried out for matching point sets using Markov chain Monte Carlo simulation. An improvement to the existing Procrustes algorithm is proposed which improves convergence rates, using occasional large jumps in the burn-in period. The Procrustes and configuration methods are compared in a simulation study and using real data, where it is of interest to estimate the strengths of matches between protein binding sites. The performance of both methods is generally quite similar, and a connection between the two models is made using a Laplace approximation

Bayesian matching of unlabeled marked point sets using random fields, with an application to molecular alignment

Statistical methodology is proposed for comparing unlabeled marked point sets, with an application to aligning steroid molecules in chemoinformatics. Methods from statistical shape analysis are combined with techniques for predicting random fields in spatial statistics in order to define a suitable measure of similarity between two marked point sets. Bayesian modeling of the predicted field overlap between pairs of point sets is proposed, and posterior inference of the alignment is carried out using Markov chain Monte Carlo simulation. By representing the fields in reproducing kernel Hilbert spaces, the degree of overlap can be computed without expensive numerical integration. Superimposing entire fields rather than the configuration matrices of point coordinates thereby avoids the problem that there is usually no clear one-to-one correspondence between the points. In addition, mask parameters are introduced in the model, so that partial matching of the marked point sets can be carried out. We also propose an adaptation of the generalized Procrustes analysis algorithm for the simultaneous alignment of multiple point sets. The methodology is illustrated with a simulation study and then applied to a data set of 31 steroid molecules, where the relationship between shape and binding activity to the corticosteroid binding globulin receptor is explored.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS486 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Power Euclidean metrics for covariance matrices with application to diffusion tensor imaging

Various metrics for comparing diffusion tensors have been recently proposed in the literature. We consider a broad family of metrics which is indexed by a single power parameter. A likelihood-based procedure is developed for choosing the most appropriate metric from the family for a given dataset at hand. The approach is analogous to using the Box-Cox transformation that is frequently investigated in regression analysis. The methodology is illustrated with a simulation study and an application to a real dataset of diffusion tensor images of canine hearts

Quality-Controlled Wind Data from the Kennedy Space Center 915 Megahertz Doppler Radar Wind Profiler Network

The National Aeronautics and Space Administration s (NASA) Kennedy Space Center (KSC) has installed a five-instrument 915-Megahertz (MHz) Doppler Radar Wind Profiler (DRWP) system that records atmospheric wind profile properties. The purpose of these profilers is to fill data gaps between the top of the KSC wind tower network and the lowest measurement altitude of the KSC 50-MHz DRWP. The 915-MHz DRWP system has the capability to generate three-dimensional wind data outputs from approximately 150 meters (m) to 6,000 m at roughly 15-minute (min) intervals. NASA s long-term objective is to combine the 915-MHz and 50-MHz DRWP systems to create complete vertical wind profiles up to 18,300 m to be used in trajectory and loads analyses of space vehicles and by forecasters on day-of-launch (DOL). This analysis utilizes automated and manual quality control (QC) processes to remove erroneous and unrealistic wind data returned by the 915-MHz DRWP system. The percentage of data affected by each individual QC check in the period of record (POR) (i.e., January to April 2006) was computed, demonstrating the variability in the amount of data affected by the QC processes. The number of complete wind profiles available at given altitude thresholds for each profiler in the POR was calculated and outputted graphically, followed by an assessment of the number of complete wind profiles available for any profiler in the POR. A case study is also provided to demonstrate the QC process on a day of a known weather event