80 research outputs found
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
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
Bayesian registration of functions and curves
Bayesian analysis of functions and curves is considered, where warping and other geometrical transformations are often required for meaningful comparisons. The functions and curves of interest are represented using the recently introduced square root velocity function, which enables a warping invariant elastic distance to be calculated in a straightforward manner. We distinguish between various spaces of interest: the original space, the ambient space after standardizing, and the quotient space after removing a group of transformations. Using Gaussian process models in the ambient space and Dirichlet priors for the warping functions, we explore Bayesian inference for curves and functions. Markov chain Monte Carlo algorithms are introduced for simulating from the posterior. We also compare ambient and quotient space estimators for mean shape, and explain their frequent similarity in many practical problems using a Laplace approximation. Simulation studies are carried out, as well as practical alignment of growth rate functions and shape classification of mouse vertebra outlines in evolutionary biology. We also compare the performance of our Bayesian method with some alternative approaches
Bayesian alignment of continuous molecular shapes using random fields
Statistical methodology is proposed for comparing
molecular shapes. In order to account for the continuous nature of molecules, classical shape analysis methods are combined with techniques used for predicting random fields in spatial statistics. Applying a modification of Procrustes analysis, Bayesian inference is carried out using Markov chain Monte Carlo methods for the pairwise alignment of the resulting molecular fields. Superimposing entire fields rather than the configuration matrices of nuclear positions thereby solves the problem that there is usually no clear one--to--one correspondence between the atoms of the two molecules under consideration. Using a similar concept, we also propose an adaptation of the generalised Procrustes analysis algorithm for the simultaneous alignment of multiple molecular fields. The methodology is applied to a dataset of 31 steroid molecules
Non-Euclidean statistics for covariance matrices with applications to diffusion tensor imaging
The statistical analysis of covariance matrix data is considered and,
in particular, methodology is discussed which takes into account the nonEuclidean
nature of the space of positive semi-definite symmetric matrices.
The main motivation for the work is the analysis of diffusion tensors in medical
image analysis. The primary focus is on estimation of a mean covariance
matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons
are made with other estimation techniques, including using the matrix
logarithm, matrix square root and Cholesky decomposition. Applications
to diffusion tensor imaging are considered and, in particular, a new measure
of fractional anisotropy called Procrustes Anisotropy is discussed
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