Leif Ellingson

Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours

Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample spaces. To perform inference on these means, we appeal to the concept of neighborhood hypotheses from functional data analysis and derive a one-sample test. We then consider the analysis of shapes of contours lying in the plane. By embedding the corresponding sample space of such shapes, which is a Hilbert manifold, into a space of Hilbert-Schmidt operators, we can define extrinsic mean shapes of random planar contours and their sample analogues. We then apply the general methods to this problem while considering the computational restrictions faced when utilizing digital imaging data. Comparisons of computational cost are provided to another method for analyzing shapes of contours.

Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis

This paper addresses much needed asymptotic and nonparametric bootstrap methodology for two-sample tests for means on Riemannian manifolds with a simply transitive group of isometries. In particular, we develop a two-sample procedure for testing the equality of the generalized Frobenius means of two independent populations on the space of symmetric positive matrices. The new method naturally leads to an analysis based on Cholesky decompositions of covariance matrices which helps to decrease computational time and does not increase dimensionality. The resulting nonparametric matrix valued statistics are used for testing if there is a difference on average at a specific voxel between corresponding signals in Diffusion Tensor Images (DTIs) in young children with dyslexia when compared to their clinically normal peers, based on data that was previously analyzed using parametric methods.

Protein surface matching by combining local and global geometric information.

Comparison of the binding sites of proteins is an effective means for predicting protein functions based on their structure information. Despite the importance of this problem and much research in the past, it is still very challenging to predict the binding ligands from the atomic structures of protein binding sites. Here, we designed a new algorithm, TIPSA (Triangulation-based Iterative-closest-point for Protein Surface Alignment), based on the iterative closest point (ICP) algorithm. TIPSA aims to find the maximum number of atoms that can be superposed between two protein binding sites, where any pair of superposed atoms has a distance smaller than a given threshold. The search starts from similar tetrahedra between two binding sites obtained from 3D Delaunay triangulation and uses the Hungarian algorithm to find additional matched atoms. We found that, due to the plasticity of protein binding sites, matching the rigid body of point clouds of protein binding sites is not adequate for satisfactory binding ligand prediction. We further incorporated global geometric information, the radius of gyration of binding site atoms, and used nearest neighbor classification for binding site prediction. Tested on benchmark data, our method achieved a performance comparable to the best methods in the literature, while simultaneously providing the common atom set and atom correspondences.

An efficient algorithm for matching protein binding sites for protein function prediction

Comparing the binding sites of proteins is effective for predicting protein functions based on their structure information. However, it is still very challenging to predict the binding ligands from the atomic structures of protein binding sites. In this study, we designed a new algorithm based on the iterative closest point (ICP) algorithm. Our algorithm aims to find the maximum number of atoms that can be superposed between two protein binding sites, where any pair of matched superposed atoms has a distance smaller than a given threshold. The search starts from similar tetrahedra between two binding sites obtained from 3D Delaunay triangulation and uses the Hungarian algorithm to find additional matched atoms. We show that our method finds more matched atoms than a leading method. For benchmark data, we use the Tanimoto Index as a similarity measure and the nearest neighbor classifier to achieve a classification performance comparable to the best methods in the literature among those that provide both the common atom set and atom correspondences.

Nonparametric Bootstrap of Sample Means of Positive-Definite Matrices with an Application to Diffusion-Tensor-Imaging Data Analysis

ABSTRACT This paper presents nonparametric two-sample bootstrap tests for means of random symmetric positive-definite (SPD) matrices according to two different metrics: the Frobenius (or Euclidean) metric, inherited from the embedding of the set of SPD metrics in the Euclidean set of symmetric matrices, and the canonical metric, which is defined without an embedding and suggests an intrinsic analysis. A fast algorithm is used to compute the bootstrap intrinsic means in the case of the latter. The methods are illustrated in a simulation study and applied to a two-group comparison of means of diffusion tensors (DTs) obtained from a single voxel of registered DT images of children in a dyslexia study.

Evaluation and prediction of polygon approximations of planar contours for shape analysis

ABSTRACT Contours may be viewed as the 2D outline of the image of an object. This type of data arises in medical imaging as well as in computer vision and can be modeled as data on a manifold and can be studied using statistical shape analysis. Practically speaking, each observed contour, while theoretically infinite dimensional, must be discretized for computations. As such, the coordinates for each contour as obtained at k sampling times, resulting in the contour being represented as a k-dimensional complex vector. While choosing large values of k will result in closer approximations to the original contour, this will also result in higher computational costs in the subsequent analysis. The goal of this study is to determine reasonable values for k so as to keep the computational cost low while maintaining accuracy. To do this, we consider two methods for selecting sample points and determine lower bounds for k for obtaining a desired level of approximation error using two different criteria. Because this process is computationally inefficient to perform on a large scale, we then develop models for predicting the lower bounds for k based on simple characteristics of the contours.

On the CLT on Low Dimensional Stratified Spaces

Noncategorical observations, when regarded as points on a stratified space, lead to a nonparametric data analysis extending data analysis on manifolds. In particular, given a probability measure on a sample space with a manifold stratification, one may define the associated Frechet function, Frechet total variance, and Frechet mean set. The sample counterparts of these parameters have a more nuanced asymptotic behaviors than in nonparametric data analysis on manifolds. This allows for the most inclusive data analysis known to date. Unlike the case of manifolds, Frechet sample means on stratified spaces may stick to a lower dimensional stratum, a new dimension reduction phenomenon. The downside of stickiness is that it yields a less meaningful interpretation of the analysis. To compensate for this, an extrinsic data analysis, that is more sensitive to input data is suggested. In this paper one explores analysis of data on low dimensional stratified spaces, via simulations. An example of extrinsic analysis on phylogenetic tree data is also given.