MBI Videos

Videos by CTW: Statistics, Geometry, and Combinatorics on Stratified Spaces arising from Biological Problems

  • Object Oriented Data Analysis
    J. S. Marron
    Object Oriented Data Analysis is the statistical analysis of populations of complex objects. In the special case of Functional Data Analysis, these data objects are curves, where standard Euclidean approaches, such as principal components analysis, have been very successful. Challenges in modern medical image analysis motivate the statistical analysis of populations of mor...
  • The geometry and statistics of geometric trees
    Aasa Feragen
    Anatomical tree-structures such as airway trees from lungs, blood vessels or dendrite trees in neurons, carry information about the organ that they are part of. Anatomical trees can be modeled as geometric trees, which are combinatorial trees whose edges are endowed with edge attributes describing their geometry. We consider edge attributes which take continuous scalar or ...
  • On Omitting and Hitting Properties for Means on Circles and Shape Spaces
    Stephan Huckemann
    The classical central limit theorem states that suitably translated and root n rescaled independent sample means tend to a multivariate Gaussian. Under certain, still rather restrictive conditions, it has been shown by Bhattacharya and Patrangenaru (2005) that the analog holds true on manifolds. One condition, namely uniqueness has been pushed to "data contained in a ...
  • Medians, means and minimax centers in Riemannian geometry: existence, uniqueness, robustness and algorithms. Application to signal detection
    Marc Arnaudon
    We give detailed results on the existence and uniqueness for medians, means and minimax centers of probability measures on Riemannian manifolds, including the case when the probability measure is supported in a regular geodesic ball and the case of generic data points in a complete manifold. Some properties of Fr'echet medians are also given, such as statistical consi...
  • Nonparametric Statistics on Manifolds- By Examples and Applications
    Rabi Bhattacharya
    The general theory of nonparametric statistics on manifolds M presented here is of relatively recent origin. It builds much of its framework on the notion of the Fre'chet mean of a probability measure Q, namely, the point on the manifold which minimizes the expected squared distance from a random variable with distribution Q. The nonparametric methods are intrinsic or...
  • Sticky central limit theorems at singularities
    Ezra Miller
    Applications to areas such as biology, medicine, and image analysis require understanding the asymptotics of distributions on stratified spaces, such as tree spaces. In the surprisingly common circumstance when Frechet (intrinsic) means of distributions on stratified spaces lie on strata of low dimension, central limit theorems can exhibit non-classical "sticky" ...
  • Bacterial trees in the Human Microbiome
    Susan Holmes
    Many studies are underway to describe the human microbiome, I will describe some of the methods used that combine phylogenetic trees and abundance data from high throughput sequencing and new microarray techniques.

    In particular various `particular metrics' have shown useful in coming to conclusions about explanatory clinical or contingent variables in ...
  • Phylogenetic networks and the real moduli space of curves
    Satyan Devadoss
    Our story is motivated by the configuration space of particles on spheres. In the 1970s, Grothendieck, Deligne, and Mumford constructed a way to keep track of particle collisions in this space using Geometric Invariant Theory. In the 1990s, Gromov and Witten utilized them as invariants arising from string field theory and quantum cohomology. We consider the real points of ...
  • Towards statistical topology: homology, persistent homology and persistence landscapes
    Peter Bubenik
    One of the principal uses of topology is to patch together local quantitative data to obtain global qualitative information not readily accessible to other methods. While the early development of topology was largely driven by applications, many later advances were motivated by strictly mathematical concerns. Now the field of applied topology is returning topology to its r...
  • Riemannian barycentres: from harmonic maps and statistical shape to the classical central limit theorem
    Wilfrid Kendall
    The subject of Riemannian barycentres has a strikingly long history, stretching back to work of Frechet and Cartan. The first part of this talk will be a review of the fundamental ideas and a discussion of the work of various probabilists and statisticians on applications of the concept to probabilistic approaches to harmonic map theory and statistical shape theory. I will...
  • Some Recent Experience with Clostridium Difficile
    Peter Kim
    C. difficile associated outbreaks have been reported worldwide, some with increased mortality and morbidity. Symptoms of this infectious disease range from mild diarrhea to severe colitis and even bowel perforation and death. The bacterium C. difficile is found with the normal bacteria comprising the intestinal flora. These can be killled by antibiotics but not the C. diff...
  • Approaching the evolution of novelty: where biology needs math and statistics
    David Houle
    The genetics and evolution of biological systems are extremely complex because of the large number of traits , and complex relationships among those traits. We use the form of fruit fly wings as a model to study the variational properties of complex biological structures. Variation is important because it controls evolutionary potential. Questions about evolutionary potent...
  • Survey of stratified spaces
    Robert MacPherson
    Stratified spaces arise in many contexts within mathematics. They are the natural class of topological spaces of "finite complexity". In many cases, they come endowed with canonical probability distributions on them.

    This will be a survey talk with examples, such as spaces of configuration of points....
  • Mean location, the two sample problem Harrie Hendriks, Mathematics, Radboud University Nijmegen
    Harrie Hendriks
    The context will be the estimation of a parameter of a probability distribution, where the parameter lies in a differentiable manifold, more specifically in a submanifold of Euclidean space. The parameter could be a Frechet mean of a probability distribution on the submanifold itself, Frechet mean with respect to the Euclidean distance. We will give an account of the two-s...
  • Manifold-valued Tuning Parameters in Regularized Estimation of Multivariate Means
    Rudolf Beran
    A multivariate k-way layout consists of observations with error on an array of vector-valued means, each of which is an unknown function of k real-valued covariates. Any decomposition of these vector means into a sum of orthogonal projections induces least squares submodel fits that serve as candidate estimators of the mean vectors. MANOVA submodel fits, nested polynomial ...
  • Geometry and statistics of data
    Hongtu Zhu
    Not available...
  • Statistics in Tree Space
    Megan Owen
    The space of metric phylogenetic trees, as constructed by Billera, Holmes, and Vogtmann, is a polyhedral cone complex. This space is non-positively curved, which ensures there is a unique shortest path (geodesic) between any two trees, and that the mean and variance of a set or distribution of trees is well-defined. Furthermore, there is a polynomial time algorithm to comp...
  • Geometry and Statistics in the Eigen-structure of Symmetric (Positive Semi-Definite) Matrices
    Armin Schwartzman
    Symmetric positive semi-definite (PSD) matrices appear as data objects in the statistical analysis of Diffusion Tensor Imaging data, where there is interest in making inferences about the eigenvalues and eigenvectors of these objects. In this talk, I present a stratification of the set of symmetric PSD matrices of arbitrary dimension according to their eigenvalues, as well...
  • The geometry and topology of projective shape space
    John Kent
    Projective geometry underlies the way in which information about a 3d scene can be deduced from (one or more) 2d camera views. A key concept in projective geometry is that of a projective invariant for a configuration of collinear or coplanar points. The collection of information in the projective invariants can be termed the "projective shape" of a configuration...
  • Topological Analysis of Variance and the Maxillary Complex
    Giseon Heo
    Persistent homology, a recent development in computational topology, has shown to be useful for analyzing high dimensional non-linear data. In this talk, we connect computational topology with the traditional analysis of variance and demonstrate this synergy on a three-dimensional orthodontic landmark data set derived from the maxillary complex. (Joint work with Jennifer G...

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