1,720,982 research outputs found
A smeary central limit theorem for manifolds with application to high-dimensional spheres
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Intrinsic Shape Analysis: Geodesic PCA for Riemannian Manifolds Modulo Isometric Lie Group Actions Rejoinder
No full textspaces
No full textAbstract The task to write on data analysis on nonstandard spaces is quite substantial, with a huge body of literature to cover, from parametric to nonparametrics, from shape spaces to Wasserstein spaces. In this survey we convey simple (e.g., Fréchet means) and more complicated ideas (e.g., empirical process theory), common to many approaches with focus on their interaction with one‐another. Indeed, this field is fast growing and it is imperative to develop a mathematical view point, drawing power, and diversity from a higher level of image Surveying many non‐Euclidean statistical problems with ingenious solutions, we uncover new ones, keeping mathematicians, statisticians, computer and data scientists busy for a while.abstraction, for example, by introducing generalized Fréchet means. While many problems have found ingenious solutions (e.g., Procrustes analysis for principal component analysis [PCA] extensions on shape spaces and diffusion on the frame bundle to mimic anisotropic Gaussians), more problems emerge, often more difficult (e.g., topology and geometry influencing limiting rates and defining generic intrinsic PCA extensions). Along this survey, we point out some open problems, that will, as it seems, keep mathematicians, statisticians, computer and data scientists busy for a while. This article is categorized under: Statistical and Graphical Methods of Data Analysis \u0026gt; Analysis of High Dimensional DataDeutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659Volkswagen Foundation http://dx.doi.org/10.13039/501100001663Felix‐Bernstein‐Institute for Mathematical Statistics in the Biosciences at the University of Göttinge
Learning torus PCA based classification for multiscale RNA backbone structure correction with application to SARS-CoV-2
No full textDiffusion means in geometric spaces
No full textWe introduce a location statistic for distributions on non-linear geometric spaces, the diffusion mean, serving as an extension and an alternative to the Fréchet mean. The diffusion mean arises as the generalization of Gaussian maximum likelihood analysis to non-linear spaces by maximizing the likelihood of a Brownian motion. The diffusion mean depends on a time parameter t, which admits the interpretation of the allowed variance of the diffusion. The diffusion t-mean of a distribution X is the most likely origin of a Brownian motion at time t, given the end-point distribution X. We give a detailed description of the asymptotic behavior of the diffusion estimator and provide sufficient conditions for the diffusion estimator to be strongly consistent. Particularly, we present a smeary central limit theorem for diffusion means and we show that joint estimation of the mean and diffusion variance rules out smeariness in all directions simultaneously in general situations. Furthermore, we investigate properties of the diffusion mean for distributions on the sphere Sm. Experimentally, we consider simulated data and data from magnetic pole reversals, all indicating similar or improved convergence rate compared to the Fréchet mean. Here, we additionally estimate t and consider its effects on smeariness and uniqueness of the diffusion mean for distributions on the sphere.</p
Stability of the cut locus and a Central Limit Theorem for Fréchet means of Riemannian manifolds
Full text linkWe obtain a central limit theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin’s Omnibus central limit theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case
Comments on: Recent advances in directional statistics
No full textInspired by this felicitous, highly concentrated and rather exhaustive review of a rapidly growing field, many larger research areas that warrant further investigation come to mind. In this comment, three areas are selected: fully satisfactory PCA on tori and polyspheres, harnessing linearity through Lie algebras underlying homogeneous spaces such as those for directional data, and statistical analysis based on critical points (e.g. mode and antimodes) of Fréchet Lp
-functions.Deutsche ForschungsgemeinschaftVolkswagen Foundation (DE
Functional inference on rotational curves under sample‐specific group actions and identification of human gait
No full textPrincipal component analysis and clustering on manifolds
No full textBig data, high dimensional data, sparse data, large scale data, and imaging data are all becoming new frontiers of statistics. Changing technologies have created this flood and have led to a real hunger for new modeling strategies and data analysis by scientists. In many cases data are not Euclidean; for example, in molecular biology, the data sit on manifolds. Even in a simple non-Euclidean manifold (circle), to summarize angles by the arithmetic average cannot make sense and so more care is needed. Thus non-Euclidean settings throw up many major challenges, both mathematical and statistical. This paper will focus on the PCA and clustering methods for some manifolds. Of course, the PCA and clustering methods in multivariate analysis are one of the core topics.
We basically deal with two key manifolds from a practical point of view, namely spheres and tori. It is well known that dimension reduction on non-Euclidean manifolds with PCA-like methods has been a challenging task for quite some time but recently there has been some breakthrough. One of them is the idea of nested spheres and another is transforming a torus into a sphere effectively and subsequently use the technology of nested spheres PCA. We also provide a new method of clustering for multivariate analysis which has a fundamental property required for molecular biology that penalizes wrong assignments to avoid chemically no go areas. We give various examples to illustrate these methods. One of the important examples includes dealing with COVID-19 data
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