529 research outputs found
spaces
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
A smeary central limit theorem for manifolds with application to high-dimensional spheres
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Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth
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Intrinsic Shape Analysis: Geodesic PCA for Riemannian Manifolds Modulo Isometric Lie Group Actions Rejoinder
No full textFunctional inference on rotational curves under sample‐specific group actions and identification of human gait
No full textConfidence tubes for curves on SO(3) and identification of subject-specific gait change after kneeling
No full textAbstract In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which might be a major risk factor for the development of knee osteoarthritis, we develop confidence tubes for curves following a perturbation model on SO(3) using the Gaussian kinematic formula which are equivariant under gait similarities and have precise coverage even for small sample sizes. Applying them to gait curves from eight volunteers undergoing kneeling tasks and adjusting for different walking speeds and marker replacement at different visits, allows us to identify at which phases of the gait cycle the gait pattern changed due to kneeling
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
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