arXiv subjects
Publications and source records attributed to Stephan F. Huckemann.
We 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.
From infancy to adulthood, human growth is anisotropic, much more along the proximal-distal axis (height) than along the medial-lateral axis (width), particularly at extremities. Detecting and modeling the rate of anisotropy in fingerprint growth, and possibly other growth patterns as well, facilitates the use of children's fingerprints for long-term biometric identification. Using standard fingerprint scanners, anisotropic growth is highly overshadowed by the varying distortions created by each imprint, and it seems that this difficulty has hampered to date the development of suitable methods, detecting anisotropy, let alone, designing models. We provide a tool chain to statistically detect, with a given confidence, anisotropic growth in fingerprints and its preferred axis, where we only require a standard fingerprint scanner and a minutiae matcher. We build on a perturbation model, a new Procrustes-type algorithm, use and develop several parametric and non-parametric tests for different hypotheses, in particular for neighborhood hypotheses to detect the axis of anisotropy, where the latter tests are tunable to measurement accuracy. Taking into account realistic distortions caused by pressing fingers on scanners, our simulations based on real data indicate that, for example, already in rather small samples (56 matches) we can significantly detect proximal-distal growth if it exceeds medial-lateral growth by only around 5 percent. Our method is well applicable to future datasets of children fingerprint time series and we provide an implementation of our algorithms and tests with matched minutiae pattern data.
The (CLT) central limit theorems for generalized Frechet means (data descriptors assuming values in stratified spaces, such as intrinsic means, geodesics, etc.) on manifolds from the literature are only valid if a certain empirical process of Hessians of the Frechet function converges suitably, as in the proof of the prototypical BP-CLT (Bhattacharya and Patrangenaru (2005)). This is not valid in many realistic scenarios and we provide for a new very general CLT. In particular this includes scenarios where, in a suitable chart, the sample mean fluctuates asymptotically at a scale $n^α$ with exponents $α < 1/2$ with a non-normal distribution. As the BP-CLT yields only fluctuations that are, rescaled with $n^{1/2}$ , asymptotically normal, just as the classical CLT for random vectors, these lower rates, somewhat loosely called smeariness, had to date been observed only on the circle (Hotz and Huckemann (2015)). We make the concept of smeariness on manifolds precise, give an example for two-smeariness on spheres of arbitrary dimension, and show that smeariness, although "almost never" occurring, may have serious statistical implications on a continuum of sample scenarios nearby. In fact, this effect increases with dimension, striking in particular in high dimension low sample size scenarios.
We extend Gaussian perturbation models in classical functional data analysis to the three-dimensional rotational group where a zero-mean Gaussian process in the Lie algebra under the Lie exponential spreads multiplicatively around a central curve. As an estimator, we introduce point-wise extrinsic mean curves which feature strong perturbation consistency, and which are asymptotically a.s. unique and differentiable, if the model is so. Further, we consider the group action of time warping and that of spatial isometries that are connected to the identity. The latter can be asymptotically consistently estimated if lifted to the unit quaternions. Introducing a generic loss for Lie groups, the former can be estimated, and based on curve length, due to asymptotic differentiability, we propose two-sample permutation tests involving various combinations of the group actions. This methodology allows inference on gait patterns due to the rotational motion of the lower leg with respect to the upper leg. This was previously not possible because, among others, the usual analysis of separate Euler angles is not independent of marker placement, even if performed by trained specialists.
For sequences of random backward nested subspaces as occur, say, in dimension reduction for manifold or stratified space valued data, asymptotic results are derived. In fact, we formulate our results more generally for backward nested families of descriptors (BNFD). Under rather general conditions, asymptotic strong consistency holds. Under additional, still rather general hypotheses, among them existence of a.s. local twice differentiable charts, asymptotic joint normality of a BNFD can be shown. If charts factor suitably, this leads to individual asymptotic normality for the last element, a principal nested mean or a principal nested geodesic, say. It turns out that these results pertain to principal nested spheres (PNS) and principal nested great subsphere (PNGS) analysis by Jung et al. (2010) as well as to the intrinsic mean on a first geodesic principal component (IMo1GPC) for manifolds and Kendall's shape spaces. A nested bootstrap two-sample test is derived and illustrated with simulations. In a study on real data, PNGS is applied to track early human mesenchymal stem cell differentiation over a coarse time grid and, among others, to locate a change point with direct consequences for the design of further studies.
In this paper we consider a novel statistical inverse problem on the Poincaré, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of $2\times2$ real matrices of determinant one via Möbius transformations. Our approach is based on a deconvolution technique which relies on the Helgason--Fourier calculus adapted to this hyperbolic space. This gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random Möbius transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincaré plane exactly describes the physical system that is of statistical interest.