Thomas Hotz

Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Frechet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).

Drift estimation for single marker switching based imaging schemes.

In recent years, the diffraction barrier in fluorescence imaging has been broken and optical nanoscopes now routinely image with resolutions of down to 20 nm, an improvement of more than 10 fold. Because this allows imaging much smaller features and because all super-resolution approaches trade off speed for spatial resolution, mechanical instabilities of the microscopes become a limiting factor. Here, we propose a fully data-driven statistical registration method for drift detection and drift correction for single marker switching (SMS) imaging schemes, including a guideline for parameter choice and quality checks of the drift analysis. The necessary assumptions about the drift are minimal, allowing a model-free approach, but more specific models can easily be integrated. We determine the resulting performance on standard SMS measurements and show that the drift determination can be routinely brought to the range of precision achievable by fiducial marker-tracking methods.

Intrinsic MANOVA for Riemannian Manifolds with an Application to Kendall's Space of Planar Shapes

We propose an intrinsic multifactorial model for data on Riemannian manifolds that typically occur in the statistical analysis of shape. Due to the lack of a linear structure, linear models cannot be defined in general; to date only one-way MANOVA is available. For a general multifactorial model, we assume that variation not explained by the model is concentrated near elements defining the effects. By determining the asymptotic distributions of respective sample covariances under parallel transport, we show that they can be compared by standard MANOVA. Often in applications manifolds are only implicitly given as quotients, where the bottom space parallel transport can be expressed through a differential equation. For Kendalls space of planar shapes, we provide an explicit solution. We illustrate our method by an intrinsic two-way MANOVA for a set of leaf shapes. While biologists can identify genotype effects by sight, we can detect height effects that are otherwise not identifiable.

What Is Special about Patients with Lung Cancer and Pulmonary Metastases in Palliative Care? Results from a Nationwide Survey

BACKGROUND Patients with advanced lung cancer constitute a special focus in palliative care not only for epidemiological or prognostic reasons, but also because their symptom burden is felt to be widespread and difficult to treat. This study describes disease-specific characteristics and the symptom burden of patients with advanced incurable lung cancer, comparing them with patients suffering from other diseaseentities. METHODS A secondary analysis of the nationwide Hospice and Palliative Care Evaluation (HOPE) was performed, by focussing on inpatient hospice and palliative care unit patients and by using descriptive methods. RESULTS From 2006 to 2008, 5487 inpatients were registered, 874 of which were diagnosed with lung cancer and 1884 with pulmonary metastases. Symptoms such as weakness, tiredness, or pain were most prevalent in all subgroups. Dyspnea was significantly more prevalent in all patients with different kinds of pulmonary tumor manifestations; confusion was significantly more prevalent in patients with lung cancer. Dyspnea could not be treated as effectively as pain or nausea. Confusion and nursing problems worsened during the observation period. Dyspnea and confusion were associated with increased risk of death during the observational period. CONCLUSION The symptom pattern of patients with lung cancer is characterized by dyspnea and confusion--symptoms that are difficult to treat until discharge and that imply a worse prognosis. Therefore, increased research on the pathophysiology and treatment of dyspnea and confusion is required, and efforts in advance care planning and anticipation of dyspnea and confusion as a critical symptom in patients with lung cancer should be reinforced.

Locally adaptive image denoising by a statistical multiresolution criterion

It is shown how to choose the smoothing parameter in image denoising by a statistical multiresolution criterion, both globally and locally. Using inhomogeneous diffusion and total variation regularization as examples for localized regularization schemes, an efficient method for locally adaptive image denoising is presented. As expected, the smoothing parameter serves as an edge detector in this framework. Numerical examples together with applications in confocal microscopy illustrate the usefulness of the approach.

Principal component geodesics for planar shape spaces

In this paper a numerical method to compute principal component geodesics for Kendalls planar shape spaces-which are essentially complex projective spaces-is presented. Underlying is the notion of principal component analysis based on geodesics for non-Euclidean manifolds as proposed in an earlier paper by Huckemann and Ziezold [S. Huckemann, H. Ziezold, Principal component analysis for Riemannian manifolds with an application to triangular shape spaces, Adv. Appl. Prob. (SGSA) 38 (2) (2006) 299-319]. Currently, principal component analysis for shape spaces is done on the basis of a Euclidean approximation. In this paper, using well-studied datasets and numerical simulations, these approximation errors are discussed. Overall, the error distribution is rather dispersed. The numerical findings back the notion that the Euclidean approximation is good for highly concentrated data. For low concentration, however, the error can be strongly notable. This is in particular the case for a small number of landmarks. For highly concentrated data, stronger anisotropicity and a larger number of landmarks may also increase the error.

Idealizing Ion Channel Recordings by a Jump Segmentation Multiresolution Filter

Based on a combination of jump segmentation and statistical multiresolution analysis for dependent data, a new approach called J-SMURF to idealize ion channel recordings has been developed. It is model-free in the sense that no a-priori assumptions about the channels characteristics have to be made; it thus complements existing methods which assume a model for the channels dynamics, like hidden Markov models. The method accounts for the effect of an analog filter being applied before the data analysis, which results in colored noise, by adapting existing muliresolution statistics to this situation. J-SMURFs ability to denoise the signal without missing events even when the signal-to-noise ratio is low is demonstrated on simulations as well as on ion current traces obtained from gramicidin A channels reconstituted into solvent-free planar membranes. When analyzing a newly synthesized acylated system of a fatty acid modified gramicidin channel, we are able to give statistical evidence for unknown gating characteristics such as subgating.

Multiscale DNA partitioning: Statistical evidence for segments.

MOTIVATION DNA segmentation, i.e. the partitioning of DNA in compositionally homogeneous segments, is a basic task in bioinformatics. Different algorithms have been proposed for various partitioning criteria such as Guanine/Cytosine (GC) content, local ancestry in population genetics or copy number variation. A critical component of any such method is the choice of an appropriate number of segments. Some methods use model selection criteria and do not provide a suitable error control. Other methods that are based on simulating a statistic under a null model provide suitable error control only if the correct null model is chosen. RESULTS Here, we focus on partitioning with respect to GC content and propose a new approach that provides statistical error control: as in statistical hypothesis testing, it guarantees with a user-specified probability [Formula: see text] that the number of identified segments does not exceed the number of actually present segments. The method is based on a statistical multiscale criterion, rendering this as a segmentation method that searches segments of any length (on all scales) simultaneously. It is also accurate in localizing segments: under benchmark scenarios, our approach leads to a segmentation that is more accurate than the approaches discussed in the comparative review of Elhaik et al. In our real data examples, we find segments that often correspond well to features taken from standard University of California at Santa Cruz (UCSC) genome annotation tracks. AVAILABILITY AND IMPLEMENTATION Our method is implemented in function smuceR of the R-package stepR available at http://www.stochastik.math.uni-goettingen.de/smuce.

Extrinsic vs Intrinsic Means on the Circle

We compare extrinsic and intrinsic means for circular data with respect to criteria such as uniqueness, computational complexity, robustness, and asymptotic relative efficiency. Moreover, we construct universal, rate-optimal confidence sets for the extrinsic mean. We conclude that neither location estimator outperforms the other in general. However, due to its simplicity, greater robustness and the existence of universal, rate-optimal confidence sets for it, the extrinsic mean appears preferable to the intrinsic mean unless one knows in advance that the underlying distribution is unimodal.

On Means and Their Asymptotics: Circles and Shape Spaces

We survey some effects that singular strata may have in the positive curvature context of circles and shape spaces when conducting (semi-)intrinsic statistical analyses. Here, the analysis of data on a stratified space is based on statistical descriptors defined in a possibly different stratified space. E.g. in geodesic principal component analysis for shape spaces, shape data are described by generalized geodesics which naturally form a shape space of their own, different from the original one. In a general context, if the descriptors are obtained as generalized Fréchet means, under rather general circumstances, a strong law of large numbers is valid. If furthermore the descriptors are sufficiently well behaved, a classical central limit theorem can be adopted. One of the crucial conditions is that hitting of singular strata as well as of cut loci, if present, must be controlled. We review the statistical role of the cut locus of intrinsic means for circles as well as that of singular strata for shape spaces (occurring where the group action is degenerate) and conclude with an identification of potential research directions.