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Anne Marie Svane

Publications and source records attributed to Anne Marie Svane.

At least 19 recordsLinked to original sources
arXiv2026

Normal and Poisson approximation for Gibbs point processes with pair potentials

We provide a Poisson approximation result for dependent thinnings of Gibbs point processes as well as qualitative and quantitative central limit theorems for geometric functionals of Gibbs point processes in increasing observation windows. The present paper extends prior work on finite-range Gibbs processes to processes with repulsive pairwise interaction of unbounded interaction range as well as processes on marked Euclidean space. The proofs rely on coupling different Gibbs processes using the disagreement coupling technique, which we generalize to infinite-volume domains under a suitable non-percolation condition. For the case of repulsive pairwise interactions, we introduce a version of disagreement coupling that constructs the Gibbs process by thinning a random connection model thus making previous approximation methods more flexible.

math.PR↗
arXiv2025

Identifying cobordisms using kernel persistence

Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times.

math.AT↗
arXiv2024

The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

Let $X=\sum_{k=1}^\infty X_k β^{-k}$ be the base-$β$ expansion of a continuous random variable $X$ on the unit interval where $β$ is the positive solution to $β^n = 1 + β+ \cdots + β^{n-1}$ for an integer $n\ge 2$ (i.e., $β$ is a generalization of the golden mean for which $n=2$). We study the asymptotic distribution and convergence rate of the scaled remainder $\sum_{k=1}^\infty X_{m+k} β^{-k}$ when $m$ tends to infinity.

math.PR↗
arXiv2024

Normal approximation for Gibbs processes via disagreement couplings

This work improves the existing central limit theorems (CLTs) for geometric functionals of Gibbs processes in three aspects. First, we derive a CLT for weakly stabilizing functionals, thereby improving on the previously used assumption of exponential stabilization. Second, we show that this CLT holds for interaction ranges up to the percolation threshold of the dominating Poisson process. This avoids imprecise branching bounds from graphical construction. Third, by constructing simultaneous couplings of several Palm processes for Gibbs functionals, we provide a quantitative CLT in terms of Kolmogorov bounds for normal approximation. An important conceptual ingredient in these advances is the extension of disagreement coupling adapted to unbounded windows and to the comparison at multiple spatial locations.

math.PR↗
arXiv2023

How many digits are needed?

Let $X_1,X_2,...$ be the digits in the base-$q$ expansion of a random variable $X$ defined on $[0,1)$ where $q\ge2$ is an integer. For $n=1,2,...$, we study the probability distribution $P_n$ of the (scaled) remainder $T^n(X)=\sum_{k=n+1}^\infty X_k q^{n-k}$: If $X$ has an absolutely continuous CDF then $P_n$ converges in the total variation metric to the Lebesgue measure $μ$ on the unit interval. Under weak smoothness conditions we establish first a coupling between $X$ and a non-negative integer valued random variable $N$ so that $T^N(X)$ follows $μ$ and is independent of $(X_1,...,X_N)$, and second exponentially fast convergence of $P_n$ and its PDF $f_n$. We discuss how many digits are needed and show examples of our results. The convergence results are extended to the case of a multivariate random variable defined on a unit cube.

math.PR↗
arXiv2023

A functional central limit theorem for the K-function with an estimated intensity function

The $K$-function is arguably the most important functional summary statistic for spatial point processes. It is used extensively for goodness-of-fit testing and in connection with minimum contrast estimation for parametric spatial point process models. It is thus pertinent to understand the asymptotic properties of estimates of the $K$-function. In this paper we derive the functional asymptotic distribution for the $K$-function estimator. Contrary to previous papers on functional convergence we consider the case of an inhomogeneous intensity function. We moreover handle the fact that practical $K$-function estimators rely on plugging in an estimate of the intensity function. This removes two serious limitations of the existing literature.

math.ST↗
arXiv2021

A $K$-function for inhomogeneous random measures with geometric features

This paper introduces a $K$-function for assessing second-order properties of inhomogeneous random measures generated by marked point processes. The marks can be geometric objects like fibers or sets of positive volume, and the presented $K$-function takes into account geometric features of the marks, such as tangent directions of fibers. The $K$-function requires an estimate of the inhomogeneous density function of the random measure. We introduce parametric estimates for the density function based on parametric models that represent large scale features of the inhomogeneous random measure. The proposed methodology is applied to simulated fiber patterns as well as a three-dimensional data set of steel fibers in concrete.

stat.ME↗
arXiv2021

A functional central limit theorem for the empirical Ripley's K-function

We establish a functional central limit theorem for Ripley's K-function for two classes of point processes. One is the class of point processes having exponential decay of correlations and further satisfying a conditional m-dependence condition. The other is a family of Gibbs point processes. We illustrate the use of our theorem for goodness-of-fit tests in simulations.

math.ST↗
arXiv2021

Currents and K-functions for Fiber Point Processes

Analysis of images of sets of fibers such as myelin sheaths or skeletal muscles must account for both the spatial distribution of fibers and differences in fiber shape. This necessitates a combination of point process and shape analysis methodology. In this paper, we develop a K-function for shape-valued point processes by embedding shapes as currents, thus equipping the point process domain with metric structure inherited from a reproducing kernel Hilbert space. We extend Ripley's K-function which measures deviations from spatial homogeneity of point processes to fiber data. The paper provides a theoretical account of the statistical foundation of the K-function and its extension to fiber data, and we test the developed K-function on simulated as well as real data sets. This includes a fiber data set consisting of myelin sheaths, visualizing the spatial and fiber shape behavior of myelin configurations at different debts.

stat.ME↗
arXiv2020

Measuring shape relations using r-parallel sets

Geometrical measurements of biological objects form the basis of many quantitative analyses. Hausdorff measures such as the volume and the area of objects are simple and popular descriptors of individual objects, however, for most biological processes, the interaction between objects cannot be ignored, and the shape and function of neighboring objects are mutually influential. In this paper, we present a theory on the geometrical interaction between objects based on the theory of spatial point processes. Our theory is based on the relation between two objects: a reference and an observed object. We generate the $r$-parallel sets of the reference object, we calculate the intersection between the $r$-parallel sets and the observed object, and we define measures on these intersections. Our measures are simple like the volume and area of an object, but describe further details about the shape of individual objects and their pairwise geometrical relation. Finally, we propose a summary statistics for collections of shapes and their interaction. We evaluate these measures on a publicly available FIB-SEM 3D data set of an adult rodent.

cs.CV↗
arXiv2019

Testing goodness of fit for point processes via topological data analysis

We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

math.ST↗
arXiv2017

Voronoi-based estimation of Minkowski tensors from finite point samples

Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows to approximate these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying object is a set of positive reach. The underlying algorithm is based on a simple expression in terms of the cells of a Voronoi decomposition associated with the image.

math.MG↗
arXiv2016

Rotational Crofton formulae for Minkowski tensors and some affine counterparts

Motivated by applications in local stereology, a new rotational Crofton formula is derived for Minkowski tensors. For sets of positive reach, the formula shows how rotational averages of intrinsically defined Minkowski tensors on sections passing through the origin are related to the geometry of the sectioned set. In particular, for Minkowski tensors of order j-1 on j-dimensional linear subspaces, we derive an explicit formula for the rotational average involving hypergeometric functions. Sectioning with lines and hyperplanes through the origin is considered in detail. We also study the case where the sections are not restricted to pass through the origin. For sets of positive reach, we here obtain a Crofton formula for the integral mean of intrinsically defined Minkowski tensors on j-dimensional affine subspaces.

math.CA↗
arXiv2016

Modelling Anisotropic Covariance using Stochastic Development and Sub-Riemannian Frame Bundle Geometry

We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hormander condition is satisfied such that the Brownian motions have smooth transition densities. We identify the most probable paths for the underlying Euclidean Brownian motion and discuss small time asymptotics of the transition densities on the manifold. The geometric setup yields an intrinsic approach to the estimation of mean and covariance in non-linear spaces.

math.DG↗
arXiv2016

On functions of bounded variation

The recently introduced concept of $\mathcal{D}$-variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from Pausinger \& Svane (J. Complexity, 2014) whether every function of bounded Hardy--Krause variation is Borel measurable and has bounded $\mathcal{D}$-variation. Moreover, we show that the space of functions of bounded $\mathcal{D}$-variation can be turned into a commutative Banach algebra.

math.CA↗
arXiv2015

Local digital algorithms applied to Boolean models

We investigate the estimation of specific intrinsic volumes of stationary Boolean models by local digital algorithms; that is, by weighted sums of $n \times\ldots \times n$ configuration counts. We show that asymptotically unbiased estimators for the specific surface area or integrated mean curvature do not exist if the dimension is at least two or three, respectively. For 3-dimensional stationary, isotropic Boolean models, we derive asymptotically unbiased estimators for the specific surface area and integrated mean curvature. For a Boolean model with balls as grains we even obtain an asymptotically unbiased estimator for the specific Euler characteristic.

math.PR↗
arXiv2014

Asymptotic variance of grey-scale surface area estimators

Grey-scale local algorithms have been suggested as a fast way of estimating surface area from grey-scale digital images. Their asymptotic mean has already been described. In this paper, the asymptotic behaviour of the variance is studied in isotropic and sufficiently smooth settings, resulting in a general asymptotic bound. For compact convex sets with nowhere vanishing Gaussian curvature, the asymptotics can be described more explicitly. As in the case of volume estimators, the variance is decomposed into a lattice sum and an oscillating term of at most the same magnitude.

math.PR↗