arXiv subjects

Johannes Krebs

Publications and source records attributed to Johannes Krebs.

12 recordsLinked to original sources

arXiv2026

On the stability of filtration functions for dependent data with applications to break detection

In this paper, we study the stability of commonly used filtration functions in topological data analysis under small perturbations of the underlying nonrandom point cloud. Relying on these stability results, we then develop a test procedure to detect and determine structural breaks in a sequence of topological data objects obtained from weakly dependent data. The proposed method applies for instance to statistics of persistence diagrams of $\mathbb{R}^d$-valued Bernoulli shift systems under the Čech or Vietoris-Rips filtration.

math.ST↗

arXiv2024

On the asymptotic normality of persistent Betti numbers

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (β^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[β^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been established in different settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime, see Yogeshwaran et al. [2017] and Hiraoka et al. [2018]. In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich [2001] of persistent Betti numbers and generalize the existing results on the asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that the multivariate asymptotic normality holds for all pairs $(r,s)$, $0\le r\le s<\infty$, and that it is not affected by percolation effects in the underlying random geometric graph.

math.PR↗

arXiv2024

Two-sample tests for relevant differences in persistence diagrams

We study two-sample tests for relevant differences in persistence diagrams obtained from $L^p$-$m$-approximable data $(\mathcal{X}_t)_t$ and $(\mathcal{Y}_t)_t$. To this end, we compare variance estimates w.r.t.\ the Wasserstein metrics on the space of persistence diagrams. In detail, we consider two test procedures. The first compares the Fr{é}chet variances of the two samples based on estimators for the Fr{é}chet mean of the observed persistence diagrams $PD(\mathcal{X}_i)$ ($1\le i\le m$), resp., $PD(\mathcal{Y}_j)$ ($1\le j\le n$) of a given feature dimension. We use classical functional central limit theorems to establish consistency of the testing procedure. The second procedure relies on a comparison of the so-called independent copy variances of the respective samples. Technically, this leads to functional central limit theorems for U-statistics built on $L^p$-$m$-approximable sample data.

math.ST↗

arXiv2023

On the Bahadur representation of sample quantiles for score functionals

We establish the Bahadur representation of sample quantiles for stabilizing score functionals in stochastic geometry and study local fluctuations of the corresponding empirical distribution function. The scores are obtained from a Poisson process. We apply the results to trimmed and Winsorized means of the score functionals and establish a law of the iterated logarithm for the sample quantiles of the scores.

math.ST↗

arXiv2022

Persistent homology based goodness-of-fit tests for spatial tessellations

Motivated by the rapidly increasing relevance of virtual material design in the domain of materials science, it has become essential to assess whether topological properties of stochastic models for a spatial tessellation are in accordance with a given dataset. Recently, tools from topological data analysis such as the persistence diagram have allowed to reach profound insights in a variety of application contexts. In this work, we establish the asymptotic normality of a variety of test statistics derived from a tessellation-adapted refinement of the persistence diagram. Since in applications, it is common to work with tessellation data subject to interactions, we establish our main results for Voronoi and Laguerre tessellations whose generators form a Gibbs point process. We elucidate how these conceptual results can be used to derive goodness of fit tests, and then investigate their power in a simulation study. Finally, we apply our testing methodology to a tessellation describing real foam data.

math.ST↗

arXiv2022

Statistical inference for intrinsic wavelet estimators of SPD matrices in a log-Euclidean manifold

In this paper we treat statistical inference for an intrinsic wavelet estimator of curves of symmetric positive definite (SPD) matrices in a log-Euclidean manifold. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our high-level wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes.

stat.ME↗

arXiv2021

On approximation theorems for the Euler characteristic with applications to the bootstrap

We study approximation theorems for the Euler characteristic of the Vietoris-Rips and Cech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance.

math.PR↗

arXiv2021

Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics

The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. The present work relates to other general results in the area of stabilizing statistics, including central limit theorems for functionals of Poisson and Binomial processes in the critical regime. Specific statistics considered include the persistent Betti numbers of Čech and Vietoris-Rips complexes over point sets in $\mathbb R^d$, along with Euler characteristics, and the total edge length of the $k$-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS). Source code is available at github.com/btroycraft/stabilizing_statistics_bootstrap.

math.ST↗

arXiv2021

On limit theorems for persistent Betti numbers from dependent data

We study persistent Betti numbers and persistence diagrams obtained from time series and random fields. It is well known that the persistent Betti function is an efficient descriptor of the topology of a point cloud. So far, convergence results for the $(r,s)$-persistent Betti number of the $q$th homology group, $β^{r,s}_q$, were mainly considered for finite-dimensional point cloud data obtained from i.i.d. observations or stationary point processes such as a Poisson process. In this article, we extend these considerations. We derive limit theorems for the pointwise convergence of persistent Betti numbers $β^{r,s}_q$ in the critical regime under quite general dependence settings.

math.PR↗

arXiv2021

Functional central limit theorems for persistent Betti numbers on cylindrical networks

We study functional central limit theorems for persistent Betti numbers obtained from networks defined on a Poisson point process. The limit is formed in large volumes of cylindrical shape stretching only in one dimension. The results cover a directed sublevel-filtration for stabilizing networks and the Cech and Vietoris-Rips complex on the random geometric graph. The presented functional central limit theorems open the door to a variety of statistical applications in topological data analysis and we consider goodness-of-fit tests in a simulation study.

math.ST↗

arXiv2020

On the law of the iterated logarithm and strong invariance principles in stochastic geometry

We study the law of the iterated logarithm (Khinchin (1924), Kolmogorov (1929)) and related strong invariance principles in stochastic geometry. As potential applications, we think of well-known functionals such as functionals defined on the $k$-nearest neighbors graph and important functionals in topological data analysis such as the Euler characteristic and persistent Betti numbers.

math.PR↗

arXiv2019

Quantifying Life Insurance Risk using Least-Squares Monte Carlo

This article presents a stochastic framework to quantify the biometric risk of an insurance portfolio in solvency regimes such as Solvency II or the Swiss Solvency Test (SST). The main difficulty in this context constitutes in the proper representation of long term risks in the profit-loss distribution over a one year horizon. This will be resolved by using least-squares Monte Carlo methods to quantify the impact of new experience on the annual re-valuation of the portfolio. Therefore our stochastic model can be seen as an example for an internal model, as allowed under Solvency II or the SST. Since our model does not rely upon nested simulations it is computationally fast and easy to implement.

q-fin.RM↗