arXiv subjects

Annika Betken

Publications and source records attributed to Annika Betken.

At least 19 recordsLinked to original sources

arXiv2026

Central limit theorems for the outputs of fully convolutional neural networks with time series input

Deep learning is widely deployed for time series learning tasks such as classification and forecasting. Despite the empirical successes, only little theory has been developed so far in the time series context. In this work, we prove that if the network inputs are generated from short-range dependent linear processes, the outputs of fully convolutional neural networks (FCNs) with global average pooling (GAP) are asymptotically Gaussian and the limit is attained if the length of the observed time series tends to infinity. The proof leverages existing tools from the theoretical time series literature. Based on our theory, we propose a generalization of the GAP layer by considering a global weighted pooling step with slowly varying, learnable coefficients.

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arXiv2026

Test for independence of long-range dependent time series using distance covariance

We apply the concept of distance covariance for testing independence of two long-range dependent time series. As test statistic we propose a linear combination of empirical distance cross-covariances. We derive the asymptotic distribution of the test statistic, and we show consistency against a very general class of alternatives. The asymptotic theory developed in this paper is based on a novel non-central limit theorem for stochastic processes with values in an $L^2$-Hilbert space. This limit theorem is of general theoretical interest which goes beyond the context of this article. Subject to the dependence in the data, the standardization and the limit distributions of the proposed test statistic vary. Since the limit distributions are unknown, we propose a subsampling procedure to determine the critical values for the proposed test, and we provide a proof for the validity of subsampling. In a simulation study, we investigate the finite-sample behavior of our test, and we compare its performance to tests based on the empirical cross-covariances. As an application of our results we analyze the cross-dependencies between mean monthly discharges of three rivers.

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arXiv2026

Symmetry Testing in Time Series using Ordinal Patterns: A U-Statistic Approach

We introduce a general framework for testing temporal symmetries in time series based on the distribution of ordinal patterns. While previous approaches have focused on specific forms of asymmetry, such as time reversal, our method provides a unified framework applicable to arbitrary symmetry tests. We establish asymptotic results for the resulting test statistics under a broad class of stationary processes. Comprehensive experiments on both synthetic and real data demonstrate that the proposed test achieves high sensitivity to structural asymmetries while remaining fully data-driven and computationally efficient.

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arXiv2025

Extending Characterizations of Multivariate Laws via Distance Distributions

We extend a theorem of Maa, Pearl, and Bartoszynski, which links equality of interpoint distance distributions to equality of underlying multivariate distributions, beyond the restrictive class of homogeneous, translation-invariant distance functions. Our approach replaces geometric assumptions on the distance with analytic conditions: volume-regularity of distance-induced balls, Lebesgue differentiability with respect to the distance, and bounded centered oscillations of densities. Under these conditions, equality of interpoint distance distributions continues to imply equality of the generating laws. The result persists under monotone continuous transformations of homogeneous, translation-invariant distances, recovering the original statement, and it extends to compact Riemannian manifolds equipped with the geodesic metric. We further develop a quantitative version of the theorem, i.e., inequalities that connect discrepancies of interpoint distance distributions to the $L^2$-distance between densities, and obtain explicit rates under Ahlfors $α$-regularity of the distance function and $β$-Hölder continuity of densities, capturing dependence on dimensionality. Several representative examples illustrate the applicability of the generalization to domain-specific distances used in modern statistics. The examples include non-homogeneous non-translation invariant distances such as Canberra, entropic distances, and the Bray--Curtis dissimilarity.

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arXiv2025

Higher-order approximation for uncertainty quantification in time series analysis

For time series with high temporal correlation, the empirical process converges rather slowly to its limiting distribution. Many statistics in change-point analysis, goodness-of-fit testing and uncertainty quantification admit a representation as functionals of the empirical process and therefore inherit its slow convergence. As a result, inference based on the asymptotic distribution of those quantities is significantly affected by relatively small sample sizes. We assess the quality of higher-order approximations of the empirical process by deriving the asymptotic distribution of the corresponding error terms. Based on the limiting distribution of the higher-order terms, we propose a novel approach to calculate confidence intervals for statistical quantities such as the median. In a simulation study, we compare coverage rates and lengths of these confidence intervals with those based on the asymptotic distribution of the empirical process and highlight some benefits of higher-order approximations of the empirical process.

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arXiv2025

Depth Patterns and their Applications in Animal Tracking

We establish a definition of ordinal patterns for multivariate data sets based on the concept of Tukey's halfspace depth. Given the definition of these \emph{depth patterns}, we are interested in the probabilities of observing specific patterns in time series. For this, we consider the relative frequency of depth patterns as natural estimators for their occurrence probabilities. Depending on the choice of reference distribution and the relation between reference and data distribution, we distinguish different settings that are considered separately. Within these settings we study statistical properties of depth pattern probabilities, establishing consistency and asymptotic normality under the assumption of weakly dependent time series. Since our concept only depends on ordinal depth information, the resulting values are robust under small perturbations and measurement errors. We emphasize the applicability of our method by analyzing the depth patterns which are found in seal pubs' movement. We use our approach in order to choose an appropriate model out of a range of two-dimensional random walks, which are commonly used in mathematical biology.

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arXiv2025

On the Robustness of Kernel Ridge Regression Using the Cauchy Loss Function

Robust regression aims to develop methods for estimating an unknown regression function in the presence of outliers, heavy-tailed distributions, or contaminated data, which can severely impact performance. Most existing theoretical results in robust regression assume that the noise has a finite absolute mean, an assumption violated by certain distributions, such as Cauchy and some Pareto noise. In this paper, we introduce a generalized Cauchy noise framework that accommodates all noise distributions with finite moments of any order, even when the absolute mean is infinite. Within this framework, we study the \textit{kernel Cauchy ridge regressor} (\textit{KCRR}), which minimizes a regularized empirical Cauchy risk to achieve robustness. To derive the $L_2$-risk bound for KCRR, we establish a connection between the excess Cauchy risk and $L_2$-risk for sufficiently large scale parameters of the Cauchy loss, which reveals that these two risks are equivalent. Furthermore, under the assumption that the regression function satisfies Hölder smoothness, we derive excess Cauchy risk bounds for KCRR, showing improved performance as the scale parameter decreases. By considering the twofold effect of the scale parameter on the excess Cauchy risk and its equivalence with the $L_2$-risk, we establish the almost minimax-optimal convergence rate for KCRR in terms of $L_2$-risk, highlighting the robustness of the Cauchy loss in handling various types of noise. Finally, we validate the effectiveness of KCRR through experiments on both synthetic and real-world datasets under diverse noise corruption scenarios.

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arXiv2025

Ordinal Patterns Based Change Points Detection

The ordinal patterns of a fixed number of consecutive values in a time series is the spatial ordering of these values. Counting how often a specific ordinal pattern occurs in a time series provides important insights into the properties of the time series. In this work, we prove the asymptotic normality of the relative frequency of ordinal patterns for time series with linear increments. Moreover, we apply ordinal patterns to detect changes in the distribution of a time series.

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arXiv2025

Median of Forests for Robust Density Estimation

Robust density estimation refers to the consistent estimation of the density function even when the data is contaminated by outliers. We find that existing forest density estimation at a certain point is inherently resistant to the outliers outside the cells containing the point, which we call \textit{non-local outliers}, but not resistant to the rest \textit{local outliers}. To achieve robustness against all outliers, we propose an ensemble learning algorithm called \textit{medians of forests for robust density estimation} (\textit{MFRDE}), which adopts a pointwise median operation on forest density estimators fitted on subsampled datasets. Compared to existing robust kernel-based methods, MFRDE enables us to choose larger subsampling sizes, sacrificing less accuracy for density estimation while achieving robustness. On the theoretical side, we introduce the local outlier exponent to quantify the number of local outliers. Under this exponent, we show that even if the number of outliers reaches a certain polynomial order in the sample size, MFRDE is able to achieve almost the same convergence rate as the same algorithm on uncontaminated data, whereas robust kernel-based methods fail. On the practical side, real data experiments show that MFRDE outperforms existing robust kernel-based methods. Moreover, we apply MFRDE to anomaly detection to showcase a further application.

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arXiv2024

Differentially Private Algorithms for Linear Queries via Stochastic Convex Optimization

This article establishes a method to answer a finite set of linear queries on a given dataset while ensuring differential privacy. To achieve this, we formulate the corresponding task as a saddle-point problem, i.e. an optimization problem whose solution corresponds to a distribution minimizing the difference between answers to the linear queries based on the true distribution and answers from a differentially private distribution. Against this background, we establish two new algorithms for corresponding differentially private data release: the first is based on the differentially private Frank-Wolfe method, the second combines randomized smoothing with stochastic convex optimization techniques for a solution to the saddle-point problem. While previous works assess the accuracy of differentially private algorithms with reference to the empirical data distribution, a key contribution of our work is a more natural evaluation of the proposed algorithms' accuracy with reference to the true data-generating distribution.

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arXiv2024

A two-sample test based on averaged Wilcoxon rank sums over interpoint distances

An important class of two-sample multivariate homogeneity tests is based on identifying differences between the distributions of interpoint distances. While generating distances from point clouds offers a straightforward and intuitive way for dimensionality reduction, it also introduces dependencies to the resulting distance samples. We propose a simple test based on Wilcoxon's rank sum statistic for which we prove asymptotic normality under the null hypothesis and fixed alternatives under mild conditions on the underlying distributions of the point clouds. Furthermore, we show consistency of the test and derive a variance approximation that allows to construct a computationally feasible, distribution-free test with good finite sample performance. The power and robustness of the test for high-dimensional data and low sample sizes is demonstrated by numerical simulations. Finally, we apply the proposed test to case-control testing on microarray data in genetic studies, which is considered a notorious case for a high number of variables and low sample sizes.

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arXiv2024

A Bootstrap Test for Independence of Time Series Based on the Distance Covariance

We present a test for independence of two strictly stationary time series based on a bootstrap procedure for the distance covariance. Our test detects any kind of dependence between the two time series within an arbitrary maximum lag $L$. In simulation studies, our test outperforms alternative testing procedures. In proving the validity of the underlying bootstrap procedure, we generalise bounds for the Wasserstein distance between an empirical measure and its marginal distribution under the assumption of $α$-mixing. Previous results of this kind only existed for i.i.d. processes.

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arXiv2023

Class Probability Matching Using Kernel Methods for Label Shift Adaptation

In domain adaptation, covariate shift and label shift problems are two distinct and complementary tasks. In covariate shift adaptation where the differences in data distribution arise from variations in feature probabilities, existing approaches naturally address this problem based on \textit{feature probability matching} (\textit{FPM}). However, for label shift adaptation where the differences in data distribution stem solely from variations in class probability, current methods still use FPM on the $d$-dimensional feature space to estimate the class probability ratio on the one-dimensional label space. To address label shift adaptation more naturally and effectively, inspired by a new representation of the source domain's class probability, we propose a new framework called \textit{class probability matching} (\textit{CPM}) which matches two class probability functions on the one-dimensional label space to estimate the class probability ratio, fundamentally different from FPM operating on the $d$-dimensional feature space. Furthermore, by incorporating the kernel logistic regression into the CPM framework to estimate the conditional probability, we propose an algorithm called \textit{class probability matching using kernel methods} (\textit{CPMKM}) for label shift adaptation. From the theoretical perspective, we establish the optimal convergence rates of CPMKM with respect to the cross-entropy loss for multi-class label shift adaptation. From the experimental perspective, comparisons on real datasets demonstrate that CPMKM outperforms existing FPM-based and maximum-likelihood-based algorithms.

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arXiv2021

Ordinal pattern dependence as a multivariate dependence measure

In this article, we show that the recently introduced ordinal pattern dependence fits into the axiomatic framework of general multivariate dependence measures, i.e., measures of dependence between two multivariate random objects. Furthermore, we consider multivariate generalizations of established univariate dependence measures like Kendall's $τ$, Spearman's $ρ$ and Pearson's correlation coefficient. Among these, only multivariate Kendall's $τ$ proves to take the dynamical dependence of random vectors stemming from multidimensional time series into account. Consequently, the article focuses on a comparison of ordinal pattern dependence and multivariate Kendall's $τ$ in this context. To this end, limit theorems for multivariate Kendall's $τ$ are established under the assumption of near-epoch dependent data-generating time series. We analyze how ordinal pattern dependence compares to multivariate Kendall's $τ$ and Pearson's correlation coefficient on theoretical grounds. Additionally, a simulation study illustrates differences in the kind of dependencies that are revealed by multivariate Kendall's $τ$ and ordinal pattern dependence.

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arXiv2020

Rank-based change-point analysis for long-range dependent time series

We consider change-point tests based on rank statistics to test for structural changes in long-range dependent observations. Under the hypothesis of stationary time series and under the assumption of a change with decreasing change-point height, the asymptotic distributions of corresponding test statistics are derived. For this, a uniform reduction principle for the sequential empirical process in a two-parameter Skorohod space equipped with a weighted supremum norm is proved. Moreover, we compare the efficiency of rank tests resulting from the consideration of different score functions. Under Gaussianity, the asymptotic relative efficiency of rank-based tests with respect to the CuSum test is 1, irrespective of the score function. Regarding the practical implementation of rank-based change-point tests, we suggest to combine self-normalized rank statistics with subsampling. The theoretical results are accompanied by simulation studies that, in particular, allow for a comparison of rank tests resulting from different score functions. With respect to the finite sample performance of rank-based change-point tests, the Van der Waerden rank test proves to be favorable in a broad range of situations. Finally, we analyze data sets from economy, hydrology, and network traffic monitoring in view of structural changes and compare our results to previous analysis of the data.

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arXiv2020

Change-point tests for the tail parameter of Long Memory Stochastic Volatility time series

We consider a change-point test based on the Hill estimator to test for structural changes in the tail index of Long Memory Stochastic Volatility time series. In order to determine the asymptotic distribution of the corresponding test statistic, we prove a uniform reduction principle for the tail empirical process in a two-parameter Skorohod space. It is shown that such a process displays a dichotomous behavior according to an interplay between the Hurst parameter, i.e., a parameter characterizing the dependence in the data, and the tail index. Our theoretical results are accompanied by simulation studies and the analysis of financial time series with regard to structural changes in the tail index.

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arXiv2020

Ordinal Patterns in Long-Range Dependent Time Series

We analyze the ordinal structure of long-range dependent time series. To this end, we use so called ordinal patterns which describe the relative position of consecutive data points. We provide two estimators for the probabilities of ordinal patterns and prove limit theorems in different settings, namely stationarity and (less restrictive) stationary increments. In the second setting, we encounter a Rosenblatt distribution in the limit. We prove more general limit theorems for functions with Hermite rank 1 and 2. We derive the limit distribution for an estimation of the Hurst parameter $H$ if it is higher than 3/4. Thus, our theorems complement results for lower values of $H$ which can be found in the literature. Finally, we provide some simulations that illustrate our theoretical results.

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arXiv2017

Testing for Change in Stochastic Volatility with Long Range Dependence

In this paper, change-point problems for long memory stochastic volatility models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM- and Wilcoxon-type test statistics is derived. In this context, a limit theorem for the two-parameter empirical process of long memory stochastic volatility time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typically influenced by long range dependence. To avoid the estimation of nuisance parameters in applications, the usage of self-normalized test statistics is proposed. The theoretical results are accompanied by simulation studies which characterize the finite sample behavior of the considered testing procedures when testing for changes in mean, in variance, and in the tail index.

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