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Claudia Kirch

Publications and source records attributed to Claudia Kirch.

At least 19 recordsLinked to original sources
arXiv2026

Fully Functional Weighted Testing for Abrupt and Gradual Location Changes in Functional Time Series

Change point tests for abrupt changes in the mean of functional data, i.e., random elements in infinite-dimensional Hilbert spaces, are either based on dimension reduction techniques, e.g., based on principal components, or directly based on a functional CUSUM (cumulative sum) statistic. The former have often been criticized as not being fully functional and losing too much information. On the other hand, unlike the latter, they take the covariance structure of the data into account by weighting the CUSUM statistics obtained after dimension reduction with the inverse covariance matrix. In this paper, as a middle ground between these two approaches, we propose an alternative statistic that includes the covariance structure with an offset parameter to produce a scale-invariant test procedure and to increase power when the change is not aligned with the first components. We obtain the asymptotic distribution under the null hypothesis for this new test statistic, allowing for time dependence of the data. Furthermore, we introduce versions of all three test statistics for gradual change situations, which have not been previously considered for functional data, and derive their limit distribution. Further results shed light on the asymptotic power behavior for all test statistics under various ground truths for the alternatives.

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arXiv2025

Detection of mean changes in partially observed functional data

We propose a test for a change in the mean for a sequence of functional observations that are only partially observed on subsets of the domain, with no information available on the complement. The framework accommodates important scenarios, including both abrupt and gradual changes. The significance of the test statistic is assessed via a permutation test. In addition to the classical permutation approach with a fixed number of permutation samples, we also discuss a variant with controlled resampling risk that relies on a random (data-driven) number of permutation samples. The small sample performance of the proposed methodology is illustrated in a Monte Carlo simulation study and an application to real data.

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arXiv2025

seMCD: Sequentially implemented Monte Carlo depth computation with statistical guarantees

Statistical depth functions provide center-outward orderings in spaces of dimension larger than one, where a natural ordering does not exist. The numerical evaluation of such depth functions can be computationally prohibitive, even for relatively low dimensions. We present a novel sequentially implemented Monte Carlo methodology for the computation of, theoretical and empirical, depth functions and related quantities (seMCD), that outputs an interval, a so-called seMCD-bucket, to which the quantity of interest belongs with a high probability prespecified by the user. For specific classes of depth functions, we adapt algorithms from sequential testing, providing finite-sample guarantees. For depth functions dependent on unknown distributions, we offer asymptotic guarantees using non-parametric statistical methods. In contrast to plain-vanilla Monte Carlo methodology the number of samples required in the algorithm is random but typically much smaller than standard choices suggested in the literature. The seMCD method can be applied to various depth functions, covering multivariate and functional spaces. We demonstrate the efficiency and reliability of our approach through empirical studies, highlighting its applicability in outlier or anomaly detection, classification, and depth region computation. In conclusion, the seMCD-algorithm can achieve accurate depth approximations with few Monte Carlo samples while maintaining rigorous statistical guarantees.

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arXiv2025

A new and flexible class of sharp asymptotic time-uniform confidence sequences

Confidence sequences are anytime-valid analogues of classical confidence intervals that do not suffer from multiplicity issues under optional continuation of the data collection. As in classical statistics, asymptotic confidence sequences are a nonparametric tool showing under which high-level assumptions asymptotic coverage is achieved so that they also give a certain robustness guarantee against distributional deviations. In this paper, we propose a new flexible class of confidence sequences yielding sharp asymptotic time-uniform confidence sequences under mild assumptions. Furthermore, we highlight the connection to corresponding sequential testing problems and detail the underlying limit theorem.

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arXiv2025

Estimation of Change Points for Non-linear (auto-)regressive processes using Neural Network Functions

In this paper, we propose a new test for the detection of a change in a non-linear (auto-)regressive time series as well as a corresponding estimator for the unknown time point of the change. To this end, we consider an at-most-one-change model and approximate the unknown (auto-)regression function by a neuronal network with one hidden layer. It is shown that the test has asymptotic power one for a wide range of alternatives not restricted to changes in the mean of the time series. Furthermore, we prove that the corresponding estimator converges to the true change point with the optimal rate OP (1/n) and derive the asymptotic distribution. Some simulations illustrate the behavior of the estimator with a special focus on the misspecified case, where the true regression function is not given by a neuronal network. Finally, we apply the estimator to some financial data.

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arXiv2024

Asymptotic considerations in a Bayesian linear model with nonparametrically modelled time series innovations

This paper considers a semiparametric approach within the general Bayesian linear model where the innovations consist of a stationary, mean zero Gaussian time series. While a parametric prior is specified for the linear model coefficients, the autocovariance structure of the time series is modeled nonparametrically using a Bernstein-Gamma process prior for the spectral density function, the Fourier transform of the autocovariance function. When updating this joint prior with Whittle's likelihood, a Bernstein-von-Mises result is established for the linear model coefficients showing the asymptotic equivalence of the corresponding estimators to those obtained from frequentist pseudo-maximum-likelihood estimation under the Whittle likelihood. Local asymptotic normality of the likelihood is shown, demonstrating that the marginal posterior distribution of the linear model coefficients shrinks at parametric rate towards the true value, and that the conditional posterior distribution of the spectral density contracts in the sup-norm, even in the case of a partially misspecified linear model.

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arXiv2024

Symmetrisation of a class of two-sample tests by mutually considering depth ranks including functional spaces

Statistical depth functions provide measures of the outlyingness, or centrality, of the elements of a space with respect to a distribution. It is a nonparametric concept applicable to spaces of any dimension, for instance, multivariate and functional. Liu and Singh (1993) presented a multivariate two-sample test based on depth-ranks. We dedicate this paper to improving the power of the associated test statistic and incorporating its applicability to functional data. In doing so, we obtain a more natural test statistic that is symmetric in both samples. We derive the null asymptotic of the proposed test statistic, also proving the validity of the testing procedure for functional data. Finally, the finite sample performance of the test for functional data is illustrated by means of a simulation study and a real data analysis on annual temperature curves of ocean drifters is executed.

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arXiv2023

Scan statistics for the detection of anomalies in M-dependent random fields with applications to image data

Anomaly detection in random fields is an important problem in many applications including the detection of cancerous cells in medicine, obstacles in autonomous driving and cracks in the construction material of buildings. Such anomalies are often visible as areas with different expected values compared to the background noise. Scan statistics based on local means have the potential to detect such local anomalies by enhancing relevant features. We derive limit theorems for a general class of such statistics over M-dependent random fields of arbitrary but fixed dimension. By allowing for a variety of combinations and contrasts of sample means over differently-shaped local windows, this yields a flexible class of scan statistics that can be tailored to the particular application of interest. The latter is demonstrated for crack detection in 2D-images of different types of concrete. Together with a simulation study this indicates the potential of the proposed methodology for the detection of anomalies in a variety of situations.

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arXiv2023

A nonparametrically corrected likelihood for Bayesian spectral analysis of multivariate time series

This paper presents a novel approach to Bayesian nonparametric spectral analysis of stationary multivariate time series. Starting with a parametric vector-autoregressive model, the parametric likelihood is nonparametrically adjusted in the frequency domain to account for potential deviations from parametric assumptions. We show mutual contiguity of the nonparametrically corrected likelihood, the multivariate Whittle likelihood approximation and the exact likelihood for Gaussian time series. A multivariate extension of the nonparametric Bernstein-Dirichlet process prior for univariate spectral densities to the space of Hermitian positive definite spectral density matrices is specified directly on the correction matrices. An infinite series representation of this prior is then used to develop a Markov chain Monte Carlo algorithm to sample from the posterior distribution. The code is made publicly available for ease of use and reproducibility. With this novel approach we provide a generalization of the multivariate Whittle-likelihood-based method of Meier et al. (2020) as well as an extension of the nonparametrically corrected likelihood for univariate stationary time series of Kirch et al. (2019) to the multivariate case. We demonstrate that the nonparametrically corrected likelihood combines the efficiencies of a parametric with the robustness of a nonparametric model. Its numerical accuracy is illustrated in a comprehensive simulation study. We illustrate its practical advantages by a spectral analysis of two environmental time series data sets: a bivariate time series of the Southern Oscillation Index and fish recruitment and time series of windspeed data at six locations in California.

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arXiv2023

Bayesian nonparametric spectral analysis of locally stationary processes

Based on a novel dynamic Whittle likelihood approximation for locally stationary processes, a Bayesian nonparametric approach to estimating the time-varying spectral density is proposed. This dynamic frequency-domain based likelihood approximation is able to depict the time-frequency evolution of the process by utilizing the moving periodogram previously introduced in the bootstrap literature. The posterior distribution is obtained by updating a bivariate extension of the Bernstein-Dirichlet process prior with the dynamic Whittle likelihood. Asymptotic properties such as sup-norm posterior consistency and L2-norm posterior contraction rates are presented. Additionally, this methodology enables model selection between stationarity and non-stationarity based on the Bayes factor. The finite-sample performance of the method is investigated in simulation studies and applications to real-life data-sets are presented.

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arXiv2022

Data Segmentation for Time Series Based on a General Moving Sum Approach

In this paper we propose new methodology for the data segmentation, also known as multiple change point problem, in a general framework including classic mean change scenarios, changes in linear regression but also changes in the time series structure such as in the parameters of Poisson-autoregressive time series. In particular, we derive a general theory based on estimating equations proving consistency for the number of change points as well as rates of convergence for the estimators of the locations of the change points. More precisely, two different types of MOSUM (moving sum) statistics are considered: A MOSUM-Wald statistic based on differences of local estimators and a MOSUM-score statistic based on a global estimator. The latter is usually computationally less involved in particular in non-linear problems where no closed form of the estimator is known such that numerical methods are required. Finally, we evaluate the methodology by means of simulated data as well as using some geophysical well-log data.

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arXiv2022

Bootstrap confidence intervals for multiple change points based on moving sum procedures

The problem of quantifying uncertainty about the locations of multiple change points by means of confidence intervals is addressed. The asymptotic distribution of the change point estimators obtained as the local maximisers of moving sum statistics is derived, where the limit distributions differ depending on whether the corresponding size of changes is local, i.e. tends to zero as the sample size increases, or fixed. A bootstrap procedure for confidence interval generation is proposed which adapts to the unknown magnitude of changes and guarantees asymptotic validity both for local and fixed changes. Simulation studies show good performance of the proposed bootstrap procedure, and some discussions about how it can be extended to serially dependent errors is provided.

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arXiv2021

Posterior consistency for the spectral density of non-Gaussian stationary time series

Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation has been proposed in Kirch et al. who prove posterior consistency for spectral density estimation in combination with the Bernstein-Dirichlet process prior for Gaussian time series. In this paper, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem of Shalizi for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood as proposed by Choudhuri, Ghosal and Roy is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.

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arXiv2021

Data segmentation algorithms: Univariate mean change and beyond

Data segmentation a.k.a. multiple change point analysis has received considerable attention due to its importance in time series analysis and signal processing, with applications in a variety of fields including natural and social sciences, medicine, engineering and finance. In the first part of this survey, we review the existing literature on the canonical data segmentation problem which aims at detecting and localising multiple change points in the mean of univariate time series. We provide an overview of popular methodologies on their computational complexity and theoretical properties. In particular, our theoretical discussion focuses on the separation rate relating to which change points are detectable by a given procedure, and the localisation rate quantifying the precision of corresponding change point estimators, and we distinguish between whether a homogeneous or multiscale viewpoint has been adopted in their derivation. We further highlight that the latter viewpoint provides the most general setting for investigating the optimality of data segmentation algorithms. Arguably, the canonical segmentation problem has been the most popular framework to propose new data segmentation algorithms and study their efficiency in the last decades. In the second part of this survey, we motivate the importance of attaining an in-depth understanding of strengths and weaknesses of methodologies for the change point problem in a simpler, univariate setting, as a stepping stone for the development of methodologies for more complex problems. We illustrate this with a range of examples showcasing the connections between complex distributional changes and those in the mean. We also discuss extensions towards high-dimensional change point problems where we demonstrate that the challenges arising from high dimensionality are orthogonal to those in dealing with multiple change points.

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arXiv2021

Moving sum data segmentation for stochastics processes based on invariance

The segmentation of data into stationary stretches also known as multiple change point problem is important for many applications in time series analysis as well as signal processing. Based on strong invariance principles, we analyse data segmentation methodology using moving sum (MOSUM) statistics for a class of regime-switching multivariate processes where each switch results in a change in the drift. In particular, this framework includes the data segmentation of multivariate partial sum, integrated diffusion and renewal processes even if the distance between change points is sublinear. We study the asymptotic behaviour of the corresponding change point estimators, show consistency and derive the corresponding localisation rates which are minimax optimal in a variety of situations including an unbounded number of changes in Wiener processes with drift. Furthermore, we derive the limit distribution of the change point estimators for local changes - a result that can in principle be used to derive confidence intervals for the change points.

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arXiv2020

Two-stage data segmentation permitting multiscale change points, heavy tails and dependence

The segmentation of a time series into piecewise stationary segments, a.k.a. multiple change point analysis, is an important problem both in time series analysis and signal processing. In the presence of multiscale change points with both large jumps over short intervals and small changes over long stationary intervals, multiscale methods achieve good adaptivity in their localisation but at the same time, require the removal of false positives and duplicate estimators via a model selection step. In this paper, we propose a localised application of Schwarz information criterion which, as a generic methodology, is applicable with any multiscale candidate generating procedure fulfilling mild assumptions. We establish the theoretical consistency of the proposed localised pruning method in estimating the number and locations of multiple change points under general assumptions permitting heavy tails and dependence. Further, we show that combined with a MOSUM-based candidate generating procedure, it attains minimax optimality in terms of detection lower bound and localisation for i.i.d. sub-Gaussian errors. A careful comparison with the existing methods by means of (a) theoretical properties such as generality, optimality and algorithmic complexity, (b) performance on simulated datasets and run time, as well as (c) performance on real data applications, confirm the overall competitiveness of the proposed methodology.

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arXiv2020

A novel change point approach for the detection of gas emission sources using remotely contained concentration data

Motivated by an example from remote sensing of gas emission sources, we derive two novel change point procedures for multivariate time series where, in contrast to classical change point literature, the changes are not required to be aligned in the different components of the time series. Instead the change points are described by a functional relationship where the precise shape depends on unknown parameters of interest such as the source of the gas emission in the above example. Two different types of tests and the corresponding estimators for the unknown parameters describing the change locations are proposed. We derive the null asymptotics for both tests under weak assumptions on the error time series and show asymptotic consistency under alternatives. Furthermore, we prove consistency for the corresponding estimators of the parameters of interest. The small sample behavior of the methodology is assessed by means of a simulation study and the above remote sensing example analyzed in detail.

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