arXiv subjects

Florian Heinrichs

Publications and source records attributed to Florian Heinrichs.

17 recordsLinked to original sources

arXiv2026

Testing for Stable Intervals in Non-Stationary Time Series

Many time series are not stable over their full observation horizon, but may contain scientifically meaningful periods during which a signal remains stable up to a prescribed tolerance. We formulate this as an existence test for stable intervals in a non-stationary regression model with dependent, locally stationary errors. For a signal $d$ derived from the mean function, including deviations from reference levels and derivatives, stability over duration $δ$ is encoded by $d_\infty=\inf_{t\in[0,1-δ]}\sup_{s\in[t,t+δ]}|d(s)|$. The hypothesis $d_\infty\geΔ$ states that no interval of length $δ$ remains within the tolerance $Δ$, while rejection provides evidence for the existence of a relevant stable period. We estimate $d$ by local linear regression and construct plug-in tests for $d_\infty$. The asymptotic distribution is determined only by near-extremal windows at which the minimax functional is attained. We formalize this localization through extremal sets and derive Gaussian and extreme value approximations for kernel estimators over possibly shrinking index sets with time-varying long-run variance. The resulting tests are consistent and have asymptotic level control. The theory also extends supremum-based relevant-change tests to time-varying long-run variance and derivative-based hypotheses. Simulations and applications to physiological and engineering time series illustrate the method.

math.ST↗

arXiv2026

A Functional Central Limit Theorem for Localized Partial Sums of Non-Stationary Time Series

A localized functional central limit theorem is established for kernel-weighted partial sum processes of piecewise locally stationary time series under geometric decay of the physical dependence measure. The localized process is shown to converge weakly to a centered Gaussian random distribution in $D'(0,1)$, and the limit extends naturally to an isonormal Gaussian process on $L^2([0,1])$. Weak convergence is further derived for processes indexed by totally bounded subsets of $L^2([0,1])$. As an application, the localized limit theory is used to construct tests for constant mean functions against linear, polynomial, and general alternatives in non-parametric regression with locally stationary errors. Simulation results and data examples illustrate the finite sample performance and practical applicability of the proposed methodology.

math.ST↗

arXiv2026

Self-Normalization for CUSUM-based Change Detection in Locally Stationary Time Series

A new bivariate partial sum process for locally stationary time series is introduced and its weak convergence to a Brownian sheet is established. This construction enables the development of a novel self-normalized CUSUM test statistic for detecting changes in the mean of a locally stationary time series. For stationary data, self-normalization relies on the factorization of a constant long-run variance and a stochastic factor. In this case, the CUSUM statistic can be divided by another statistic proportional to the long-run variance, so that the latter cancels, avoiding estimation of the long-run variance. Under local stationarity, the partial sum process converges to $\int_0^t σ(x) d B_x$ and no such factorization is possible. To overcome this obstacle, a bivariate partial-sum process is introduced, allowing the construction of self-normalized test statistics under local stationarity. Weak convergence of the process is proven, and it is shown that the resulting self-normalized tests attain asymptotic level $α$ under the null hypothesis of no change, while being consistent against abrupt, gradual, and multiple changes under mild assumptions. Simulation studies show that the proposed tests have accurate size and substantially improved finite-sample power relative to existing approaches. Two data examples illustrate practical performance.

math.ST↗

arXiv2026

A Note on Physical Dependence and Mixing Conditions for Triangular Arrays

Under mild structural assumptions and regularity conditions on the marginal and conditional densities, an explicit bound on the $β$-mixing coefficients in terms of the physical dependence measure is provided. Consequently, weak physical dependence implies $β$-mixing and strong mixing for triangular arrays, complementing Hill (2025), who proved the converse implication under moment assumptions.

math.PR↗

arXiv2025

EEG-EyeTrack: A Benchmark for Time Series and Functional Data Analysis with Open Challenges and Baselines

A new benchmark dataset for functional data analysis (FDA) is presented, focusing on the reconstruction of eye movements from EEG data. The contribution is twofold: first, open challenges and evaluation metrics tailored to FDA applications are proposed. Second, functional neural networks are used to establish baseline results for the primary regression task of reconstructing eye movements from EEG signals. Baseline results are reported for the new dataset, based on consumer-grade hardware, and the EEGEyeNet dataset, based on research-grade hardware.

eess.SP↗

arXiv2025

A Note on Local Linear Regression for Time Series in Banach Spaces

This work extends local linear regression to Banach space-valued time series for estimating smoothly varying means and their derivatives in non-stationary data. The asymptotic properties of both the standard and bias-reduced Jackknife estimators are analyzed under mild moment conditions, establishing their convergence rates. Simulation studies assess the finite sample performance of these estimators and compare them with the Nadaraya-Watson estimator. Additionally, the proposed methods are applied to smooth EEG recordings for reconstructing eye movements and to video analysis for detecting pedestrians and abandoned objects.

math.ST↗

arXiv2025

Consumer-grade EEG-based Eye Tracking

Electroencephalography-based eye tracking (EEG-ET) leverages eye movement artifacts in EEG signals as an alternative to camera-based tracking. While EEG-ET offers advantages such as robustness in low-light conditions and better integration with brain-computer interfaces, its development lags behind traditional methods, particularly in consumer-grade settings. To support research in this area, we present a dataset comprising simultaneous EEG and eye-tracking recordings from 113 participants across 116 sessions, amounting to 11 hours and 45 minutes of recordings. Data was collected using a consumer-grade EEG headset and webcam-based eye tracking, capturing eye movements under four experimental paradigms with varying complexity. The dataset enables the evaluation of EEG-ET methods across different gaze conditions and serves as a benchmark for assessing feasibility with affordable hardware. Data preprocessing includes handling of missing values and filtering to enhance usability. In addition to the dataset, code for data preprocessing and analysis is available to support reproducibility and further research.

eess.SP↗

arXiv2025

Sequential Outlier Detection in Non-Stationary Time Series

A novel method for sequential outlier detection in non-stationary time series is proposed. The method tests the null hypothesis of ``no outlier'' at each time point, addressing the multiple testing problem by bounding the error probability of successive tests, using extreme value theory. The asymptotic properties of the test statistic are studied under the null hypothesis and alternative. The finite sample properties of the new detection scheme are investigated by means of a simulation study, and the method is compared with alternative procedures which have recently been proposed in the statistics and machine learning literature.

math.ST↗

arXiv2024

OML-AD: Online Machine Learning for Anomaly Detection in Time Series Data

Time series are ubiquitous and occur naturally in a variety of applications -- from data recorded by sensors in manufacturing processes, over financial data streams to climate data. Different tasks arise, such as regression, classification or segmentation of the time series. However, to reliably solve these challenges, it is important to filter out abnormal observations that deviate from the usual behavior of the time series. While many anomaly detection methods exist for independent data and stationary time series, these methods are not applicable to non-stationary time series. To allow for non-stationarity in the data, while simultaneously detecting anomalies, we propose OML-AD, a novel approach for anomaly detection (AD) based on online machine learning (OML). We provide an implementation of OML-AD within the Python library River and show that it outperforms state-of-the-art baseline methods in terms of accuracy and computational efficiency.

cs.LG↗

arXiv2024

Detecting 5G Narrowband Jammers with CNN, k-nearest Neighbors, and Support Vector Machines

5G cellular networks are particularly vulnerable against narrowband jammers that target specific control sub-channels in the radio signal. One mitigation approach is to detect such jamming attacks with an online observation system, based on machine learning. We propose to detect jamming at the physical layer with a pre-trained machine learning model that performs binary classification. Based on data from an experimental 5G network, we study the performance of different classification models. A convolutional neural network will be compared to support vector machines and k-nearest neighbors, where the last two methods are combined with principal component analysis. The obtained results show substantial differences in terms of classification accuracy and computation time.

eess.SP↗

arXiv2024

GT-PCA: Effective and Interpretable Dimensionality Reduction with General Transform-Invariant Principal Component Analysis

Data analysis often requires methods that are invariant with respect to specific transformations, such as rotations in case of images or shifts in case of images and time series. While principal component analysis (PCA) is a widely-used dimension reduction technique, it lacks robustness with respect to these transformations. Modern alternatives, such as autoencoders, can be invariant with respect to specific transformations but are generally not interpretable. We introduce General Transform-Invariant Principal Component Analysis (GT-PCA) as an effective and interpretable alternative to PCA and autoencoders. We propose a neural network that efficiently estimates the components and show that GT-PCA significantly outperforms alternative methods in experiments based on synthetic and real data.

stat.ML↗

arXiv2023

Monitoring Machine Learning Models: Online Detection of Relevant Deviations

Machine learning models are essential tools in various domains, but their performance can degrade over time due to changes in data distribution or other factors. On one hand, detecting and addressing such degradations is crucial for maintaining the models' reliability. On the other hand, given enough data, any arbitrary small change of quality can be detected. As interventions, such as model re-training or replacement, can be expensive, we argue that they should only be carried out when changes exceed a given threshold. We propose a sequential monitoring scheme to detect these relevant changes. The proposed method reduces unnecessary alerts and overcomes the multiple testing problem by accounting for temporal dependence of the measured model quality. Conditions for consistency and specified asymptotic levels are provided. Empirical validation using simulated and real data demonstrates the superiority of our approach in detecting relevant changes in model quality compared to benchmark methods. Our research contributes a practical solution for distinguishing between minor fluctuations and meaningful degradations in machine learning model performance, ensuring their reliability in dynamic environments.

cs.LG↗

arXiv2023

Functional Neural Networks: Shift invariant models for functional data with applications to EEG classification

It is desirable for statistical models to detect signals of interest independently of their position. If the data is generated by some smooth process, this additional structure should be taken into account. We introduce a new class of neural networks that are shift invariant and preserve smoothness of the data: functional neural networks (FNNs). For this, we use methods from functional data analysis (FDA) to extend multi-layer perceptrons and convolutional neural networks to functional data. We propose different model architectures, show that the models outperform a benchmark model from FDA in terms of accuracy and successfully use FNNs to classify electroencephalography (EEG) data.

cs.LG↗

arXiv2020

A Portmanteau-type test for detecting serial correlation in locally stationary functional time series

The Portmanteau test provides the vanilla method for detecting serial correlations in classical univariate time series analysis. The method is extended to the case of observations from a locally stationary functional time series. Asymptotic critical values are obtained by a suitable block multiplier bootstrap procedure. The test is shown to asymptotically hold its level and to be consistent against general alternatives.

math.ST↗

arXiv2020

A distribution free test for changes in the trend function of locally stationary processes

In the common time series model $X_{i,n} = μ(i/n) + \varepsilon_{i,n}$ with non-stationary errors we consider the problem of detecting a significant deviation of the mean function $μ$ from a benchmark $g (μ)$ (such as the initial value $μ(0)$ or the average trend $\int_{0}^{1} μ(t) dt$). The problem is motivated by a more realistic modelling of change point analysis, where one is interested in identifying relevant deviations in a smoothly varying sequence of means $ (μ(i/n))_{i =1,\ldots ,n }$ and cannot assume that the sequence is piecewise constant. A test for this type of hypotheses is developed using an appropriate estimator for the integrated squared deviation of the mean function and the threshold. By a new concept of self-normalization adapted to non-stationary processes an asymptotically pivotal test for the hypothesis of a relevant deviation is constructed. The results are illustrated by means of a simulation study and a data example.

math.ST↗

arXiv2020

Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators

Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly non-stationary time series and at (2) identifying regions where the mean exhibits a piecewise constant behavior. In many applications however, it is more reasonable to assume that the mean changes gradually in a smooth way. Those gradual changes may either be non-relevant (i.e., small), or relevant for a specific problem at hand, and the present paper presents statistical methodology to detect the latter. More precisely, we consider the common nonparametric regression model $X_{i} = μ(i/n) + \varepsilon_{i}$ with possibly non-stationary errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function $μ$ from a functional $g (μ)$ (such as the value $μ(0)$ or the integral $\int_{0}^{1} μ(t) dt$) is smaller than a given threshold on a given interval $[x_{0},x_{1}] \subseteq [0,1]$. A test for this type of hypotheses is developed using an appropriate estimator, say $\hat d_{\infty, n}$, for the maximum deviation $ d_{\infty}= \sup_{t \in [x_{0},x_{1}]} |μ(t) - g( μ) |$. We derive the limiting distribution of an appropriately standardized version of $\hat d_{\infty,n}$, where the standardization depends on the Lebesgue measure of the set of extremal points of the function $μ(\cdot)-g(μ)$. A refined procedure based on an estimate of this set is developed and its consistency is proved. The results are illustrated by means of a simulation study and a data example.

math.ST↗

arXiv2018

Detecting deviations from second-order stationarity in locally stationary functional time series

A time-domain test for the assumption of second order stationarity of a functional time series is proposed. The test is based on combining individual cumulative sum tests which are designed to be sensitive to changes in the mean, variance and autocovariance operators, respectively. The combination of their dependent $p$-values relies on a joint dependent block multiplier bootstrap of the individual test statistics. Conditions under which the proposed combined testing procedure is asymptotically valid under stationarity are provided. A procedure is proposed to automatically choose the block length parameter needed for the construction of the bootstrap. The finite-sample behavior of the proposed test is investigated in Monte Carlo experiments and an illustration on a real data set is provided.

math.ST↗