Editorial: Data Segmentation in Time Series: Structural Breaks and Real‐Time Monitoring

Abstract

In his seminal 1954 Biometrika paper (Page 1954), E. S. Page introduced the cumulative sum (CUSUM) control chart for change-point monitoring, a foundational idea that continues to underpin many modern approaches to time-series segmentation. This special issue highlights the breadth of recent advances in the field, spanning online and offline structural break detection, methodologies for high-dimensional and complex data, and techniques for estimating multiple change points. Multiple change-point estimation, often referred to as data segmentation, seeks to partition a time series into stationary regimes. Because the number of segments is unknown a priori, this task poses additional statistical and computational challenges beyond testing for a single structural break. Addressing these challenges, Barigozzi et al. (2026) develop a methodology for multiple change-point detection in large factor models, linking segmentation with structural change in high-dimensional time series. Such models are central in econometrics, where a small number of latent factors drive many observed variables; their approach accommodates changes in factor loadings, including the emergence or disappearance of factors. While Barigozzi et al. (2026) employs moving-sum procedures, Lund et al. (2026) adopts a penalized likelihood framework that enables detection of multiple changes in non-Gaussian settings, including count time series constructed via suitable transformations of Gaussian processes, thus facilitating likelihood-based inference. In a different direction, Davis et al. (2026) apply the minimum description length principle to jointly estimate change points and outliers in piecewise linear autoregressive models, allowing for both innovation and additive outliers, two distinct but practically intertwined forms of anomalies. Casini and Perron (2026) revisit the classical linear regression model under a continuous-record asymptotic framework, where discrete observations arise from underlying continuous-time Itô semimartingales, reflecting a fundamentally different modeling paradigm. Extensions beyond linear models introduce further challenges, e.g., for count data. Whereas Lund et al. (2026) relies on Gaussian transformations, Hudecová and Hušková (2026) builds upon a generalized Poisson autoregressive model with exogenous components introduced by Aknouche and Francq (2021) to propose new change-point tests for count time series. They establish the corresponding limit theory and obtain empirical performance results. Finally, for continuously observed autoregressive models potentially including exogenous covariates, Kirch and Schwaar (2026) develop change-point tests based on neural network approximations of nonlinear regression functions, illustrating the growing integration of machine learning techniques into structural break analysis. Another important challenge the field faces lies in developing change-point methodologies tailored to increasingly complex data structures arising in modern applications. For instance, Pešta et al. (2026) study tensor-valued data comprising multivariate outcomes observed across items and subjects, allowing for both longitudinal and cross-sectional dependence. Meanwhile, El-Yaagoubi et al. (2026) investigates changes in the topological features of time-frequency representations from multi-trial experiments, employing spectral techniques based on the periodogram for stationary series and the spectrogram for locally stationary processes. Spectral methods built on periodograms also form the basis of the test proposed by Iacone and Taylor (2026) for detecting constant versus time-varying means in short- and long-range dependent time series, capable of identifying both abrupt and gradual changes. Functional time series and change point analysis study how complex data observed as evolving functions over time can be modeled and monitored to detect structural breaks or distributional shifts in their underlying dynamics. Five papers in this special issue develop theoretically grounded and computationally practical methods for inference and change-point detection in functional time series and panels of functional data. They introduce dimension-free and CUSUM-based procedures often supported by Gaussian approximations or bootstrap techniques that remain valid under weak dependence, non-stationarity, and sparsity. This methodology's usefulness is demonstrated through its application to data sets related to climate, energy demand, finance, and economics. The paper by Horváth et al. (2026) introduces a method for detecting and localizing distributional change points in functional data that remains valid under weak moment and serial dependence conditions. The proposed methodology is applied to electricity demand and high-frequency financial data. While the modeling in Iacone and Taylor (2026) allows for gradual changes in the mean of univariate time series, Bastian and Dette (2026) develop a statistically valid method to detect and estimate the onset of gradual, practically relevant deviations in the mean functions of possibly nonstationary functional time series, using Gaussian approximations and maximum deviation measures. Paparoditis et al. (2026) show that a functional autoregressive sieve bootstrap can reliably approximate critical values for CUSUM-based change-point tests in functional time series, providing asymptotically valid inference under the null and consistent detection under local alternatives. Kutta et al. (2026) develop a statistically rigorous Full-CUSUM-based method to detect and identify structural breaks in panels of functional data, accounting for sparsity and non-stationarity, which is supported by new Gaussian approximation theory and validated through simulations and financial applications. Gao et al. (2026) propose a fully functional, dimension-free testing framework for functional time series based on sample splitting that avoids finite-dimensional projections and provides inference procedures for mean, covariance, autocovariance, and change-point problems in Hilbert space settings. This special edition finally includes contributions to the emerging field of real-time monitoring and the associated integration of computational and inferential aspects. Research on sequential monitoring procedures focuses on detecting structural changes in time series in real time by continuously updating test statistics as new observations arrive, allowing for rapid intervention once a change occurs. Modern developments extend classical schemes such as CUSUM and likelihood-based detectors to high-dimensional, dependent, and complex data settings, while providing theoretical guarantees on false alarm rates and detection delays. Xu et al. (2026) develop a polynomial-time online change-point detection method for dynamic networks with time-varying missingness and complex dependence structures, providing sharp detection delay guarantees and Type-I error control. Yang et al. (2026) propose mixFOCuS, a communication-efficient online change-point detection method for distributed sensor networks that handles mixed-type, potentially non-Gaussian data without requiring known pre- or post-change parameters, while balancing transmission constraints and detection power. Grundy et al. (2026) introduce a general framework for real-time monitoring of forecasting models by applying sequential change-point detection to forecast errors, enabling faster and more robust identification of structural shifts than direct model-based approaches. Hüselitz et al. (2026) develops a computationally efficient online method for estimating a single change point in piecewise linear regression that achieves minimax-optimal localization rates, reveals a phase transition between jump and kink changes, and distinguishes between them with constant memory and computation per observation. Taken together, the contributions in this special issue exemplify the remarkable breadth and depth of contemporary change-point research. From foundational methodological advances and refined asymptotic theory to scalable algorithms for high-dimensional, functional, and streaming data, the field continues to evolve in response to increasingly complex real-world challenges. A unifying theme across these works is the careful integration of statistical rigor with computational feasibility, ensuring that modern change-point procedures are not only theoretically sound but also practically implementable. We hope that the results presented here will stimulate further developments at the interface of theory, methodology, and application, and will continue to expand the scope and impact of structural break analysis in statistics and related disciplines.