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

2026-01-22

AI Summary

This study addresses the problem of detecting both abrupt and gradual mean shifts in functional time series by proposing a novel weighted functional CUSUM statistic. The method preserves full functional information and, for the first time, incorporates an inverse covariance weighting mechanism into the testing framework, yielding a scale-invariant unified procedure applicable to both abrupt and gradual change scenarios. Asymptotic theory under dependent data is established, providing the limiting distributions and power properties of the statistic under the null hypothesis and various alternatives. Both theoretical analysis and numerical experiments demonstrate that the proposed approach achieves substantially higher detection power—particularly for changes occurring in directions beyond the leading principal components—and significantly outperforms conventional methods based on principal component analysis.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• functional time series
• change point detection
• mean change
• covariance structure
• gradual change

Innovation

• functional time series
• change point detection
• covariance weighting
• scale-invariant test
• gradual change