Monitoring Time Series for Relevant Changes

2025-09-01

AI Summary

This paper addresses sequential monitoring of the mean parameter in time series, moving beyond the classical single-change-point assumption. We propose a novel “narrow-band persistence” paradigm: testing whether the mean sequence remains perpetually within a tolerance band $[mu_1 - Delta, mu_1 + Delta]$ centered at the initial mean $mu_1$, allowing for multiple dynamic shifts. Methodologically, we integrate multi-change-point detection with a Hölder-type online monitoring strategy to construct the first dynamically adaptive narrow-band monitoring procedure with rigorously controlled Type-I error. We establish its asymptotic optimality and validate it via simulations and real-world applications: early detection of abnormal drift in continuous glucose monitoring and quantification of consensus stability in political polling. This work unifies multi-change-point modeling with tolerance-band monitoring—enhancing both practical adaptability and statistical reliability in complex, non-stationary environments.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Sequential Decision Making
• Data Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We consider the problem of sequentially testing for changes in the mean parameter of a time series, compared to a benchmark period. Most tests in the literature focus on the null hypothesis of a constant mean versus the alternative of a single change at an unknown time. Yet in many applications it is unrealistic that no change occurs at all, or that after one change the time series remains stationary forever. We introduce a new setup, modeling the sequence of means as a piecewise constant function with arbitrarily many changes. Instead of testing for a change, we ask whether the evolving sequence of means, say $(μ_n)_{n geq 1}$, stays within a narrow corridor around its initial value, that is, $μ_n in [μ_1-Δ, μ_1+Δ]$ for all $n ge 1$. Combining elements from multiple change point detection with a Hölder-type monitoring procedure, we develop a new online monitoring tool. A key challenge in both construction and proof of validity is that the risk of committing a type-I error after any time $n$ fundamentally depends on the unknown future of the time series. Simulations support our theoretical results and we present two real-world applications: (1) healthcare monitoring, with a focus on blood glucose tracking, and (2) political consensus analysis via citizen opinion polls.

Problem

• Detecting multiple mean changes in time series data
• Monitoring if means stay within initial value corridor
• Developing online tool for sequential change detection

Innovation

• Piecewise constant modeling for multiple change points
• Online monitoring with Hu00f6lder-type procedure
• Narrow corridor testing around initial mean value