Online detection of distributional changes for time series in metric spaces

2026-08-26

AI Summary

本文提出一种基于两样本U统计量的在线检测框架，用于检测度量空间中序列依赖数据的分布变化，并通过蒙特卡洛校准实现快速检测。

Abstract

We propose an online testing framework for detecting distributional changes in serially dependent data with values in a separable metric space. Based on two-sample $U$-statistics, the framework encompasses sequential analogs of energy distance and maximum mean discrepancy (MMD) procedures while accommodating temporal dependence. We establish asymptotic theory for finite and open-ended monitoring horizons that characterizes the full asymptotic run-length distribution under $H_0$ and yields asymptotic false-alarm control. We further establish new spectral approximation results for kernel matrices formed from serially dependent observations, and use them to construct a feasible Monte Carlo calibration procedure. Our flexible window construction encompasses classical, Page-type, and full-scan historical-baseline monitoring and can achieve short detection delays for both early and late changepoints, without requiring sub-Gaussianity or high-order moments of the raw observations. Simulations show reliable false-alarm control across linear, nonlinear, high-dimensional, and functional time-series models and further demonstrate that, over a broad range of alternatives and changepoint locations, the proposed method can achieve substantially shorter delays than recent procedures designed specifically for rapid detection. Applications to foreign exchange rates, electricity-market curves, and daily air transportation networks illustrate the methodology across scalar, functional, and network-valued time series.

Problem

• time series
• distributional changes
• metric spaces
• sequential dependence

Innovation

• online detection
• U-statistics
• temporal dependence
• spectral approximation
• Monte Carlo calibration