Optimal Change Point Detection and Inference in the Spectral Density of General Time Series Models

TL;DR

This work develops a nonparametric, Wold-decomposition-based framework for offline change-point detection in the spectral density of general time series. A two-stage approach first identifies a near-optimal change-point estimator by AR($p$) fitting on split segments and then refines it to an optimal estimator with provable localization and an asymptotic distribution, enabling confidence intervals. The method accommodates heavy-tailed observations via sub-Weibull assumptions and yields a refined localization rate, with an explicit limiting distribution involving Brownian-motion-based processes. Empirical evaluations on EEG seizure data and surveillance video demonstrate superior detection accuracy and valid inference compared with existing methods, highlighting broad applicability to real-world nonstationary signals. The work thus provides a principled, inferential tool for detecting and quantifying shifts in temporal dependence structures across domains.

Abstract

This paper addresses the problem of detecting change points in the spectral density of time series, motivated by EEG analysis of seizure patients. Seizures disrupt coherence and functional connectivity, necessitating precise detection. Departing from traditional parametric approaches, we utilize the Wold decomposition, representing general time series as autoregressive processes with infinite lags, which are truncated and estimated around the change point. Our detection procedure employs an initial estimator that systematically searches across time points. We examine the localization error and its dependence on time series properties and sample size. To enhance accuracy, we introduce an optimal rate method with an asymptotic distribution, facilitating the construction of confidence intervals. The proposed method effectively identifies seizure onset in EEG data and extends to event detection in video data. Comprehensive numerical experiments demonstrate its superior performance compared to existing techniques.

Figures (17)

• Figure 1: Spectral densities of selected simulation scenarios for before and after change points. The values of coefficients for all the scenarios are as follows: Scenario (I) $\theta=-0.9$ and $\phi=-0.5$, Scenario (II) $\theta=-0.9$ and $\phi=0.5$, Scenario (III) $\phi=-0.9$, Scenario (IV) $\theta=-0.9$, and Scenario (V) $\phi=-0.5$.
• Figure 2: Distributions of estimated change points for all the methods, where the "true" values are specified with the dashed lines per case. Note that number 12 is our proposed method. Additionally, methods 1 and 3 have significant number of missing values. From top-left panel to bottom-right panel, the missing percentages are: (1) PELT $(99,98,99,98)\%$ and (3) ARC $(78,73,84,67)\%$.
• Figure 3: ($90\%$, $95\%$, $99\%$) coverage probability in Scenario I with $T=1000$.
• Figure 4: EEG data set over 110 seconds with their estimated change points using our method. The $\% 80$ CIs are overlaid on the plots through shaded parts. In the last panel, we describe the spectral densities before and after the estimated change points for all the channels. Note that we used additional points (162,228) for the description of spectral density of after, and before is simply related to the interval of (1,$\lfloor T \tilde{\tau} \rfloor$).
• Figure D.1: AB and RMSE of $\lfloor T\hat{\tau} \rfloor$ and $\lfloor T\tilde{\tau} \rfloor$ for Scenario I with $T=1000$.

Theorems & Definitions (38)

• Definition 2.1: Orlicz norms
• Definition 2.2: Sub-Weibull($\gamma$) random variable and norm
• Definition 2.3
• Remark 1
• Remark 2
• Theorem 3.1
• Lemma 4.1
• Theorem 4.1
• Remark 3
• Theorem 4.2