[Paper Review] Data Segmentation for Time Series Based on a General Moving Sum Approach

Claudia Kirch, Kerstin Reckruehm|arXiv (Cornell University)|Jul 15, 2022

Advanced Statistical Methods and Models4 citations

TL;DR

This paper proposes a general moving sum (MOSUM) framework for time series data segmentation that detects multiple change points in complex models, including non-linear and non-Gaussian processes like Poisson autoregressions. It introduces two MOSUM statistics—Wald and score—proving consistency and convergence rates for change point estimators under general estimating equations, with the MOSUM-score variant offering computational advantages and robustness in non-linear settings.

ABSTRACT

In this paper we propose new methodology for the data segmentation, also known as multiple change point problem, in a general framework including classic mean change scenarios, changes in linear regression but also changes in the time series structure such as in the parameters of Poisson-autoregressive time series. In particular, we derive a general theory based on estimating equations proving consistency for the number of change points as well as rates of convergence for the estimators of the locations of the change points. More precisely, two different types of MOSUM (moving sum) statistics are considered: A MOSUM-Wald statistic based on differences of local estimators and a MOSUM-score statistic based on a global estimator. The latter is usually computationally less involved in particular in non-linear problems where no closed form of the estimator is known such that numerical methods are required. Finally, we evaluate the methodology by means of simulated data as well as using some geophysical well-log data.

Motivation & Objective

• To develop a unified framework for detecting multiple change points in time series across diverse models, including non-linear and non-Gaussian processes.
• To establish theoretical consistency and convergence rates for change point estimators under general estimating equations.
• To compare and contrast two MOSUM-based methods—MOSUM-Wald and MOSUM-score—for robustness and computational efficiency.
• To validate the methodology empirically using simulated data and real geophysical well-log data.
• To enable application in complex settings such as Poisson autoregressive models and neural network-based regressions.

Proposed method

• The method uses a general framework based on estimating equations to define model parameters and detect structural changes in time series.
• Two MOSUM statistics are proposed: MOSUM-Wald, based on differences of local parameter estimators, and MOSUM-score, based on a global estimator.
• The MOSUM-score procedure uses a single global estimator, reducing computational cost, especially in non-linear models requiring numerical optimization.
• Consistency of the number of change points and localization rates are proven under high-level regularity and moment conditions.
• The framework allows for model misspecification by detecting differences in best-approximating parameters across segments.
• Post-processing via information criteria is suggested for future refinement, building on existing work in the mean-change context.

Experimental results

Research questions

• RQ1 Can a general MOSUM framework consistently estimate the number and locations of multiple change points across diverse time series models?
• RQ2 How do MOSUM-Wald and MOSUM-score statistics compare in terms of computational cost and robustness to local optima in non-linear models?
• RQ3 What are the theoretical convergence rates for change point estimators under general estimating equations?
• RQ4 How does model misspecification affect the detection performance of the proposed methods?
• RQ5 Can the MOSUM-score procedure effectively generate change point candidates across multiple bandwidths with minimal computational overhead?

Key findings

• The MOSUM-score statistic achieves consistency in change point detection with significantly lower computational cost than MOSUM-Wald, especially in non-linear models requiring numerical optimization.
• The MOSUM-Wald procedure is sensitive to multiple local optima in parameter space, leading to potential misidentification of change points in complex models like neural networks.
• The MOSUM-score method is robust to multimodal parameter landscapes and maintains stable performance even when local estimators are unreliable.
• Theoretical consistency and localization rates are established under general high-level assumptions, which are verified for smooth estimating functions under standard moment conditions.
• Empirical results on simulated and geophysical well-log data confirm the method's ability to detect structural breaks in non-Gaussian and non-linear time series.
• The choice of covariance estimator affects small-sample performance, particularly in false alarm rates and detectability near change points, though asymptotic properties remain robust.