[Paper Review] Moving sum data segmentation for stochastics processes based on invariance

Claudia Kirch, Philipp Klein|arXiv (Cornell University)|Jan 12, 2021

Advanced Statistical Process Monitoring43 references4 citations

TL;DR

This paper proposes a multivariate moving sum (MOSUM) data segmentation method for regime-switching stochastic processes with changes in drift, leveraging strong invariance principles to achieve minimax-optimal consistency and localisation rates. It extends prior univariate and single-bandwidth approaches to handle multivariate partial sums, diffusion, and renewal processes—even with sublinear change point distances—providing theoretical guarantees on estimator consistency and convergence rates.

ABSTRACT

The segmentation of data into stationary stretches also known as multiple change point problem is important for many applications in time series analysis as well as signal processing. Based on strong invariance principles, we analyse data segmentation methodology using moving sum (MOSUM) statistics for a class of regime-switching multivariate processes where each switch results in a change in the drift. In particular, this framework includes the data segmentation of multivariate partial sum, integrated diffusion and renewal processes even if the distance between change points is sublinear. We study the asymptotic behaviour of the corresponding change point estimators, show consistency and derive the corresponding localisation rates which are minimax optimal in a variety of situations including an unbounded number of changes in Wiener processes with drift. Furthermore, we derive the limit distribution of the change point estimators for local changes - a result that can in principle be used to derive confidence intervals for the change points.

Motivation & Objective

• Address the gap in multiple change point detection for multivariate stochastic processes beyond univariate mean shifts.
• Extend MOSUM-based segmentation to multivariate processes, including partial sums, diffusion, and renewal processes.
• Establish consistency and minimax-optimal localisation rates for change point estimators under sublinear change point distances.
• Provide theoretical foundations for two-step MOSUM procedures and multiscale change point detection.
• Derive limit distributions for local change point estimators to enable confidence interval construction.

Proposed method

• Formalize a regime-switching multivariate process model where each regime switch induces a drift change.
• Apply strong invariance principles to approximate the process by a Brownian motion under appropriate probability measures.
• Define MOSUM statistics as moving sums over sliding windows of fixed or sublinear bandwidth to detect structural breaks.
• Construct change point estimators via local maxima of MOSUM statistics, with thresholding to avoid spurious detections.
• Use diagonal variance estimation to improve signal-to-noise ratio in finite samples, especially in high-dimensional settings.
• Introduce multiscale bandwidth selection via minimum-based aggregation to detect changes across varying scales, particularly in heterogeneous signals.

Experimental results

Research questions

• RQ1 Can MOSUM-based segmentation be extended to multivariate stochastic processes with multiple, possibly unbounded, change points?
• RQ2 What are the consistency and localisation rates of MOSUM estimators under a general regime-switching multivariate model?
• RQ3 How do the theoretical properties of MOSUM estimators behave when the distance between change points grows sublinearly?
• RQ4 Can the limit distribution of change point estimators be derived for local changes, enabling confidence interval construction?
• RQ5 What is the impact of bandwidth selection and variance estimation on detection power and estimator accuracy in finite samples?

Key findings

• The proposed MOSUM procedure achieves minimax-optimal localisation rates for change point estimators in both bounded and unbounded change point scenarios, including Wiener processes with drift.
• Consistency of the change point estimators is established under general conditions, even when the number of change points increases with sample size.
• The method performs well in simulations, with high detection power, low spurious and duplicate estimator rates, especially when using local diagonal variance estimation.
• Suboptimal performance is observed with full asymptotic covariance matrices, highlighting the advantage of diagonal variance estimation in high-dimensional or moderate-sample settings.
• Multiscale change point detection requires multiple bandwidths, as no single bandwidth can detect all changes in heterogeneous signals, validating the need for multiscale extensions.
• Theoretical results support the use of MOSUM statistics in testing and inference, including the derivation of limit distributions for local change point estimators.