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Choosing the Right Norm for Change Point Detection in Functional Data

Patrick Bastian

TL;DR

This paper addresses change point detection in the mean functions of functional time series under an AMOC framework and proposes an $L^1$-norm based CUSUM approach. It establishes strong and weak invariance principles for $L^1$-valued data, develops bootstrap procedures for both classical and relevant hypotheses, and introduces a power-enhancement component to boost detection against sparse alternatives. Theoretical guarantees include consistency of change-point estimators, bootstrap validity, and directional Hadamard differentiability of the $\|\cdot\|_{\infty,1}$ norm, complemented by extensive finite-sample and real-data demonstrations (e.g., Melbourne temperature curves). Collectively, the results show that the $L^1$-based method offers robustness to heavy tails and performs well for dense signals, with practical interpretability through the area-between-curves interpretation, while offering mechanisms to handle sparse, spike-like changes via the power enhancement.

Abstract

We consider the problem of detecting a change point in a sequence of mean functions from a functional time series. We propose an $L^1$ norm based methodology and establish its theoretical validity both for classical and for relevant hypotheses. We compare the proposed method with currently available methodology that is based on the $L^2$ and supremum norms. Additionally we investigate the asymptotic behaviour under the alternative for all three methods and showcase both theoretically and empirically that the $L^1$ norm achieves the best performance in a broad range of scenarios. We also propose a power enhancement component that improves the performance of the $L^1$ test against sparse alternatives. Finally we apply the proposed methodology to both synthetic and real data.

Choosing the Right Norm for Change Point Detection in Functional Data

TL;DR

This paper addresses change point detection in the mean functions of functional time series under an AMOC framework and proposes an -norm based CUSUM approach. It establishes strong and weak invariance principles for -valued data, develops bootstrap procedures for both classical and relevant hypotheses, and introduces a power-enhancement component to boost detection against sparse alternatives. Theoretical guarantees include consistency of change-point estimators, bootstrap validity, and directional Hadamard differentiability of the norm, complemented by extensive finite-sample and real-data demonstrations (e.g., Melbourne temperature curves). Collectively, the results show that the -based method offers robustness to heavy tails and performs well for dense signals, with practical interpretability through the area-between-curves interpretation, while offering mechanisms to handle sparse, spike-like changes via the power enhancement.

Abstract

We consider the problem of detecting a change point in a sequence of mean functions from a functional time series. We propose an norm based methodology and establish its theoretical validity both for classical and for relevant hypotheses. We compare the proposed method with currently available methodology that is based on the and supremum norms. Additionally we investigate the asymptotic behaviour under the alternative for all three methods and showcase both theoretically and empirically that the norm achieves the best performance in a broad range of scenarios. We also propose a power enhancement component that improves the performance of the test against sparse alternatives. Finally we apply the proposed methodology to both synthetic and real data.
Paper Structure (19 sections, 14 theorems, 138 equations, 2 figures, 5 tables)

This paper contains 19 sections, 14 theorems, 138 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. $L^1$ valued random variables
  3. Change Point Detection: Classical case
  4. Change Point Detection: Relevant case
  5. Power Comparison and Enhancement
  6. Finite Sample Performance
  7. Norm Comparison for the Classical Hypotheses
  8. Relevant Hypotheses: Synthetic Data
  9. Real Data Application
  10. Proofs
  11. Some Facts about Cotype 2 Spaces
  12. Theorem \ref{['Thm:L1Inv']}: Strong invariance principle for $L^1$ valued time series
  13. Theorem \ref{['Thm:BootCons']}: Weak invariance principle for the classical bootstrap
  14. Theorem \ref{['Thm:CPCon']}: Consistency of Change Point Estimator
  15. Theorem \ref{['Thm:BootTest']}
  16. ...and 4 more sections

Key Result

Theorem 2.2

Suppose that assumptions A1) to A4) hold. We then have that, on a possibly larger probability space, there exists a $L^1$ valued Brownian motion $B$ with covariance operator $C$ such that for some $\gamma>0$.

Figures (2)

  • Figure 1: Plots of $\Phi_c$ for different choices of $c$.
  • Figure 2: Plots of the mean estimator before (mu1) and after(mu2) the estimated change point for the Melbourne temperature time series.

Theorems & Definitions (22)

  • Theorem 2.2
  • Remark 2.3
  • Corollary 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3
  • Theorem 5.1
  • Corollary 5.2
  • ...and 12 more