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Multiscale Change Point Detection for Functional Time Series

Tim Kutta, Holger Dette, Shixuan Wang

TL;DR

The paper tackles the challenge of detecting and localizing multiple mean changes in Banach-space valued time series, including functional data, under non-stationarity and dependence. It introduces a flexible multiscale Hölder-type scan statistic and the MultiScan algorithm, enabling both weak and strong localization with controllable error through quantiles of a Hölder-space limit $L^ ho$. The methodology encompasses changes in distribution and extends to panels, supported by Gaussian-approximation results in Hölder spaces, Monte Carlo validation, and an empirical VIX analysis that demonstrates practical localization of volatility-regime shifts. The approach offers robustness to heavy tails and dependency structures while preserving localization precision, with broad applicability to FDA problems and financial time series.

Abstract

We study the problem of detecting and localizing multiple changes in the mean parameter of a Banach space-valued time series. The goal is to construct a collection of narrow confidence intervals, each containing at least one (or exactly one) change, with globally controlled error probability. Our approach relies on a new class of weighted scan statistics, called Hölder-type statistics, which allow a smooth trade-off between efficiency (enabling the detection of closely spaced, small changes) and robustness (against heavier tails and stronger dependence). For Gaussian noise, maximum weighting can be applied, leading to a generalization of optimality results known for scalar, independent data. Even for scalar time series, our approach is advantageous, as it accommodates broad classes of dependency structures and non-stationarity. Its primary advantage, however, lies in its applicability to functional time series, where few methods exist and established procedures impose strong restrictions on the spacing and magnitude of changes. We obtain general results by employing new Gaussian approximations for the partial sum process in Hölder spaces. As an application of our general theory, we consider the detection of distributional changes in a data panel. The finite-sample properties and applications to financial datasets further highlight the merits of our method.

Multiscale Change Point Detection for Functional Time Series

TL;DR

The paper tackles the challenge of detecting and localizing multiple mean changes in Banach-space valued time series, including functional data, under non-stationarity and dependence. It introduces a flexible multiscale Hölder-type scan statistic and the MultiScan algorithm, enabling both weak and strong localization with controllable error through quantiles of a Hölder-space limit . The methodology encompasses changes in distribution and extends to panels, supported by Gaussian-approximation results in Hölder spaces, Monte Carlo validation, and an empirical VIX analysis that demonstrates practical localization of volatility-regime shifts. The approach offers robustness to heavy tails and dependency structures while preserving localization precision, with broad applicability to FDA problems and financial time series.

Abstract

We study the problem of detecting and localizing multiple changes in the mean parameter of a Banach space-valued time series. The goal is to construct a collection of narrow confidence intervals, each containing at least one (or exactly one) change, with globally controlled error probability. Our approach relies on a new class of weighted scan statistics, called Hölder-type statistics, which allow a smooth trade-off between efficiency (enabling the detection of closely spaced, small changes) and robustness (against heavier tails and stronger dependence). For Gaussian noise, maximum weighting can be applied, leading to a generalization of optimality results known for scalar, independent data. Even for scalar time series, our approach is advantageous, as it accommodates broad classes of dependency structures and non-stationarity. Its primary advantage, however, lies in its applicability to functional time series, where few methods exist and established procedures impose strong restrictions on the spacing and magnitude of changes. We obtain general results by employing new Gaussian approximations for the partial sum process in Hölder spaces. As an application of our general theory, we consider the detection of distributional changes in a data panel. The finite-sample properties and applications to financial datasets further highlight the merits of our method.

Paper Structure

This paper contains 22 sections, 8 theorems, 135 equations, 6 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related literature
  3. A multiscale method for change point detection
  4. Notations and mathematical preliminaries
  5. Statistical methodology
  6. Theoretical analysis
  7. Convergence in the Hölder space
  8. An application to changes in distribution
  9. Monte Carlo Simulations
  10. Independent functional time series
  11. Dependent functional time series
  12. Distribution functions in a data panel
  13. Empirical Applications
  14. Main analysis
  15. Sensitivity analysis
  16. ...and 7 more sections

Key Result

Theorem 2.4

Suppose that Assumption ass_1 is satisfied and let $\mathbf{N}_N$ be either $\mathbf{N}^{all}$ or $\mathbf{N}_N^\theta$. Then, for any $\alpha \in (0,1)$, the output of MultiScan$(\emptyset, \mathbf{N}_N, q_{1-\alpha})$ satisfies the weak localization property e:weakl. Furthermore, if $K=0$, then e:

Figures (6)

  • Figure 1: Frequency of $n_{(n,h)}^{max}$ with $N=300$ for the i.i.d. error. Top left: $H_{A,1}$; Top right: $H_{A,2}$; Bottom left: $H_{A,3}$; Bottom right: $H_{A,4}$.
  • Figure 2: Frequency of $n_{(n,h)}^{max}$ with $N=300$ for the dependent DGP. Top left: $H_{A,1}$; Top right: $H_{A,2}$; Bottom left: $H_{A,3}$; Bottom right: $H_{A,4}$.
  • Figure 3: Frequency of $n_{(n,h)}^{max}$ with $N=300$. Top: $H_{A,1}^*$; Bottom: $H_{A,2}^*$.
  • Figure 4: Intraday VIX Curves in 5-minute frequency
  • Figure 5: Detected changes for VIX of three intraday frequencies
  • ...and 1 more figures

Theorems & Definitions (11)

  • Example 2.3
  • Theorem 2.4
  • Theorem 2.6
  • Proposition 2.8
  • Proposition 2.9
  • Theorem 2.11
  • Theorem 3.2
  • Lemma 6.1
  • proof : Proof of Lemma \ref{['lem:conv']}
  • Lemma 6.2
  • ...and 1 more