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Non-parametric multiple change-point detection

Andreas Anastasiou, Piotr Fryzlewicz

TL;DR

The paper introduces Non-Parametric Isolate-Detect (NPID), a framework for consistent, offline detection of multiple distributional change-points in a non-parametric setting. NPID isolates each true change-point within subintervals to reduce interference from nearby changes and uses a non-parametric ECDF-based CUSUM contrast aggregated by mean-dominant norms, with a thresholding or information-criterion stopping rule. The authors establish optimal-rate consistency under mild assumptions and compare NPID against state-of-the-art methods, showing superior performance across a wide range of change types and data-generating processes; they also provide an R implementation and practical variants for speed and robustness. The approach is capable of handling general distributional changes and is extensible to data in arbitrary metric spaces via VC-class theory. Overall, NPID offers a theoretically sound, computationally efficient, and practically versatile tool for non-parametric change-point analysis with strong empirical performance on simulations and real data, including micro-array, financial, and epidemiological applications.

Abstract

We introduce a methodology, labelled Non-Parametric Isolate-Detect (NPID), for the consistent estimation of the number and locations of multiple change-points in a non-parametric setting. The method can handle general distributional changes and is based on an isolation technique preventing the consideration of intervals that contain more than one change-point, which enhances the estimation accuracy. As stopping rules, we propose both thresholding and the optimization of an information criterion. In the scenarios tested, which cover a broad range of change types, NPID outperforms the state of the art. An R implementation is provided.

Non-parametric multiple change-point detection

TL;DR

The paper introduces Non-Parametric Isolate-Detect (NPID), a framework for consistent, offline detection of multiple distributional change-points in a non-parametric setting. NPID isolates each true change-point within subintervals to reduce interference from nearby changes and uses a non-parametric ECDF-based CUSUM contrast aggregated by mean-dominant norms, with a thresholding or information-criterion stopping rule. The authors establish optimal-rate consistency under mild assumptions and compare NPID against state-of-the-art methods, showing superior performance across a wide range of change types and data-generating processes; they also provide an R implementation and practical variants for speed and robustness. The approach is capable of handling general distributional changes and is extensible to data in arbitrary metric spaces via VC-class theory. Overall, NPID offers a theoretically sound, computationally efficient, and practically versatile tool for non-parametric change-point analysis with strong empirical performance on simulations and real data, including micro-array, financial, and epidemiological applications.

Abstract

We introduce a methodology, labelled Non-Parametric Isolate-Detect (NPID), for the consistent estimation of the number and locations of multiple change-points in a non-parametric setting. The method can handle general distributional changes and is based on an isolation technique preventing the consideration of intervals that contain more than one change-point, which enhances the estimation accuracy. As stopping rules, we propose both thresholding and the optimization of an information criterion. In the scenarios tested, which cover a broad range of change types, NPID outperforms the state of the art. An R implementation is provided.
Paper Structure (20 sections, 4 theorems, 104 equations, 10 figures, 7 tables)

This paper contains 20 sections, 4 theorems, 104 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Methodology and Theory
  3. Methodology
  4. Theoretical behaviour of NPID
  5. Information Criterion approach
  6. Computational complexity and practicalities
  7. Computational cost
  8. Parameter choice
  9. Variants
  10. Simulations
  11. Real data examples
  12. Micro-array data
  13. Major American stock indices.
  14. The COVID-19 outbreak in the UK
  15. Discussion
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\left\lbrace X_t \right\rbrace_{t=1,\ldots,T}$ follow model our_model and assume that (A1) holds. Let $N$ and $r_j, j=1,\ldots,N$ be the number and locations of the change-points, while $\hat{N}$ and $\hat{r}_j, j=1,\ldots,\hat{N}$ are their estimates (sorted in increasing order) when NPID is e

Figures (10)

  • Figure 1: The isolation and detection process for the example in \ref{['example']}. The right and left expanding intervals are coloured in red and blue, respectively. The green dashed line is the first isolation interval at each phase.
  • Figure 2: Examples of data sequences, used in simulations. The change-point locations are indicated with red, vertical, solid lines.
  • Figure 3: Change-point detection for the first individual in the micro-array data set. The NPID-estimated change-point locations are marked with red solid vertical lines.
  • Figure 4: Change-point detection for Individual 4 in the micro-array data set. The estimated change-point locations are given with dashed vertical lines. Top row: The results for the NPID and ECP methods. Bottom row: The results for the NMCD and NWBS methods.
  • Figure 5: Change-point detection for Individual 39 in the micro-array data set. The estimated change-point locations are given with dashed vertical lines. Top row: The results for the NPID and ECP methods. Bottom row: The results for the NMCD and NWBS methods.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof