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Changepoint Detection As Model Selection: A General Framework

Michael Grantham, Xueheng Shi, Bertrand Clarke

TL;DR

Changepoint Detection As Model Selection delivers a general, information-criteria-driven framework for changepoint analysis built on L0-model selection. The centerpiece is Iteratively Reweighted Fused Lasso (IRFL), which adaptively reweights penalties within the generalized lasso to converge toward an $\ell_0$-like solution and robustly recover true changepoints across mean shifts, trends, seasonality, and dependence structures. The dissertation unifies heuristic, exact, and adaptive segmentation under a single framework, demonstrates strong performance in extensive simulations, and validates the approach on real data such as Mauna Loa CO$_2$ records and medical/imaging applications. Overall, IRFL provides a flexible, interpretable, and scalable tool for structural-change detection with broad applicability in time series and image analysis.

Abstract

This dissertation presents a general framework for changepoint detection based on L0 model selection. The core method, Iteratively Reweighted Fused Lasso (IRFL), improves upon the generalized lasso by adaptively reweighting penalties to enhance support recovery and minimize criteria such as the Bayesian Information Criterion (BIC). The approach allows for flexible modeling of seasonal patterns, linear and quadratic trends, and autoregressive dependence in the presence of changepoints. Simulation studies demonstrate that IRFL achieves accurate changepoint detection across a wide range of challenging scenarios, including those involving nuisance factors such as trends, seasonal patterns, and serially correlated errors. The framework is further extended to image data, where it enables edge-preserving denoising and segmentation, with applications spanning medical imaging and high-throughput plant phenotyping. Applications to real-world data demonstrate IRFL's utility. In particular, analysis of the Mauna Loa CO2 time series reveals changepoints that align with volcanic eruptions and ENSO events, yielding a more accurate trend decomposition than ordinary least squares. Overall, IRFL provides a robust, extensible tool for detecting structural change in complex data.

Changepoint Detection As Model Selection: A General Framework

TL;DR

Changepoint Detection As Model Selection delivers a general, information-criteria-driven framework for changepoint analysis built on L0-model selection. The centerpiece is Iteratively Reweighted Fused Lasso (IRFL), which adaptively reweights penalties within the generalized lasso to converge toward an -like solution and robustly recover true changepoints across mean shifts, trends, seasonality, and dependence structures. The dissertation unifies heuristic, exact, and adaptive segmentation under a single framework, demonstrates strong performance in extensive simulations, and validates the approach on real data such as Mauna Loa CO records and medical/imaging applications. Overall, IRFL provides a flexible, interpretable, and scalable tool for structural-change detection with broad applicability in time series and image analysis.

Abstract

This dissertation presents a general framework for changepoint detection based on L0 model selection. The core method, Iteratively Reweighted Fused Lasso (IRFL), improves upon the generalized lasso by adaptively reweighting penalties to enhance support recovery and minimize criteria such as the Bayesian Information Criterion (BIC). The approach allows for flexible modeling of seasonal patterns, linear and quadratic trends, and autoregressive dependence in the presence of changepoints. Simulation studies demonstrate that IRFL achieves accurate changepoint detection across a wide range of challenging scenarios, including those involving nuisance factors such as trends, seasonal patterns, and serially correlated errors. The framework is further extended to image data, where it enables edge-preserving denoising and segmentation, with applications spanning medical imaging and high-throughput plant phenotyping. Applications to real-world data demonstrate IRFL's utility. In particular, analysis of the Mauna Loa CO2 time series reveals changepoints that align with volcanic eruptions and ENSO events, yielding a more accurate trend decomposition than ordinary least squares. Overall, IRFL provides a robust, extensible tool for detecting structural change in complex data.
Paper Structure (40 sections, 8 theorems, 246 equations, 44 figures, 2 tables, 12 algorithms)

This paper contains 40 sections, 8 theorems, 246 equations, 44 figures, 2 tables, 12 algorithms.

Table of Contents

  1. Changepoint Problems
  2. Oil Drilling
  3. Global Warming Surge
  4. Global Warming Data with Autocorrelation
  5. Liver transplantation
  6. Model Selection and Objective Criteria
  7. Dissertation Organization
  8. Segmentation Algorithms: Heuristic and Exact Approaches
  9. Heuristic Methods
  10. Exact Methods for Segmentation
  11. Changepoint Detection via Model Selection
  12. The Lasso and Fused Lasso
  13. The Adaptive Lasso and Adaptive Fused Lasso
  14. Generalized Lasso (Genlasso)
  15. Motivation for IRFL
  16. ...and 25 more sections

Key Result

Proposition 1

Let $\mathcal{T}_s=\{\tau:1=\tau_0<\tau_1<\cdots<\tau_m<\tau_{m+1}=s\}$ denote the set of changepoint vectors for $y_{1:(s-1)}$, and define Then for any $s\ge 2$,

Figures (44)

  • Figure 1: Example plot illustrating changes in nuclear magnetic resonance and corresponding changepoints in rock strata. In this plot, shifts in NMR responses indicate variations in these properties, corresponding to distinct layers or changes in the rock strata, highlighted by the changing mean NMR response.
  • Figure 2: Example plot of a naïve linear regression of global mean land and ocean temperature.
  • Figure 3: Plot of residuals from linear model in Figure \ref{['fig:GGIS_trend']}.
  • Figure 4: By including a detected changepoint before running the regression, one can see that the estimate of trend radically changes.
  • Figure 5: There is speculation of a "surge" in the rate of global warming (a trend shift) at approximately the year 2015.
  • ...and 39 more figures

Theorems & Definitions (21)

  • Proposition 1: Optimal partition recursion
  • proof
  • Proposition 2: PELT Pruning Criterion killick-2012-pelt
  • proof
  • Remark
  • Proposition 3: CPOP recursion fearnhead-2019-cpop
  • proof
  • Proposition 4: Segment Neighbourhood Recursion Maidstone-2017-algorithms
  • proof
  • Proposition 5: Recursive Structure in pSN Rigaill-2010-PrunedSN
  • ...and 11 more