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Exact GLR changepoint detection is now fast enough to run online

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2026-08-01 11:10 UTC pith:MTVFGRNR

load-bearing objection A solid, useful software paper for the focus changepoint family; the abstract's unconditional 'exact, without approximations' claim is too broad because the high-dimensional projection mode is approximate. the 3 major comments →

arxiv 2607.19961 v1 pith:MTVFGRNR submitted 2026-07-22 stat.ME

focus and focus-cpt: Fast Online Changepoint Detection in R and Python

Gaetano Romano , Kes Ward , Yuntang Fan , Guillem Rigaill , Vincent Runge , Idris A. Eckley , Paul Fearnhead This is my paper
classification stat.ME MSC 62L1062M10
keywords changepoint detectiononline algorithmsGeneralised Likelihood Ratioconvex hull pruningexponential familyR packagePython packageautoregressive detection
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents focus and focus-cpt, R and Python packages that implement the focus family of online changepoint detection algorithms. The central claim is that the Generalised Likelihood Ratio statistic for a single changepoint can be computed exactly, with no approximation, while the per-iteration cost grows only logarithmically with the length of the observed sequence (log(n)^d for d-dimensional data). This is achieved by pruning the set of candidate changepoint locations to the vertices of a convex hull built from the data, because any candidate lying inside the hull can never be the maximizer. If correct, this means practitioners can run exact likelihood-ratio changepoint detection in real time on streams of counts, means, variances, and other exponential-family data, rather than resorting to approximate or costly methods.

Core claim

The paper establishes that the GLR statistic for a single changepoint can be maintained exactly in an online fashion by exploiting a convex-hull geometry: each candidate changepoint τ is mapped to a point P(τ) = (τ, Σ_{t≤τ} T(y_t)) in dimension d+1, and the log-likelihood difference for a change at τ is a scalar product of P(τ) with a model-dependent vector. Proposition 1 shows that if P(τ′) lies in the interior of the convex hull of all candidate points, then τ′ can never be the maximiser for any pre- and post-change parameters and can be permanently discarded. Hence the active candidate set is contained in the hull vertices, whose expected size grows only as O(log(n)^d). The package implem

What carries the argument

The key object is the geometric map from changepoint candidates to points P(τ) = (τ, cumulative sufficient statistics) in R^{d+1}, together with the convex hull of those points. The log-likelihood of a change at τ is the scalar product of P(τ) with a model-dependent vector ψ(θ0, θ1), so the argmax over τ is exactly the support point of the hull in the direction ψ. Pruning candidates that lie in the interior of the hull removes all points that cannot win, and the number of hull vertices bounds the per-iteration cost. This geometry is what carries the exactness and the logarithmic complexity claim.

Load-bearing premise

The claimed logarithmic per-iteration cost rests on the expected number of convex-hull vertices growing only as log(n)^d, and on the pruning rule correctly discarding every candidate whose point lies in the interior of the hull; if a stream produces a hull with linearly many vertices, or if the projection shortcut in high dimensions drops a vertex that carries the true change, the speed or exactness is lost.

What would settle it

Run the univariate detector on a deterministically convex sequence (e.g., y_t = exp(t)) and count the number of active candidates after each update; if the count grows linearly or per-step wall-clock time grows linearly, the log-cost claim is false for that stream. Separately, compare the full-hull and projection-based statistics on a high-dimensional stream with a change confined to dimensions not covered by the chosen projections; a large discrepancy in the statistic or a missed detection would show the approximation sacrifices exactness.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Real-time monitoring of counts, means, variances, and other exponential-family streams can now use exact GLR statistics at each time step instead of approximations, at logarithmic per-observation cost.
  • Offline threshold calibration becomes practical: focus_offline computes full statistic traces in one pass, enabling Monte Carlo-based false-positive-rate thresholds without rerunning the algorithm for every threshold.
  • Because R and Python share the same C++ backend, results are identical across interfaces, which reduces replication risk and makes the package a reliable common tool for practitioners.
  • For high-dimensional streams, the projection-based hull approximation offers a large speedup at the cost of a small difference in the statistic, as illustrated by the paper's comparison on six-dimensional data.
  • The AR(p) detector accounts for the transient residual behaviour after a change, preserving detection power near the end of the stream where naive pre-whitening would lose it.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The convex-hull pruning insight is not limited to GLR: any sequential test whose statistic is a maximum of linear functions over candidates could reuse the same hull maintenance to update its argmax exactly in sublinear time.
  • The projection-based approximation for high dimensions may miss changes whose signal is concentrated entirely in dimensions not jointly projected; the exposed dim_indexes argument turns this into a modelling choice, but the paper does not quantify worst-case power loss.
  • The lambda weighting mechanism generalises the detector to covariate-scaled rates (e.g., Poisson data with time-varying background), which suggests the same geometric pruning could support non-stationary exposure models without algorithmic change.
  • If a stream's convex hull grows linearly rather than logarithmically, the speed guarantee would degrade gracefully but not hold; a window-limited fallback or a size cap on the candidate set would be a natural robustness extension.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper describes an R and Python package (focus / focus-cpt) implementing the focus family of online changepoint detectors. The core algorithmic idea, imported from previous work by the same group, is to reduce the set of candidate changepoints to geometric extreme points of a data-dependent convex hull, so that the generalized likelihood ratio statistic can be evaluated exactly at a claimed per-iteration cost of roughly log(n)^d for a d-dimensional stream. The package covers one-parameter exponential family models (Gaussian, Poisson, Binomial, Exponential, Gamma), a nonparametric eCDF-based detector, and a detector for autoregressive data, and it provides both online and offline interfaces backed by a shared C++ implementation. The manuscript gives an algorithmic overview, a detailed interface description, and several simulated and real-data examples. The central advertised contribution is that the implementation delivers exact GLR statistics without approximations while remaining fast enough for real-time use.

Significance. If the claims are properly qualified, this is a useful software contribution: it unifies several methodological papers into one maintained package with R and Python interfaces, a shared C++ backend, and reproducible example code on GitHub. The package is available on CRAN and PyPI, which is a concrete strength. However, the headline exactness claim is currently overstated: the body of the paper explicitly introduces an approximate high-dimensional mode that does not compute the full exact GLR statistic. The complexity guarantee is also stated inconsistently and is inherited from an expected-size result rather than established here. These issues are fixable, but they affect the central claims of the paper and should be addressed before publication.

major comments (3)
  1. [Abstract; Section 2.3; Section 4.2] The abstract and Section 1 state that focus computes the GLR statistic 'exactly, without introducing approximations' at O(log(n)^d) cost. Section 2.3, however, introduces for d>5 a projection-based mode: data are projected onto overlapping subsets of d-tilde <= 5 coordinates, hulls are computed in each projection, and the union of the lower-dimensional hull vertices is used as the candidate set. A vertex of the full (d+1)-dimensional hull whose supporting hyperplane uses all coordinates need not be a vertex of any projection, so this candidate set can miss the GLR maximizer. Section 4.2 confirms the issue: all.equal reports a mean relative difference of 0.003782673 between the full-hull and projection-based statistics. Exactness therefore holds only for the full-hull mode; the abstract's unconditional 'without introducing approximations' claim is not supported as written and should be qu
  2. [Section 2.3; Section 1] The paper advertises two incompatible complexity statements. The abstract and Section 1 say the per-iteration cost is approximately log(n)^d, while Section 2.3 says the univariate pruning is amortised O(1); for d=1 these differ by a factor of log n. More importantly, the O(log(n)^d) growth is cited from Corollary 1 of Pishchagina et al. (2025) as an expected hull-size result, but the present paper gives no statement of the conditions under which this expectation holds. Because real-time performance is central to the paper's motivation, either reproduce the relevant bound with its assumptions or clearly state that the efficiency guarantee is inherited and is expected/average-case rather than worst-case.
  3. [Section 2.2, Proposition 1] Proposition 1 prunes only points in the interior of the convex hull, yet the text immediately concludes that D_n is contained in the set of hull vertices. This inference is not immediate: a point on a facet of the hull but not a vertex is not interior, and it can be an argmax of <P(tau), psi> only when the same maximum value is also attained by a hull vertex. To support the stated exactness for general exponential-family models, the paper should either prove that omitting boundary non-vertices cannot change the value of max_tau q_tau, or specify that the algorithm retains such points or resolves ties in a way that preserves the maximum. As written, the proposition alone does not establish the claimed reduction to vertices.
minor comments (5)
  1. [Section 1] Typo: 'unvariate' should be 'univariate'.
  2. [Section 4.1] Only 500 Monte Carlo replications are used to estimate a 99th percentile threshold. This gives a noisy estimate; please report a standard error or use more replications.
  3. [Section 4.2] The text says the projection-based approximation produces 'extremely similar results' despite a reported mean relative difference of 0.0038. Please quantify this more carefully, e.g., with the maximum relative difference and the effect on detection times, since the difference is exactly the kind of approximation error the abstract denies.
  4. [Figure 6] The x-axis of Figure 6 appears to be game index, but the text refers to '640 games' and later to 1000; please clarify the sample size and axis labeling.
  5. [Appendix A] The Python and R examples intentionally produce different stopping times due to different RNGs. This is fine, but the discrepancy may confuse readers; consider a sentence noting that the algorithms are identical and the difference is purely the random seed/stream.

Circularity Check

0 steps flagged

No significant circularity: the derivation is either self-contained or rests on independently published geometry results, and the projection-mode caveat is an accuracy limitation rather than a circular step.

full rationale

I walked the claimed derivation chain. The paper's core contribution is an implementation; it does not attempt a from-scratch proof of the focus/GLR geometry. Section 2.1 defines the GLR statistic by standard likelihood maximization. Proposition 1 then gives a self-contained convex-hull pruning proof: any interior point P(tau') is dominated for every (theta0, theta1) by a convex combination of hull points. The logarithmic per-iteration cost is inherited from Corollary 1 of Pishchagina et al. (2025), an externally published theoretical bound. Citing that bound, even with overlapping authorship, is legitimate evidence rather than circular reasoning, because the bound is not assumed true by fitting it here and is not equivalent to the paper's own outputs. No model parameter is fitted to a subset of the target data and then renamed a prediction; the Monte Carlo thresholds in Section 4.1 are calibration/operating parameters, not predicted GLR values. The only tension is that the abstract states that the algorithms compute GLR 'exactly ... without introducing approximations,' while Section 2.3 explicitly describes an 'approximate algorithm' for d>5 using projections and Section 4.2 reports a mean relative difference of 0.003782673 between full-hull and projection-based statistics. This is an internal consistency/scope-of-claim concern, not a circularity: the exact claim is restricted to the full-hull mode, and the high-dimensional mode is openly labeled approximate. I found no self-definitional reduction, no fitted-input-called-prediction, no uniqueness theorem imported to force the answer, and no ansatz smuggled in by self-citation. Therefore the paper receives a circularity score of 0.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claim leans on standard convex geometry and on results inherited from earlier focus papers; the only hand-tuned numbers are user-facing defaults and example thresholds. No new entities are introduced.

free parameters (4)
  • pruning_mult = 2 (default)
    User-controlled multiplier for QuickHull trigger threshold; chosen 'in practice' (§2.3) and not derived; affects candidate set size and runtime, but not the central claim.
  • pruning_offset = 1 (default)
    User-controlled offset for QuickHull trigger; same as above.
  • Monte Carlo threshold = 31.221 in §4.1; 32.796 in §5.1
    Thresholds calibrated by simulation quantiles to control false-positive rate in examples; not part of algorithm derivation.
  • anomaly_intensity = NULL (disabled); 1.5 in examples
    User-set minimum signal intensity to suppress baseline drift; not fitted, but a hand-picked knob for anomaly mode.
axioms (5)
  • standard math A point in the interior of a convex hull can be written as a convex combination of d+2 hull points, so it cannot maximize a linear functional.
    Used in proof of Proposition 1 (Section 2.2).
  • domain assumption Data are generated i.i.d. from a one-parameter exponential family before and after the change, with independence across dimensions.
    The GLR and log^d complexity claims are for this model class; violations (e.g., dependence, non-exponential family) are outside the core claim.
  • domain assumption The expected number of convex hull vertices of the P(τ) random walk is O(log(n)^d), as stated in Corollary 1 of Pishchagina et al. (2025).
    This is the source of the per-iteration complexity claim; the paper does not reprove it.
  • domain assumption Pruning never discards a candidate that can be the GLR maximizer: D_n ⊆ D_{n-1} ∪ {n} and vertices of the hull suffice.
    Central to exactness; proof is sketched in Proposition 1 but inherited from prior work.
  • domain assumption For the AR(p) detector, AR coefficients and residual variance are known or estimated over a training period; Fan et al. (2026) provides the extension.
    Section 2.4 and 4.5; if coefficients are mis-estimated, the GLR is not exact for the true model.

pith-pipeline@v1.3.0-alltime-deepseek · 25094 in / 11296 out tokens · 103928 ms · 2026-08-01T11:10:16.064494+00:00 · methodology

0 comments
read the original abstract

We present an R and Python package for fast online changepoint detection in univariate and multivariate data streams for a variety of models. The package implements the focus family of algorithms, which compute the Generalised Likelihood Ratio test for a single changepoint exactly and efficiently, with a per-iteration cost of approximately $\log(n)^d$ for a d-dimensional sequence, without introducing approximations. This is achieved by exploiting a connection between the location of the changepoint candidates and the geometry of the data. The package supports a broad range of models from the natural exponential family, including Gaussian, Poisson, Binomial, Exponential and Gamma distributions, as well as a non-parametric detector based on the empirical cumulative distribution function and a detector for autoregressive data.

Figures

Figures reproduced from arXiv: 2607.19961 by Gaetano Romano, Guillem Rigaill, Idris A. Eckley, Kes Ward, Paul Fearnhead, Vincent Runge, Yuntang Fan.

Figure 1
Figure 1. Figure 1: The trace of the statistics for one simulated time series using the R [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The trace of the statistics for one simulated 3-dimensional multivariate time series [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of statistics with and without minimum change size under a slowly [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of sum and max statistics for two different change scenarios. Left: [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The AR(2) process with mean shift and the resulting detection statistic using the [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The NBA plus-minustest data (top) and the resulting test statistics (bottom). [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Online detection of a gamma-ray burst from Fermi-GBM data. Top: radiation [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The trace of fluorescence intensity over time from a calcium imaging recording of [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The trace of the statistics for one simulated time series using the Python interface. [PITH_FULL_IMAGE:figures/full_fig_p038_9.png] view at source ↗

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