Pith. sign in

REVIEW 2 major objections 5 minor 4 references

Akaike information criterion for segmented regression models

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For segmented regression, the AIC penalty is 2 per continuous change-point and 6 per discontinuous one, derived from the original Kullback-Leibler definition of AIC.

desk verdict The paper's theoretical AIC derivation for continuous and discontinuous change-points is real, but the simulations are sloppy and the scope of the 6 penalty is overstated beyond the shrinking-jump regime. read the letter →

arxiv 2506.08760 v1 pith:7GO5PYTN submitted 2025-06-10 stat.ME

Kazuki Nakajima , Yoshiyuki Ninomiya This is my paper
classification stat.ME MSC 62J0262F12
keywords AICsegmentedregressionchange-pointanalysisjoinpointmodelselectionpiecewiseBrownianmotionasymptotictheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper derives an Akaike information criterion for segmented regression models from the original definition of AIC: the criterion estimates the Kullback-Leibler divergence between the true distribution and the fitted model by adding an asymptotic bias correction to the maximized log-likelihood. The central result is that the bias equals $p(m+1)$ when the regression function is continuous at the change-points and $3m+p(m+1)$ when it is discontinuous, giving penalties $2p(m+1)$ and $6m+2p(m+1)$. In concrete terms, a continuous change-point costs the same as one ordinary parameter, 2, while a discontinuous change-point costs 6. This matters because only BIC-type criteria with model-selection consistency have been available, and those impose the same penalty on both kinds of change-points; the authors show that their AIC tends to produce smaller divergence than BIC in simulations, and that on UK COVID-19 new-infection data the continuous-versus-discontinuous choice changes the estimated trend.

What carries the argument

The machinery is the asymptotic-bias identity $c_{\mathrm{limit}} = \mathrm{dlim}_{n\to\infty}\{\sum_i g(\hat\xi;x_i,y_i)-\sum_i g(\hat\xi;\tilde x_i,\tilde y_i)\}$, the difference between the maximized log-likelihood and the same quantity at an independent copy, whose expectation supplies the AIC penalty. In the continuous case the work is done by the asymptotic normality of $\hat\theta$ and $\hat\tau$, which makes each change-point estimator a linear function of the score. In the discontinuous case the work is done by the two-sided random walk $Q^{(k)}(\tau^{(k)})$ with negative drift; under Assumption 2 this converges to a two-sided Brownian motion $\sigma W_s-\sigma^2|s|/2$, a Gaussian process with independent increments and variance $|s|$ on both sides of zero. The expected value of $\sup_s V(s)-\tilde V(\arg\sup_s V(s))$ is 3 for each change-point, and the factor 2 in AIC doubles it to the penalty 6.

What would settle it

Simulate discontinuous segmented regression with fixed, non-shrinking jump sizes over a grid of sample sizes and estimate the exact bias $E[c_{\mathrm{limit}}]$ by Monte Carlo: under the paper's Assumption 2 the bias should approach $3m+p(m+1)$, so any systematic departure that grows with jump magnitude would falsify the use of penalty 6 outside the local-asymptotics regime.

Watch

Extended reading notes

Core claim

Under Assumption 1, for a generalized linear segmented model, the maximum likelihood estimators are consistent; when the regression function is continuous at the breakpoints, $\sqrt{n}(\hat\theta-\theta^*)$ and $\sqrt{n}(\hat\tau-\tau^*)$ are asymptotically normal, so the change-point parameters behave like regular parameters and the asymptotic bias of the maximum log-likelihood is $p(m+1)$. When the regression function is discontinuous, the change-point estimators instead converge through a two-sided random walk, and under Assumption 2—true jump sizes of order $\alpha_n^{-1/2}$ with $\alpha_n=o(n)$—the bias becomes $3m+p(m+1)$. The proposed criteria are therefore $-2\log L(\hat\xi)+2p(m+1)$ for continuous segmented regression and $-2\log L(\hat\xi)+6m+2p(m+1)$ for discontinuous segmented regression; for a mixture of $m_1$ continuous and $m_2$ discontinuous change-points the penalty is $2p(m+1)+6m_2$.

Load-bearing premise

The discontinuous-case penalty of 6 rests on the assumption that the true jump sizes shrink with the sample size; if real-world jumps stay fixed in size, the same AIC correction may not minimize Kullback-Leibler divergence.

Editorial extensions

If this is right

  • In continuous joinpoint regression, the standard AIC penalty formula $2p(m+1)$ is valid, so treating change-point locations as ordinary parameters is justified when the goal is minimizing predictive divergence.
  • In discontinuous segmented regression, applying the ordinary penalty 2 to each change-point will overfit; the correct per-jump penalty is 6, so naive AIC is systematically too liberal.
  • Using the extended penalty $2p(m+1)+6m_2$, model selection can simultaneously determine the number of change-points and whether each one is continuous or discontinuous.
  • Because BIC-type criteria target consistency rather than divergence, AIC and BIC will sometimes select different models; the paper's COVID-19 analysis shows such disagreement on real data.
  • The gap between the continuous penalty 2 and discontinuous penalty 6 means that deciding whether a structural break is a jump or a smooth slope change is itself a model-selection problem with a well-defined AIC answer.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same Brownian-argmax mechanism carries other non-regular parameters selected by optimizing over a nuisance domain—threshold indicators, knot locations, regime switches—those parameters should also carry penalties larger than 2 when the target is predictive divergence.
  • The paper's own Table 1 shows the discontinuous bias approximation degrades as jump sizes grow, so a testable extension is a jump-size-dependent penalty that reduces to 6 under local asymptotics but grows for fixed, large jumps.
  • A practical implication the authors do not spell out: trend-surveillance software should report both the continuous and discontinuous AIC values before declaring a structural break, because the 6-versus-2 difference changes what counts as evidence.
  • The same derivation is likely to extend to partial-likelihood and survival settings, but that extension is not established by this paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper derives Akaike information criteria for segmented regression models in two settings: regression functions that are continuous at the change-points and those that are discontinuous. In the continuous case, the authors prove that the change-point estimator is asymptotically normal and that the AIC penalty for each change-point is 2, so the total penalty is 2p(m+1). In the discontinuous case, following the change-point asymptotics of Ninomiya (2015), they show under a local shrinking-jump assumption (Assumption 2) that the per-change-point penalty is 6, giving a total penalty of 6m+2p(m+1). The paper presents Monte Carlo experiments and a real-data analysis on COVID-19 infection counts, arguing that the proposed AIC yields lower Kullback-Leibler divergence than BIC and than a naive AIC, and that model selection can differ between continuous and discontinuous formulations.

Significance. If the theoretical results hold, the paper makes a substantive contribution by providing a KL-optimal model selection criterion for segmented regression, extending earlier work on change-point AIC to generalized linear and nonlinear piecewise-continuous models. The distinction between penalties of 2 and 6 for continuous versus discontinuous change-points is practically important for fields such as epidemiology, where both types of models are used. A strength of the paper is that the penalties are derived, not tuned: the proofs of the key bias evaluations (Theorems 3 and 5) are given in the text and appendix, and the asymptotic normality for the continuous case is extended beyond the earlier linear-setting results. However, the numerical validation has a serious flaw in the continuous-case simulation design, and the scope of the discontinuous-case penalty is narrower than the abstract suggests.

major comments (2)
  1. [Section 5, Tables 1, 2, and 4] The true parameter vectors used for the 'continuous case' simulations do not satisfy the continuity condition at the change-points. For example, Table 2 lists τ* = 0.5 and θ* = (0, 0, 2.5, -1.25); at x = 0.5 the left-hand limit is 0 and the right-hand limit is 2.5 - 1.25·0.5 = 1.875, so the generated data are discontinuous. The same inconsistency appears in the continuous columns of Table 1 and throughout Table 4. Thus the numerical experiments labeled 'continuous' are actually fitting a discontinuous model and cannot validate the continuous-case AIC (9). The authors must correct the true parameter vectors (for instance by imposing θ1^{[k+1]} = θ1^{[k]} + (θ2^{[k]} - θ2^{[k+1]}) τ^{[k]*}) and rerun the continuous-case experiments.
  2. [Section 4, Theorem 5 and Assumption 2; Section 7] The penalty of 6 per discontinuous change-point is derived under the local asymptotics of Assumption 2, where the true jump size shrinks as α_n^{-1/2} with α_n = o(n). For fixed jump magnitudes, the asymptotic bias of the change-point estimator is not a universal constant; it depends on the jump size and error distribution. The numerical experiments in Section 5 use fixed jumps, so they do not test the 6 penalty in the regime in which it was derived. The abstract and Section 7 present 6 as 'the' penalty for discontinuous segmented regression without this caveat. I recommend qualifying the claim explicitly and either adding simulations under shrinking-jump designs or providing a heuristic or additional argument for the fixed-jump setting.
minor comments (5)
  1. [Section 5, first paragraph] The stated penalty for continuous-case AIC naive, 8m+4, appears inconsistent with the model's parameter dimension p=2: assigning a penalty of 6 to each of m change-points plus 2p(m+1) for the regression parameters gives 4(m+1)+6m = 10m+4. Please check the formula and the values used in the tables.
  2. [Table 3] The two rows for n=300 with θ* = (0,0,1.2,0) and θ* = (0,0,1.8,0) are identical to the corresponding n=100 rows, which is likely a copy-paste error; the correct n=300 results should be reported.
  3. [Section 4, proof of Theorem 5] The evaluation E[sup_s V(s) - Ṽ(argmax_s V(s))] = 3/2 is cited from Ninomiya (2015) but not reproduced; including this calculation in the appendix would make the paper more self-contained, particularly because the definition of the bias differs slightly from that reference.
  4. [Throughout] There are several typographical errors: 'AIC navie' in Section 6, 'where where' near equation (10), and inconsistent spelling of 'naive'.
  5. [Section 6] The statement that 'it is more natural that they should be continuous unless there is some reason' is an interpretive remark; consider clarifying that this is a subject-matter preference, not a statistical conclusion from the model comparison.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the AIC penalties are derived from asymptotic bias calculations, with only a minor same-author citation for a standard Brownian-motion expectation.

full rationale

The paper's central penalties are derived, not fitted. In the continuous case, Theorem 3 proves E[climit] = p(m+1) from the asymptotic normality of the estimators, and the penalty 2p(m+1) follows from E[(w - w_tilde)^T T^{-1} w] = p(m+1); no constant is tuned to data. In the discontinuous case, Theorem 4 derives E[climit] = sum_k E[sup_tau Q^(k)(tau) - Q_tilde^(k)(argsup_tau Q^(k)(tau))] + p(m+1) with proofs given in the appendix, and Theorem 5 invokes Assumption 2 to pass to a two-sided Brownian motion functional. The only imported step is the final expectation E[sup_s V^(k)(s)] = E[-V_tilde^(k)(argsup_s V^(k)(s))] = 3/2, which the paper takes from Ninomiya (2015). That is a same-author citation, but it is a parameter-free, externally published mathematical calculation about two-sided Brownian motion, not a result that depends on the present paper's fitted values or assumptions; the surrounding asymptotic reduction is carried out in this manuscript. The numerical experiments are checks rather than calibrations: Table 1 compares Monte Carlo bias with the theoretical penalty, and Tables 2-5 compare KL divergence and selection rates without fitting the penalty. The scope limitation concerning Assumption 2 (shrinking jumps) versus the fixed jumps used in simulations is a correctness/domain-of-validity concern, not a circularity concern. Therefore the derivation chain is essentially self-contained, with one minor same-author citation that is not load-bearing enough to constitute circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The derivation rests on standard GLM regularity, the change-point asymptotics in Assumptions 1 and 2, and the Brownian-motion result for the two-sided random walk. No parameters are fitted to data; the penalties are derived analytically.

assumptions (6)
  • domain assumption Assumption 1: in a neighborhood of each true change-point, x has continuous positive density, and either the regression function or its derivative is discontinuous at the change-point.
    Needed for consistency and for the continuous-case asymptotics; quoted from Section 2.
  • domain assumption Assumption 2: true jump sizes shrink as theta[k+1]* - theta[k]* = alpha_n^{-1/2} Delta[k]_{theta*} with alpha_n = o(n).
    Used in Section 4 to turn the random-walk bias into the constant 3 per change-point; without it the penalty is not a simple constant.
  • domain assumption The model family is correctly specified: the true distribution is of the form (2) for some xi*.
    The AIC derivation is a bias correction of the maximum log-likelihood under the model; the paper does not address misspecification.
  • standard math Regularity of the GLM: a, b, phi are C^2 and the parameter space is compact, with a'' bounded away from zero.
    Used in the proof of Theorem 1 and Lemma 1; stated in Section 2.
  • standard math Empirical process machinery (Lemma 19.34 of van der Vaart) for uniform convergence of the score processes.
    Invoked in the proofs of Lemmas 1 and 2.
  • standard math Brownian motion scaling and the identity E[sup_s V(s)] = E[-V(argmax)] = 3/2 for two-sided Brownian motion with linear negative drift.
    Used in the proof of Theorem 5, following Ninomiya (2015).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Akaike information criterion for segmented regression models." pith.science (2026). https://pith.science/paper/7GO5PYTN

@misc{pith2026250608760,
  author       = {Pith},
  title        = {Pith review of: Akaike information criterion for segmented regression models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7GO5PYTN}},
  note         = {Machine review of arXiv:2506.08760}
}
read the original abstract

In segmented regression, when the regression function is continuous at the change-points that are the boundaries of the segments, it is also called joinpoint regression, and the analysis package developed by \cite{KimFFM00} has become a standard tool for analyzing trends in longitudinal data in the field of epidemiology. In addition, it is sometimes natural to expect the regression function to be discontinuous at the change-points, and in the field of epidemiology, this model is used in \cite{JiaZS22}, which is considered important due to the analysis of COVID-19 data. On the other hand, model selection is also indispensable in segmented regression, including the estimation of the number of change-points; however, it can be said that only BIC-type information criteria have been developed. In this paper, we derive an information criterion based on the original definition of AIC, aiming to minimize the divergence between the true structure and the estimated structure. Then, using the statistical asymptotic theory specific to the segmented regression, we confirm that the penalty for the change-point parameter is 6 in the discontinuous case. On the other hand, in the continuous case, we show that the penalty for the change-point parameter remains 2 despite the rapid change in the derivative coefficients. Through numerical experiments, we observe that our AIC tends to reduce the divergence compared to BIC. In addition, through analyzing the same real data as in \cite{JiaZS22}, we find that the selection between continuous and discontinuous using our AIC yields new insights and that our AIC and BIC may yield different results.

Figures

Figures reproduced from arXiv: 2506.08760 by the authors.

Figure 1
Figure 1. Results obtained by allowing the regression function to be continuous at the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Results obtained by allowing the regression function to be continuous at the [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

  1. [1]

    Acharya, J., Diakonikolas, I., Li, J., and Schmidt, L. (2016). Fast algorithms for segmented regression, InInternational Conference on Machine Learning, PMLR, 2878–2886. Aue, A., H¨ ormann, S., Horvath, L., and Reimherr, M. (2009). Break detection in the covariance structure of multivariate time series models,The Annals of Statistics,37, 4046–4087. Aue, A...

    work page 2016

  2. [464]

    Siahkamari, A., Gangrade, A., Kulis, B., and Saligrama, V. (2020). Piecewise linear regression via a difference of convex functions, InInternational Conference on Machine Learning, PMLR, 8895–8904. van der Vaart, A. W. (2000).Asymptotic Statistics: Cambridge University Press. Wood, S. N., Pya, N., and S¨ afken, B. (2016). Smoothing parameter and model sel...

    work page 2020

  3. [597]

    Kim, H.-J., Chen, H.-S., Byrne, J., Wheeler, B., and Feuer, E. J. (2022). Twenty years since Joinpoint 1.0: Two major enhancements, their justification, and impact,Statistics in Medicine,41, 3102–3130. Kim, J. and Kim, H.-J. (2008). Asymptotic results in segmented multiple regression, Journal of Multivariate Analysis,99, 2016–2038. (2016). Consistent mode...

    work page 2022

  4. [1563]

    (1992).Asymptotic inference for segmented regression models: PhD Thesis, Uni- versity of British Columbia

    Wu, S. (1992).Asymptotic inference for segmented regression models: PhD Thesis, Uni- versity of British Columbia. Zhang, N. R. and Siegmund, D. O. (2007). A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data,Biometrics,63, 22–32. 30

    work page 1992

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.