Uniform confidence bands for joint angles across different fatigue phases

Holger Dette; Patrick Bastian; Rupsa Basu

REVIEW 5 major objections 6 minor 11 references

Uniform confidence bands for joint angles across different fatigue phases

T0 review · 5 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper constructs simultaneous uniform confidence bands for the mean curves of the segments between relevant change points in a functional time series, using a multiplier bootstrap that remains valid when the change locations and…

desk verdict A useful but under-proved extension of Dette et al. (2020): the bootstrap bands are plausible, but the main theorem relies on unstated rates and a separation condition that the proof sketch does not supply. read the letter →

arxiv 2502.08430 v1 pith:ENMWEH2A submitted 2025-02-12 stat.ME

Patrick Bastian , Rupsa Basu , Holger Dette This is my paper · ORCID

classification stat.ME MSC 62G1562G2062M1062R10

keywords uniformconfidencebandsfunctionaltimeserieschangepointdetectionmultiplierbootstraplongrunvariancefatiguebiomechanicsrangeofmotionkneeangledata

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 extends post-hoc change point analysis for functional time series to confidence statements. It constructs uniform confidence bands for the segment mean curve of each phase where the change in the mean exceeds a threshold $\Delta$, so that all relevant segments are covered simultaneously with asymptotic probability at least $1-\alpha$ in the generic case. The motivation is biomechanical: locating fatigue states in a runner's lower-extremity joint angles and then describing the range of motion within each state. The bands are built from a multiplier bootstrap on block sums, after using existing estimators for change points and the relevant change set. If the asymptotic result holds, practitioners can compare rest, pre-fatigue and fatigued movement patterns with simultaneous statistical guarantees.

What carries the argument

The load-bearing mechanism is a multiplier bootstrap for the studentized statistic $T_n = \max_{i\in\hat I} \sqrt{\hat n_i}\, \|(\hat\mu_i - \mu_i)/\hat\sigma\|_\infty$, where $\hat I$ is the estimated set of relevant change points. Bootstrap copies $\hat\mu_i^*$ are formed by multiplying block sums of centered residuals $Y_{n,j} = X_{n,j} - \hat\mu_i$ by independent standard normals, then normalizing by the long-run variance estimator $\hat\sigma^2$ of (15); the $(1-\alpha)$ quantile of these copies sets the band width in (11). The construction inherits its validity from the preliminary estimators: the proof uses consistency of the change point and relevant-set estimators to replace estimated segments by true segments, and consistency of $\hat\sigma^2$ to replace $\hat\sigma$ by $\sigma$. The block length $L$ and bandwidth $c$ are tuning parameters.

What would settle it

Simulate functional time series that satisfy the assumed conditions, with known change points and one jump size exactly equal to $\Delta$, and compute empirical coverage of the bands over many replications; Theorem 1 predicts coverage at least $1-\alpha-\beta$ in that case and exactly $1-\alpha$ when no jump equals $\Delta$, so a systematic shortfall would refute the claim.

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Extended reading notes

Core claim

Theorem 1 is the central claim: under Conditions (A1)--(A4) of Dette and Kokot (2022), a consistent long-run variance estimator, and consistent estimators of the change points and relevant set from Bastian et al. (2024), the bands in (11) satisfy $\liminf_n P(\cap_{i\in I} \{\forall t: \mu_i^-(t)\le \mu_i(t)\le \mu_i^+(t)\}) \ge 1-\alpha-\beta$, with equality to $1-\alpha$ when no true jump size equals $\Delta$. The bootstrap quantile of the studentized sup-statistic $T_n$, taken over all estimated relevant segments, therefore produces simultaneous bands rather than pointwise ones. The proof replaces estimated change points and estimated relevant indices by their true values, then uses weak invariance principles and the consistency of the variance estimator to show that the bootstrap copies converge jointly with $T_n$.

Load-bearing premise

The bands inherit their validity from the unstated conditions (A1)--(A4) of Dette and Kokot (2022) and from fast-enough consistency of the change point and relevant-set estimators; if those assumptions fail, the promised coverage is not guaranteed.

Editorial extensions

If this is right

The bands in (11) give simultaneous, asymptotically valid coverage over all relevant fatigue segments of one runner, so statements about the range of motion in rest versus fatigued phases hold jointly rather than curve-by-curve.
If no observed jump equals the threshold $\Delta$, the asymptotic coverage equals $1-\alpha$; if one jump equals $\Delta$, coverage stays at least $1-\alpha-\beta$, accounting for the preliminary selection of relevant changes.
For the knee-angle data, the bands separate rest, pre-fatigue and fatigue movement patterns, and the fatigue phase shows reduced knee bending at the second peak, matching the protection-mechanism interpretation in the biomechanics literature.
The method turns the biomechanical question into a three-step pipeline---detect changes, keep those with mean jump above $\Delta$, build bands---so the same recipe applies to hip and ankle angles or to other athletes.

Reading between the lines

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

Because the bands are simultaneous over relevant segments, one could invert them to obtain simultaneous confidence intervals for the difference $\mu_i - \mu_0$ between fatigue phases, directly testing whether range of motion changes at each point of the stride.
The data example chooses $\Delta$ from the first and last 5% of the run, which makes the target parameter data-dependent; a rigorous extension would account for that selection in the coverage statement.
For real-time monitoring, one could recompute the bands after each new block of strides; the asymptotic conditions leave open how large $n$ and the block length $L$ must be for nominal coverage in finite samples.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper does something sensible: it takes the uniform-band machinery from Dette et al. (2020) for a single segment and extends it to the multiple-segment case after change point detection, with a relevance threshold. That is a legitimate incremental contribution, and it is clearly motivated by the biomechanics application. The writing is honest about leaning on earlier work, and the data examples are illustrative without being oversold.

The soft spot is exactly where the stress-test points. Theorem 1 is the whole paper, and its proof is a three-step sketch. Step 1 replaces estimated change points by true ones in the definitions of nhat_i and muhat_i. That is only valid if the change point estimators are accurate enough, e.g. fractional error o_P(n^{-1/2}); the paper cites Theorem 4.1 of Bastian et al. (2024) but does not state the rate it supplies, nor the Conditions (A1)-(A4) it needs. Without that rate, the substitution can leave a bias that does not vanish. Step 2 claims Ihat = I with probability 1-o(1), but the only condition stated is that no jump size equals Delta exactly. In the triangular-array model, jump sizes may depend on n, so a jump of size Delta + n^{-1/2} is never exactly Delta, yet the relevant-set estimator may not stabilize. Equality in (13) therefore does not follow from the stated assumptions. This is a real gap, not a manufactured one.

I also share the reader's note that there are no simulations or code, which is a genuine omission for an inferential method. The consistency lemma for the long-run variance is also proved only up to a reference to a supplement, though that part is likely fine. The application uses a data-dependent Delta, which is reasonable in context but further obscures the separation condition.

All of this is addressable. The core idea is plausible and the paper is a clear, readable contribution for statisticians working on functional time series and change point analysis, and for biomechanists who want simultaneous bands. But as written, the coverage guarantee is not established. A serious referee should not desk-reject it; they should send it back for a completed proof, stated conditions, and a simulation study.

Referee Report

5 major / 6 minor

Summary. The paper proposes uniform confidence bands for segment means of a functional time series after multiple change point detection, targeting only segments whose adjacent mean jumps exceed a threshold Δ. The construction combines three ingredients: change point estimates and an estimated relevant-change set from Bastian et al. (2024), a long-run variance estimator of functional type, and a multiplier block bootstrap. The central theoretical statement, Theorem 1, claims that the bands defined in (11) have asymptotic simultaneous coverage at least 1−α−β, and exactly 1−α when no jump size equals Δ. The proof is only a three-step sketch that invokes external results from Bastian et al. (2024), Dette and Kokot (2022), and Dette et al. (2020). An application to knee-angle data of runners under fatigue illustrates the method. The paper is a short note: the motivation is clear and the idea is plausible, but the theorem as stated and proved is not currently verifiable from the manuscript.

Significance. If the claimed coverage result is correct, the paper would provide a practically useful post-hoc inference tool for biomechanical fatigue analysis, extending the change point methodology of Bastian et al. (2024) to simultaneous confidence statements about segment means. The combination of change point selection with bootstrap inference is of genuine interest, and the real-data examples demonstrate a relevant application. However, the significance is currently limited by the absence of a complete, self-contained proof: the central theorem depends on unstated conditions and unstated localization rates, and there is no simulation study or finite-sample check. The application is illustrative rather than a validation of the coverage property.

major comments (5)

[§2, proof of Theorem 1, Step 1] The proof replaces estimated change points by true change points in the definitions of n_hat_i and mu_hat_i, but no localization rate is stated. For the replacement to be asymptotically negligible, one needs max_i |s_hat_i − s_i| = o_P(n^{−1/2}); otherwise the misallocated boundary block contributes a bias of order sqrt(n) |s_hat_i − s_i| times the jump size to sqrt(n_hat_i)(mu_hat_i − mu_i). The citation to Theorem 4.1 of Bastian et al. (2024) is insufficient unless that theorem is stated together with the rate it provides and the conditions under which it holds. This is load-bearing because Step 1 is the first step of the proof of (12) and (13).
[§2, Theorem 1, condition for (13)] The condition that ||mu_{i+1} − mu_i||_∞ is not equal to Delta for all i does not suffice for the equality statement (13) under the triangular-array model (1)–(2). Because the segment means may depend on n, a jump of size Delta + n^{−1/2} is not equal to Delta for any n, yet the relevant-change estimator cannot stabilize, so P(I_hat = I) need not tend to 1. The proof's Step 2 therefore needs a separation condition, for example all relevant jumps at least Delta + ε_n and all non-relevant jumps at most Delta − ε_n with ε_n large enough for consistent detection, or an explicit assumption that the mu_i are fixed and hence a positive gap exists.
[§2, Theorem 1 and Lemma 1] The paper repeatedly invokes Conditions (A1)–(A4) of Dette and Kokot (2022) without stating them, and the proof of Lemma 1 dismisses the effect of estimated means by saying that the general result follows by straightforward approximation arguments. Since Theorem 1 and Lemma 1 are the theoretical core, the manuscript should either reproduce these conditions or state the precise assumptions on the noise process epsilon, including the mixing and moment conditions and the sense in which the process is valued in C[0,1]. The approximation step in Lemma 1 also requires the same change point localization rates as Step 1 and should be spelled out.
[§2, eqs. (6), (7), (11) and Algorithm 1] There is an internal inconsistency in the critical value used for the bands. The statistic T_n in (6) and its bootstrap version T*_n in (10) are sup-norm statistics, so the nominal (1−α) simultaneous bands should use the (1−α) quantile of T_n. Equations (7) and (11), however, use q_{1−α/2} (and (11) uses the bootstrap version q_hat*_{1−α/2}), while Algorithm 1 line 11 computes q_hat*_{1−α}. With the quantile as written in (11), the equality in (13) does not follow; the coverage would be at a different level or only an inequality. Please reconcile the definition of the band and the quantile.
[§2, Theorem 1 statement] The event in (12) and (13) is written as the intersection over all i in I of events involving mu_hat_i^pm, but (11) defines mu_hat_i^pm only for i in I_hat. If I_hat differs from I, the probability statement is not formally well-defined. The bands should be defined for all candidate indices (with arbitrary values outside I_hat) or the intersection should be over I ∩ I_hat, with the negligible difference handled explicitly in the proof.

minor comments (6)

[Algorithm 1] In line 1, the set of estimated change points is written as {s_hat_1, ..., s_hat_k_hat}, but the estimated number of change points is denoted m_hat elsewhere; the notation should be consistent.
[Algorithm 1, line 8] The sum in line 8 runs to floor(n s_hat_{i+1}), whereas (8) and (9) use floor(n s_hat_{i+1}) − 1; the algorithm and the formula should agree.
[§2, eq. (7)] Equation (7) uses sqrt(n) in the denominator while the final bands in (11) use sqrt(n_hat_i); since (7) is not used in the theorem, either align it with (11) or remove it to avoid confusion.
[Throughout] There are several typographical errors: 'theses results' should be 'these results', 'chnage' should be 'change', and the sentence 'In this paper will explicate' is missing a subject.
[§2, eq. (4)] The set I includes 0 for notational convenience, but mu_0 is not otherwise defined before its use in the expression mu_1 − mu_0; please define mu_0 explicitly.
[§3] The paper contains no simulation study or finite-sample assessment of the coverage of the proposed bands; at least a small Monte Carlo experiment matching the model in (1)–(2) would substantially strengthen the practical claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bootstrap bands rely on standard invariance principles and on published consistency results for change-point estimators; the data-driven threshold is a selection device, not an input that reproduces the output.

full rationale

The claimed derivation chain is: estimate change points and the relevant set via Bastian et al. (2024), form segment means, standardize by a consistent long-run variance estimator, and calibrate a simultaneous quantile by a multiplier bootstrap. Theorem 1 is conditional on Conditions (A1)-(A4) of Dette and Kokot (2022) and on the consistency of the change-point and relevant-set estimators (Theorems 4.1 and 4.2 of Bastian et al. (2024)). These cited results are published, externally checkable, and not derived from the present paper's bands; citing them is normal mathematical dependency, not circularity. The application chooses Delta from the first and last 5% of the data, which makes the selected segment set data-dependent in practice, but Theorem 1 treats Delta as fixed, and the bands themselves are not constructed from Delta. The proof is explicitly a sketch and omits the rate conditions needed for replacing estimated change points by true values, which is a rigor gap concerning coverage rather than an instance of a result reducing to its own input. No equation in the paper is equivalent by construction to an input, and no fitted parameter is renamed as a prediction. Therefore the paper is not circular.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on unstated regularity conditions (A1-A4) from Dette and Kokot (2022), on the piecewise-constant mean model, and on the consistency of change point estimators from Bastian et al. (2024). No new physical or statistical entities are introduced. Tuning parameters Delta, L, c, and beta are inputs to the procedure rather than quantities derived within the theory.

free parameters (4)

Relevance threshold Delta = 6.6 (runner A right), 5.9 (runner A left), 8 (runner B left)
Used to define the relevant change set I; in the application it is computed from the data as the sup norm difference between first and last 5% segment means, while Theorem 1 treats Delta as fixed.
Bootstrap block length L
Required by the multiplier bootstrap in Algorithm 1; no selection rule is given and the theorem does not state conditions on L for the block bootstrap to work.
Bandwidth c for long-run variance estimator
Lemma 1 requires c to infinity and c^3/n to 0, but the application does not state how c is chosen.
Significance level beta for relevant change detection
Used in Algorithm 2 to estimate I; the coverage guarantee includes beta, but the paper does not state that beta must go to 0 for the strengthened equality result.

assumptions (4)

domain assumption Conditions (A1)-(A4) of Dette and Kokot (2022) hold for the functional time series.
Invoked in Theorem 1 and Lemma 1 but not stated in this paper; these conditions presumably cover stationarity, mixing, and moment bounds for the error process.
domain assumption The mean function is piecewise constant in j with unknown change points s_1,...,s_m.
Equation (2); this is the core model assumption that turns fatigue phases into segments with constant mean curves.
domain assumption The change point estimators and relevant-set estimator from Bastian et al. (2024) are consistent at the required rate.
The proof of Theorem 1 relies on Theorem 4.1 and 4.2 of Bastian et al. (2024) to replace estimated change points by true values and to show Ihat = I with probability 1-o(1).
domain assumption sigma(t)^2 > 0 for all t in [0,1].
Stated before Theorem 1 to avoid division by zero in the bootstrap statistic.

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Cite this review

Pith. "Pith review of Uniform confidence bands for joint angles across different fatigue phases." pith.science (2026). https://pith.science/paper/ENMWEH2A

@misc{pith2026250208430,
  author       = {Pith},
  title        = {Pith review of: Uniform confidence bands for joint angles across different fatigue phases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENMWEH2A}},
  note         = {Machine review of arXiv:2502.08430}
}

abstract

We develop uniform confidence bands for the mean function of stationary time series as a post-hoc analysis of multiple change point detection in functional time series. In particular, the methodology in this work provides bands for those segments where the jump size exceeds a certain threshold $\Delta$. In \cite{bastian2024multiplechangepointdetection} such exceedences of $\Delta$ were related to fatigue states of a running athlete. The extension to confidence bands stems from an interest in understanding the range of motion (ROM) of lower-extremity joints of running athletes under fatiguing conditions. From a biomechanical perspective, ROM serves as a proxy for joint flexibility under varying fatigue states, offering individualized insights into potentially problematic movement patterns. The new methodology provides a valuable tool for understanding the dynamic behavior of joint motion and its relationship to fatigue.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 8 canonical work pages

[1]

Bastian, P., Basu, R., and Dette, H. (2024). Multiple change point detection in functional data with applications to biomechanical fatigue data . The Annals of Applied Statistics , 18(4):3109 -- 3129

work page 2024
[2]

Fatigue detection via sequential testing of biomechanical data using martingale statistic

Basu, R. and Proksch, K. (2023). Fatigue detection via sequential testing of biomechanical data using martingale statistic. https://arxiv.org/abs/2306.01566

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[3]

and Kokot, K

Dette, H. and Kokot, K. (2022). Detecting relevant differences in the covariance operators of functional time series: a sup-norm approach. Annals of the Institute of Statistical Mathematics , 74:195--231

work page 2022
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Dette, H., Kokot, K., and Aue, A. (2020). Functional data analysis in the Banach space of continuous functions . The Annals of Statistics , 48(2):1168 -- 1192

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Yoshihara, K.-i. (1978). Moment inequalities for mixing sequences . Kodai Mathematical Journal , 1(2):316 -- 328

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Zandbergen, M. A., Marotta, L., Bulthuis, R., Buurke, J. H., Veltink, P. H., and Reenalda, J. (2023). Effects of level running-induced fatigue on running kinematics: A systematic review and meta-analysis. Gait & posture , 99:60--75

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Bastian, P., Basu, R., and Dette, H. (2024). Multiple change point detection in functional data with applications to biomechanical fatigue data . The Annals of Applied Statistics , 18(4):3109 -- 3129

work page 2024

[2] [2]

Fatigue detection via sequential testing of biomechanical data using martingale statistic

Basu, R. and Proksch, K. (2023). Fatigue detection via sequential testing of biomechanical data using martingale statistic. https://arxiv.org/abs/2306.01566

work page Pith review arXiv 2023

[3] [3]

and Kokot, K

Dette, H. and Kokot, K. (2022). Detecting relevant differences in the covariance operators of functional time series: a sup-norm approach. Annals of the Institute of Statistical Mathematics , 74:195--231

work page 2022

[4] [4]

Dette, H., Kokot, K., and Aue, A. (2020). Functional data analysis in the Banach space of continuous functions . The Annals of Statistics , 48(2):1168 -- 1192

work page 2020

[5] [5]

Yoshihara, K.-i. (1978). Moment inequalities for mixing sequences . Kodai Mathematical Journal , 1(2):316 -- 328

work page 1978

[6] [6]

A., Marotta, L., Bulthuis, R., Buurke, J

Zandbergen, M. A., Marotta, L., Bulthuis, R., Buurke, J. H., Veltink, P. H., and Reenalda, J. (2023). Effects of level running-induced fatigue on running kinematics: A systematic review and meta-analysis. Gait & posture , 99:60--75

work page 2023

[7] [7]

In: Broy, M., Dener, E

Broy, M.: Software engineering --- from auxiliary to key technologies. In: Broy, M., Dener, E. (eds.) Software Pioneers, pp. 10-13. Springer, Heidelberg (2002)

work page 2002

[8] [8]

In: The Dictionary of Substances and Their Effects

Dod, J.: Effective substances. In: The Dictionary of Substances and Their Effects. Royal Society of Chemistry (1999) Available via DIALOG. \\ http://www.rsc.org/dose/title of subordinate document. Cited 15 Jan 1999

work page 1999