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REVIEW 2 major objections 5 minor 33 references

A multivariate time series alone can reveal the two-dimensional input–response plane through which a stable system transiently amplifies, and how close it sits to the amplification threshold.

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2026-08-02 01:07 UTC pith:RV2CA3OG

load-bearing objection The inference pipeline and synthetic ladder are genuinely useful, but the load-bearing threshold Kc(Δ) is asserted, not derived, and direct checks show it is not the actual amplification threshold—so R's advertised 'distance to threshold' semantics do not hold as stated. the 2 major comments →

arxiv 2607.14786 v1 pith:RV2CA3OG submitted 2026-07-16 physics.data-an cs.CGnlin.CD

Inferring Non-Normal Amplification Geometry from Multivariate Time Series

V.R. Saiprasad , V. Troude , D. Sornette This is my paper
classification physics.data-an cs.CGnlin.CD
keywords non-normal dynamicstransient amplificationoperator inferenceinput-response planemoving-window diagnosticsmultivariate time seriesridge regressioneigenvector non-orthogonality
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 argues that a multivariate time series can reveal the geometry through which a stable linear system transiently amplifies small disturbances, even when the full system operator is too large to estimate entry by entry. The authors propose fitting a local linear operator, projecting it onto a dominant two-dimensional input–response plane, and summarizing the plane with three scalars: the eigenvalue splitting Δ, the non-orthogonality index K, and the ratio R=K/Kc(Δ) that locates the geometry relative to the claimed two-dimensional transient-amplification threshold. Their benchmarks show that R and the plane are recoverable from a small fraction of the N^2 samples that full operator recovery would require, and moving-window applications to uterine, seizure, freezing-of-gait, and push-up recordings show interpretable geometry changes around known events. If the claim holds, the method gives a data-efficient diagnostic that complements eigenvalue-based stability analysis, potentially useful for early-warning and event monitoring.

Core claim

The central discovery is that the target of inference need not be the full N-dimensional operator: it can be a two-dimensional subspace that carries the input–response geometry. From a sliding window, the method estimates a local one-step linear map, then uses either an optimization that maximizes directed transverse coupling or a commutator construction to find the plane Q=[br,bn]. The projected 2x2 operator yields Δ, K, and R=K/Kc(Δ). The paper's synthetic evidence is that R and the plane are identifiable with a few percent of the N^2 samples needed for entrywise operator recovery, and that the extracted response direction aligns with burst activity in empirical recordings. The authors cla

What carries the argument

The central object is the projection of the fitted operator onto a two-dimensional orthonormal plane Q, giving Γ=Q^T F Q. The ratio R=K/Kc(Δ) normalizes the reduced non-normality index K by the threshold Kc(Δ)=[√(1−Δ²)/(1−√(1−Δ²))]^{1/2}, which the paper takes as the two-dimensional transient-amplification threshold. Two plane-extraction methods carry the argument: optimization-based directed-coupling extraction (M2), which searches for an orthonormal input/output pair maximizing transverse action, and commutator-based plane extraction (M3), which uses the symmetric commutator F F^T − F^T F to locate non-normal imbalance. The eigenbasis-SVD method (M1) is used as a fragile baseline.

Load-bearing premise

The entire reading of R=K/Kc(Δ) rests on the assertion that Kc(Δ) in Eq. (17) is the two-dimensional transient-amplification threshold, but the paper never derives it, its own Appendix B shows the exact threshold is scale-dependent, and a direct numerical check for the paper's canonical matrix fails, so R is a monotone relative diagnostic rather than an absolute distance to the true threshold.

What would settle it

Compute max_t ||F^t e_2|| for the paper's canonical matrix (A.2) with a=e^{−1}, b=e^{−2}: at the claimed threshold Kc=2.80 the norm peaks around 0.67, below 1, so no transient amplification occurs where the paper says R=1; any single (a,b) exhibiting the same mismatch settles that Kc is not the threshold.

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

If this is right

  • Reduced geometry diagnostics can be tracked in moving windows from a single trajectory, requiring far fewer samples than full operator identification.
  • R rises around elevated uterine activity, seizure onsets, and freezing-of-gait onset, suggesting non-normal amplification geometry is common in stable-but-active physiological states.
  • The scale-free ratio R allows comparison of amplification-prone geometry across different systems, recording modalities, and event types.
  • Because R can move while eigenvalues remain stable, the method complements critical-slowing-down indicators by detecting geometric rather than spectral precursors.

Where Pith is reading between the lines

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

  • If R is genuinely recoverable from a few percent of N^2 samples, a natural consequence the authors leave implicit is its use as a cheap screening tool in high-dimensional monitoring before full model identification is feasible.
  • The paper's own Appendix B concedes that the exact threshold is scale-dependent; a testable extension is to replace the analytical Kc(Δ) with a numerically computed per-window exact threshold and compare R>1 verdicts.
  • The claimed threshold Kc(Δ) is asserted rather than derived, so the empirical interpretation of R>1 as 'above the amplification threshold' should be read as a relative diagnostic unless the exact 2D threshold confirms the calibration.

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

2 major / 5 minor

Summary. The paper proposes non-normal directional response inference: estimate a local linear operator from multivariate time series, extract a dominant two-dimensional input–response plane by one of three methods (M1, M2, M3), project the fitted operator onto that plane, and summarize the reduced dynamics by Δ (eigenvalue splitting), K (non-orthogonality index), and R = K/Kc(Δ), where Kc(Δ) is called the two-dimensional transient-amplification threshold. The authors validate R recovery through a careful synthetic ladder — fixed-parameter scans, M/N/T dependence, moving-window tracking, and an embedded-block benchmark with broad singular-value spectra — and apply the same pipeline to EHG, seizure EEG, freezing-of-gait, and push-up inertial data. The central claim is that the reduced geometry, and in particular R, can be recovered from finite data even when the full operator is poorly estimated. The paper ships a reproducibility package and is generally careful in its empirical hedging. However, the threshold Kc(Δ) in Eq. (17) is asserted without derivation and Appendix B explicitly concedes it is not the exact delayed-peak threshold q^disc_{c,peak}; a direct two-point check on the paper's own canonical matrix shows that q = Kc does not correspond to the onset of transient amplification. This undermines the advertised "distance to the threshold" semantics of R.

Significance. If the threshold issue can be repaired, the paper would be a valuable methodological contribution: it targets a real and under-served problem, its synthetic validation ladder is unusually thorough, it includes an external embedded-block benchmark, and the empirical analyses are appropriately conservative and reproducible. The recovery experiments are not circular in the derivation sense, because R_true and Δ_true are computed from the known generating operator F. The paper also makes its code and data-processing pipeline available, which is a concrete strength. The drawback is that the central interpretable quantity R is normalized by a threshold that, on the evidence in the manuscript, is not the threshold for transient amplification. Since the exact threshold for the canonical 2×2 dynamics depends on the absolute scale of the eigenvalues and not only on Δ, the "scale-free threshold" concept cannot be rescued by a trivial correction; the authors must either derive a different, correctly grounded threshold or remove the threshold interpretation from R and revise the abstract claims accordingly.

major comments (2)
  1. [§3.4, Eq. (17); Appendices A and B] Kc(Δ) is never derived and is contradicted by the paper's own exact calculation. For the canonical matrix (A.2) with a=e^{-1}, b=e^{-2}, one has Δ≈0.462, and Eq. (17) gives Kc≈2.80. The exact delayed-peak threshold (A.25) is q^disc_{c,peak}≈4.26, and even the one-step threshold (A.8) is q_{c,1}≈3.96. At q=Kc=2.80, σmax(F)<1 and max_t ||F^t e_2||≈0.67, so no transient amplification exists at the claimed threshold. Appendix B (B.10–B.11) explicitly concedes that Kc(Δ) is not q^disc_{c,peak}, but no derivation is supplied in its place. Moreover, because the exact threshold depends on the absolute scale of (a,b) rather than only on Δ, no function Kc(Δ) alone can be the true two-dimensional amplification threshold; this is a logical gap, not merely a numerical discrepancy.
  2. [§5 and §6; Figs. 3–5, 7–16] The 'threshold' semantics propagate through the entire validation and application. R_true in the synthetic benchmarks is computed from the same unverified Eq. (17), so the recovery experiments establish internal consistency of an estimator, not that R=1 marks the onset of transient growth. The 'subcritical/supercritical' labeling in Fig. 3 and the R=1 reference lines in Figs. 4, 5, 7, 9, 10, 12–15 are therefore ungrounded. The figures show that R is a monotone, reproducible relative diagnostic, but the abstract's claim that R measures 'how close this plane lies to the threshold for transient amplification' is not established. The revision must either derive a correct threshold (which will necessarily be scale-dependent, destroying the claimed scale-free property) or explicitly redefine R as a scale-normalized non-normality index without threshold interpretation, and remove the threshold
minor comments (5)
  1. [§5.6 caption] The caption contains a duplicated reference '(23,23)' in the definition of the principal-angle error; it should be Eq. (23) only.
  2. [Eq. (65)] The notation R^2_{br→m,j} for a coefficient of determination is easy to confuse with the reduced ratio R=K/Kc(Δ). The text notes the distinction, but a different symbol (e.g., ρ² or r²) would prevent misreading.
  3. [Fig. 6] The M=1 insets are shown only for the intermediate regime R_true≈1. For the single-trajectory data-efficiency claim, it would be informative to show M=1 performance also for strongly supercritical regimes, where the plane is easier to identify.
  4. [References] Reference [21] is missing volume/page details ('Communications Physics (2026)'), and reference [33] is listed as 'Software (2026)' without a journal or repository version identifier beyond the DOI; both should be completed.
  5. [Eq. (A.25)] The superscript 'disc' in q^disc_{c,peak} is introduced but the text later uses q_{c,peak}; please define one consistent notation.

Circularity Check

1 steps flagged

K_c(Δ) threshold in Eq. (17) is stipulated rather than derived, making the advertised 'distance to threshold' meaning of R definitional; the rest of the inference chain is self-contained.

specific steps
  1. self definitional [§3.4 Eq. (17); Appendix B Eqs. (B.10)–(B.11); Appendix A Eq. (A.25)]
    "The reduced threshold K_c(Δ) is obtained from the two-dimensional real-eigenvalue calculation summarized in Appendix A. In this setting, K_c(Δ)= [√(1−Δ²)/(1−√(1−Δ²))]^{1/2} ... This threshold is not the same object as q^disc_{c,peak}. The latter is an exact delayed-peak threshold for a specified matrix and a specified input direction. In contrast, K_c(Δ) is a scale-normalized reduced threshold expressed in the basis-invariant diagnostics Δ and K."

    The abstract and §1 advertise R=K/K_c(Δ) as measuring 'how close this plane lies to the threshold for transient amplification'. That load-bearing semantic is placed entirely on K_c. Eq. (17) is never derived; Appendix B explicitly concedes K_c is not the exact threshold q^disc_{c,peak} of Eq. (A.25), and because Appendix B shows K=q in the canonical model (B.9), R=1 means q=K_c(Δ) by definition. E.g. with the paper's own a=e^{-1}, b=e^{-2}, Δ=0.462 gives K_c=2.80, whereas q^disc_{c,peak}≈4.26 and at q=2.80 max_t||F^t e_2||≈0.67. Thus 'R=1 marks the reduced threshold' is a stipulated normalization, not a derivation, so the threshold-distance semantics of the central claim are definitional.

full rationale

The derivation chain is otherwise largely self-contained. R_true and Δ_true in the synthetic benchmarks are computed from the known generating operator F, the fitted operator comes from ridge regression on finite data, and the plane-extraction comparison is not forced: M1 systematically fails while M2 and M3 recover the plane, so the finite-sample recovery content is genuine. The empirical demonstrations are correlations with independently annotated events, not predictions derived from fitted parameters. Self-citations [21,22,30,31] are motivational or interpretational and are not load-bearing for the reduction. The one significant circular/definitional element is the threshold K_c(Δ): it is named the 'threshold for transient amplification' but is only stipulated in Eq. (17), and Appendix B concedes it differs from the exact threshold. This undermines the semantic claim that R measures distance to a threshold, though R remains a monotone relative diagnostic and the estimation machinery retains independent content; hence score 3.

Axiom & Free-Parameter Ledger

8 free parameters · 5 axioms · 0 invented entities

No new physical entities are postulated: the input direction b_n and response direction b_r are data-derived constructs, not new forces, mediators, or conserved quantities, so the invented-entities ledger is empty. The free-parameter burden is concentrated in the empirical protocol (window lengths, shrinkage, quantile cutoffs, screening rules), which is dataset-specific and partly unquantified. The main ad hoc element is the Kc(Δ) threshold curve (Eq. 17), which is asserted as the transient-amplification threshold but is not derived in Appendix A and does not match the exact threshold of the paper's own canonical model.

free parameters (8)
  • Ridge regularization λ = validation-selected (synthetic); 'dataset-specific' (empirical)
    Controls bias-variance of the fitted operator F̂ and hence the extracted plane; §3.2, §5.3, §6.2.
  • Isotropic shrinkage α = α=0.10 (EHG), 0.05 (seizure EEG), 0.04 (push-ups)
    Stabilizes short-window regressions; dataset-specific per §6.3-6.6, Eq. (56).
  • Window length W / step = 60 s/5 s EHG; 40 s/2 s EEG; 3 s/0.5 s FOG; ~1 s push-up check
    Each empirical analysis uses its own windowing; results change with W (push-up association washes out for windows ≥2 s).
  • Activity/baseline quantile cutoffs = top 20% vs bottom 50% (EHG, Eq. 67); top 15% high-acceleration (push-up, §6.6)
    Defines the high-vs-baseline comparison from which the main empirical effect sizes are computed.
  • FOG onset interval = first 4 s of each annotated freezing episode
    The interval where the R rise is claimed; changing this definition changes the reported onset effect (§6.5).
  • Push-up cycle-removal hyperparameters = K=3 harmonics, 6 s window, 0.2-1.2 Hz phase band, 1-15 Hz residual band
    Chosen for the robustness check that the R-high/low difference survives cycle removal (§6.6).
  • Window screening thresholds = not quantified
    Windows are excluded when 'too poorly conditioned', 'numerically unstable beyond what regularization repairs', or 'too degenerate' (§6.2) — the quantitative criteria are not given, so the reported R distributions depend on unrecorded exclusions.
  • Circular-shift null constants = B=1000 shifts, minimum shift 3 s
    Defines the push-up timing null distribution and its p-value floor of 9.99e-4 (§6.6).
axioms (5)
  • domain assumption Within each sliding window the dynamics are adequately approximated by a stable linear VAR(1) with additive Gaussian noise, x_{k+1} = F x_k + η_k, ρ(F) < 1 (Eq. 8).
    The entire inference — fitting F̂ and projecting onto a plane — assumes local linearity; nonlinearity, non-Gaussian noise, or within-window nonstationarity would make the geometry an artifact of the approximation. Acknowledged as a limitation in §7.
  • domain assumption The dominant transient-amplification mechanism is concentrated in a two-dimensional input-response subspace (Eq. 12; §3.3).
    The method reduces every system to a 2D plane; §5.6 defends this for an embedded 2D block amid broad spectra, but the paper does not establish that empirical systems' amplification geometry is 2D.
  • ad hoc to paper Kc(Δ) in Eq. (17) is the threshold for transient amplification in the reduced 2D real-eigenvalue dynamics.
    Asserted in §3.4 and the abstract; Appendix A derives a different scale-dependent threshold (A.25), and Appendix B (B.10) reasserts the normalized curve while conceding it is not the exact object. Numerically, the canonical matrix with a=e^{−1}, b=e^{−2} has Kc=2.80 but the exact input-direction threshold is ≈4.26; at q=2.80 no transient growth occurs (max_t||F^t e_2||=0.67).
  • domain assumption The reduced 2×2 operator has real eigenvalues whenever the R=1 semantics are invoked.
    Kc is derived for a 'two-dimensional real-eigenvalue calculation' (§3.4); complex-eigenvalue windows in empirical sliding windows are neither counted nor separately treated (§6), leaving R's meaning undefined in that case.
  • domain assumption Empirical windows that pass screening are stationary and well-conditioned enough for the linear fit to carry geometric meaning.
    §6.2 excludes 'too poorly conditioned' or 'too degenerate' windows using qualitative criteria; the validity of the reported R distributions depends on these unformalized exclusions.

pith-pipeline@v1.3.0-alltime-deepseek · 36332 in / 50051 out tokens · 407165 ms · 2026-08-02T01:07:22.865045+00:00 · methodology

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read the original abstract

Across hydrodynamics, ecology, neuroscience, network dynamics, non-Hermitian physics, and socio-economic systems, asymptotically stable dynamics can exhibit large transient amplifications that are invisible to eigenvalue-based analyses. The mechanism is geometric rather than spectral: perturbations entering along one direction may be expressed transiently along another, allowing asymptotic decay to coexist with strong transient or noise-driven amplification. We introduce non-normal directional response inference, a data-driven method for detecting this geometry from multivariate time series when the governing operator is unknown. A local linear operator is estimated from sliding windows and projected onto the dominant two-dimensional input-response subspace. The reduced dynamics are summarized by the eigenvalue splitting $\Delta$, eigenvector non-orthogonality $K$, and the scale-free ratio $R=K/K_c(\Delta)$, where $K_c(\Delta)$ is the two-dimensional threshold for transient amplification. Controlled benchmarks show that the reduced geometry, particularly $R$, can be recovered from finite data even when the full high-dimensional operator is poorly estimated. Tests across sample size, dimension, training horizon, spectral structure, and non-stationarity confirm that the relevant response geometry requires far fewer observations than full-matrix recovery. Applied in moving windows to electrohysterogram, seizure EEG, freezing-of-gait, and unstable push-up inertial recordings, the method reveals systematic changes around known physiological or behavioral episodes through shifts in $R$, changes in $\Delta$, or stronger projection of fluctuations onto the inferred response direction. It thus exposes interpretable changes in local response geometry without framing the problem as supervised event detection.

Figures

Figures reproduced from arXiv: 2607.14786 by D. Sornette, V.R. Saiprasad, V. Troude.

Figure 1
Figure 1. Figure 1: Finite-time amplification in a stable non-normal system. The non-normal system is x˙ = Ann x, with Ann = [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the inference workflow. A multivariate time series {xk} is used to estimate a local one-step linear operator Fb. A two-dimensional orthonormal basis Q = [br, bn] is then extracted using one of the methods introduced in section 4. The fitted operator is projected onto this plane to obtain the reduced matrix Γ = Q ⊤ FbQ. The scalar diagnostics computed from Γ are the reduced eigenvalue splitting… view at source ↗
Figure 3
Figure 3. Figure 3: Synthetic VAR(1) trajectories across values of the ratio R. Trajectories are generated from xk+1 = F(κ)xk + ηk, with N = 200, spectral radius ρ(F) = 0.95, ηk ∼ N(0, σ2 IN), σ = 0.05, and T = 600 time steps. The same noise realization is used for the four representative cases. The operator F(κ) is constructed according to (44) with fixed orthogonal matrices U, V, a fixed stable diagonal spectrum Λ, and a va… view at source ↗
Figure 4
Figure 4. Figure 4: Mean-field correlation with the response coordinate across the non-normal transition. The horizontal axis is the normalized reduced non￾normality ratio R = K/Kc(∆), computed from the known reduced operator Q ⊤F(κ)Q. The vertical axis is the absolute correlation |corr(mk,br ⊤ xk)| between the mean field mk = N −11 ⊤ xk and the response coordinate br ⊤ xk. The parame￾ter scan uses the same synthetic VAR(1) c… view at source ↗
Figure 5
Figure 5. Figure 5: Inferred reduced non-normality ratio R = K/Kc(∆) and response plane from estimated operators, for plane-extraction methods M1, M2, and M3. Synthetic trajectories are generated from the VAR(1) model xk+1 = F(κ)xk + ηk, with N = 100, ρ(F) = 0.95, ηk ∼ N(0, σ2 IN), and σ = 0.05. For each value of κ, the fitted operator Fb is estimated by concatenating M = 25 independent synthetic training trajectories, each o… view at source ↗
Figure 6
Figure 6. Figure 6: Dependence of reduced non-normal geometry recovery on number M of samples, state dimension N and training horizon T. Synthetic trajectories are generated from the VAR(1) model xk+1 = F(κ)xk + ηk, with ρ(F) = 0.95, ηk ∼ N(0, σ2 IN), and σ = 0.05. For each scaling variable, the internal anisotropy parameter κ is chosen to obtain three reference regimes: Rtrue ≈ 0.2, Rtrue ≈ 1, and Rtrue ≈ 10, where Rtrue = K… view at source ↗
Figure 7
Figure 7. Figure 7: Time-varying synthetic test with moving windows. Trajectories are generated from the time-varying VAR(1) model ( [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Inferred response plane and ratio R = K/Kc(∆) for operators with broad singular-value spectra. The full operator is F = O diag(BR, Dβ) O ⊤ (50), where O is random orthogonal, BR is a stable two-dimensional non-normal block with prescribed reference ratio R⋆, and Dβ = diag(c j−β ) is a stable diagonal background with c = 0.74 and exponent β ∈ {0.25, 0.75, 1.25, 1.75, 2.25}. The benchmark uses N = 80, M = 18… view at source ↗
Figure 9
Figure 9. Figure 9: Moving-window EHG diagnostics in one representative recording. One multichannel EHG recording analyzed with 60 s moving windows advanced in 5 s steps. Diagnostics are plotted at the window end time (54). The signals were bandpass filtered between 0.08 and 2.0 Hz, converted to µV, and downsampled to 10 Hz. Local generators were estimated by ridge-regularized least squares and stabilized by the shrinkage in … view at source ↗
Figure 10
Figure 10. Figure 10: EHG diagnostics aligned to detected activity peaks in the rep￾resentative recording. The activity peaks detected in fig. 9 are aligned at time zero, using 23 peaks and a [−100, 100] s window around each. Orange curves denote optimization-based directed-coupling extraction (M2), and green curves denote commutator-based plane extraction (M3). Shaded bands show the across-peak variability. (a) Peak-aligned a… view at source ↗
Figure 11
Figure 11. Figure 11: Subject-level summary of EHG diagnostics during high-activity and baseline windows. The analysis includes 123 recordings from 45 sub￾jects. High-activity windows are the top 20% of the smoothed activity envelope within each recording, baseline windows the bottom 50%, as defined in (67). Recording-level quantities are summarized within each subject by their median. Blue denotes optimization-based directed-… view at source ↗
Figure 12
Figure 12. Figure 12: Moving-window EEG diagnostics in one representative seizure recording. The recording is CHB-MIT file chb01_03.edf. The EEG was bandpass filtered between 0.5 and 40 Hz, notch filtered at 60 Hz, downsampled to 32 Hz, and analyzed with 40 s moving windows advanced in 2 s steps. Diagnostics are plotted at the onset-relative window endpoint τend defined in (71). The shaded interval marks the annotated seizure.… view at source ↗
Figure 13
Figure 13. Figure 13: Seizure-aligned EEG diagnostics pooled across patients. The co￾hort analysis uses the successfully analyzed seizure recordings from the CHB￾MIT dataset. Recordings were preprocessed as in fig. 12 and analyzed with 40 s moving windows advanced in 2 s steps. Each seizure was aligned to its anno￾tated onset using (70), and diagnostics are plotted at the onset-relative window endpoint τend defined in (71). Cu… view at source ↗
Figure 14
Figure 14. Figure 14: Moving-window diagnostics in one representative freezing-of-gait (FOG) event. The figure shows Daphnet recording S01R01, ankle event 03. The state is formed from the three ankle acceleration channels in (73). Diagnostics are plotted at the window end time tend defined in (54). Orange curves denote optimization-based directed-coupling extraction (M2), and green curves denote commutator-based plane extracti… view at source ↗
Figure 15
Figure 15. Figure 15: Onset and baseline distributions of reduced diagnostics. The figure summarizes freezing-of-gait events from the Daphnet recordings. For each event, the onset value is computed from the onset interval and the baseline value from matched windows outside the annotated freezing episode. Orange denotes onset windows and blue denotes baseline windows. White dashed hor￾izontal lines mark the median and quartiles… view at source ↗
Figure 16
Figure 16. Figure 16: Diagnostics for rhythmic unstable push-up motion using the LW protocol (https://logicworkoutapp.com). The analyzed signal is a three-axis Apple Watch user acceleration; the channels were demeaned. The reduced diagnostics are fitted to the acceleration x(t) (73); high-acceleration samples are the upper 15% of the acceleration envelope and low-acceleration samples the remainder. Panels (a,b) show one repres… view at source ↗

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Reference graph

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