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arxiv: 2604.22123 · v1 · submitted 2026-04-23 · 📊 stat.AP

Recognition: unknown

Modeling Physical Activity Change as Smooth Transformations: Temporal and Amplitude Patterns Associated with Physical Function in Older Women

Andrea Z. LaCroix, Charles Kooperberg, Chongzhi Di, Jingjing Zou, Lindsay Dillon, Loki Natarajan, Marcia L. Stefanick, Michael J. LaMonte, Phyllis A. Richey, Ramon Casanova, Rong W. Zablocki, Sheri J. Hartman, Steve Nguyen, Yacun Wang

Pith reviewed 2026-05-08 13:03 UTC · model grok-4.3

classification 📊 stat.AP
keywords physical activityaccelerometerRiemannian deformationmultivariate functional PCAphysical functionolder womenlongitudinal changediurnal patterns
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The pith

Increases in physical activity throughout the day, identified through Riemannian deformation of diurnal curves, are associated with better physical function in older women.

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

The paper models how daily physical activity patterns evolve over time by treating each day's minute-level accelerometer readings as a smooth curve and changes between visits as a Riemannian deformation that jointly tracks shifts in timing and magnitude. These deformations are then summarized with multivariate functional principal component analysis to extract dominant modes of variation across participants. The leading mode, which captures a broad increase or decrease in activity across the day and accounts for roughly 21 percent of the variability, shows a clear positive link to physical function scores. Overall deformation energy interacts with the observation period, indicating that the extent of pattern change relates more strongly to function in later intervals. This framework is presented as yielding more interpretable phenotypes than conventional totals such as average daily steps.

Core claim

Longitudinal change in diurnal physical activity is represented as Riemannian deformations of smooth curves between baseline and follow-up visits. These deformations are parameterized by initial momenta and decomposed via MFPCA, with the first principal component explaining 22.4 percent of variability in the first interval and 20.8 percent in the second and corresponding to uniform activity increase or decrease across the day. Participant scores on this component are positively associated with physical function (p < 0.0001), while deformation energy shows a period-dependent association (interaction p = 0.003).

What carries the argument

Riemannian deformation of smooth activity curves, jointly capturing timing and magnitude changes, then decomposed by multivariate functional principal component analysis to yield principal component scores and deformation energy.

If this is right

  • The first principal component of deformations represents an overall rise or fall in activity across the waking day and tracks with physical function.
  • Deformation energy, the total amount of pattern change, relates more strongly to physical function in the later follow-up interval.
  • Top principal components together explain at least 90 percent of the variability in how activity patterns deform between visits.
  • These modes give clinically interpretable descriptions of redistribution in activity timing and amplitude beyond single-number summaries.

Where Pith is reading between the lines

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

  • If the deformation modes prove robust, interventions could target specific timing shifts rather than simply increasing overall volume.
  • The same Riemannian-plus-MFPCA pipeline could be tested on other accelerometer-derived outcomes such as sleep-wake transitions or sedentary bout lengths.
  • External validation against performance-based mobility measures would clarify whether the smooth-curve approximation preserves signals relevant to real-world function.

Load-bearing premise

Riemannian deformations of smooth curves accurately capture clinically meaningful changes in physical activity timing and magnitude without the modeling steps themselves creating the observed associations.

What would settle it

A direct comparison in the same cohort showing that standard summary metrics such as total daily activity counts or mean step volume predict physical function as strongly as the leading MFPCA component and deformation energy would undermine the claim of added clinical value.

read the original abstract

Background: Minute-level accelerometer data capture rich diurnal physical activity (PA) patterns, but conventional summary metrics obscures clinically meaningful changes accumulated across a day. Building on Riemannian framework, we integrate multivariate functional principal component analysis (MFPCA) to identify main modes of PA change in older women and examine associations with physical function (PF). Method: A subset participant from OPACH as baseline and two WHISH follow-ups (W1, W2), yielded 3 accelerometer measurements; each participant's diurnal PA at each visit was represented as a smooth curve. Change between consecutive visits (defined as periods: baseline-W1, W1-W2) was modeled as a Riemannian deformation (RD) jointly capturing changes in PA timing and magnitude. Deformations were parameterized by initial momenta and summarized using MFPCA; participant-level changes were characterized by principal component (PC) scores and deformation energy (DE), a metric of overall pattern change. Associations with PF were assessed using linear mixed models. Results: Mean deformation in both periods showed overall downward shifts in PA magnitude with temporal redistribution between 10am and 7pm. Top 15 PCs explained >= 90% of variability in both periods; PC1 represented a pattern of PA increase/decrease throughout the day, explaining 22.4% (baseline-W1) and 20.8% (W1-W2). Among complete data (N=1157), an increase in PA in the mode of PC1 was positively associated with PF (p <0.0001). The interaction between DE and period was significantly associated with PF (p=0.003). Conclusions: Modeling longitudinal PA change as RDs and summarizing variability via MFPCA produced clinically interpretable phenotypes of diurnal PA change beyond standard metrics. The leading deformation mode was significantly associated with PF, and DE showed a stronger association with PF in the later period.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper claims that diurnal physical activity (PA) changes in older women, derived from minute-level accelerometer data across three visits in the OPACH/WHISH cohorts, can be modeled as Riemannian deformations (RDs) that jointly capture timing and magnitude shifts; these deformations are then summarized via multivariate functional principal component analysis (MFPCA) to yield principal component (PC) scores and deformation energy (DE). Linear mixed models on complete cases (N=1157) show that higher PC1 scores (reflecting overall PA increase) are positively associated with physical function (PF) (p<0.0001), while the DE×period interaction is also significant (p=0.003). The authors conclude that this yields clinically interpretable phenotypes beyond standard metrics.

Significance. If the Riemannian deformation and MFPCA steps faithfully recover clinically relevant PA dynamics without introducing artifacts that correlate with PF, the work would provide a valuable extension of functional data methods to longitudinal accelerometer analysis, enabling more nuanced phenotypes for aging research. The reported associations suggest that overall PA magnitude changes (PC1) and total pattern deviation (DE) track with PF, potentially guiding interventions focused on diurnal redistribution.

major comments (3)
  1. [Results] Results section (and abstract): The headline associations (PC1 with PF, p<0.0001; DE×period interaction, p=0.003) are obtained from linear mixed models using derived PC scores and DE as predictors, yet no model diagnostics, residual analyses, sensitivity to outliers, or checks for multicollinearity among the MFPCA-derived covariates are reported. Given that PC scores and DE are constructed directly from the same deformation data, this omission is load-bearing for interpreting the statistical significance.
  2. [Methods] Methods section: The manuscript truncates to the top 15 PCs explaining >=90% variability and uses these for the PF models, but provides no sensitivity analyses varying the number of retained components, the smoothing bandwidth in the RD step, or the choice of Riemannian metric. Because the free parameter (number of PCs) directly influences which modes enter the association models, the robustness of the PC1-PF link cannot be assessed.
  3. [Methods] Methods/Results: No benchmark comparison is presented against simpler alternatives such as standard FPCA on log-transformed count curves or conventional summary metrics (e.g., total daily PA, peak timing). Without this, it remains unclear whether the reported PF associations are driven by the Riemannian deformation itself or would arise under conventional functional data pipelines, which is central to the claim that RDs produce “clinically interpretable phenotypes beyond standard metrics.”
minor comments (3)
  1. [Abstract] Abstract: The sentence “conventional summary metrics obscures” contains a subject-verb agreement error (“obscure”).
  2. [Results] Results: The complete-case N=1157 is stated, but the manuscript should explicitly describe the missing-data mechanism and any imputation or sensitivity checks performed on the full cohort before restricting to complete cases.
  3. [Methods] Methods: The definition and computation of “deformation energy (DE)” should be given explicitly (e.g., as an integral of the squared initial momentum or reference to the precise formula), rather than only described qualitatively.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which identify key areas for strengthening the statistical rigor and comparative context of our work. We address each major comment below and will incorporate revisions to enhance transparency and robustness.

read point-by-point responses

  1. Referee: [Results] Results section (and abstract): The headline associations (PC1 with PF, p<0.0001; DE×period interaction, p=0.003) are obtained from linear mixed models using derived PC scores and DE as predictors, yet no model diagnostics, residual analyses, sensitivity to outliers, or checks for multicollinearity among the MFPCA-derived covariates are reported. Given that PC scores and DE are constructed directly from the same deformation data, this omission is load-bearing for interpreting the statistical significance.

    Authors: We agree that explicit model diagnostics are essential for validating the reported associations. In the revised manuscript, we will add a dedicated supplementary section presenting residual plots, Q-Q plots, variance inflation factor (VIF) calculations to assess multicollinearity between PC scores and DE, and sensitivity analyses that exclude influential outliers identified via Cook's distance. These additions will directly support the reliability of the p-values and strengthen the interpretation of PC1 and the DE×period interaction. revision: yes

  2. Referee: [Methods] Methods section: The manuscript truncates to the top 15 PCs explaining >=90% variability and uses these for the PF models, but provides no sensitivity analyses varying the number of retained components, the smoothing bandwidth in the RD step, or the choice of Riemannian metric. Because the free parameter (number of PCs) directly influences which modes enter the association models, the robustness of the PC1-PF link cannot be assessed.

    Authors: We concur that sensitivity to modeling choices is important. The revised manuscript will include supplementary analyses demonstrating the stability of the PC1-PF association when retaining 10, 15, or 20 components (covering 85-95% variance). We will also report results under alternative smoothing bandwidths in the Riemannian deformation step and provide a brief justification for the chosen metric, drawing on prior literature for its suitability to positive count data. These checks will confirm that the leading mode and its association with physical function are not artifacts of the specific parameter settings. revision: yes

  3. Referee: [Methods] Methods/Results: No benchmark comparison is presented against simpler alternatives such as standard FPCA on log-transformed count curves or conventional summary metrics (e.g., total daily PA, peak timing). Without this, it remains unclear whether the reported PF associations are driven by the Riemannian deformation itself or would arise under conventional functional data pipelines, which is central to the claim that RDs produce “clinically interpretable phenotypes beyond standard metrics.”

    Authors: We recognize that direct benchmarking would better isolate the contribution of the Riemannian deformation framework. While the core innovation lies in jointly modeling timing and amplitude shifts via deformations (which standard log-FPCA does not explicitly separate), we will add a concise comparison in the revised Results and Discussion. This will include re-fitting the linear mixed models using (i) total daily PA and peak timing as predictors and (ii) PC scores from standard FPCA applied to log-transformed activity curves, allowing readers to evaluate whether the reported associations with physical function are unique to the deformation approach or replicable under conventional pipelines. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper derives participant-level PC scores and deformation energy from MFPCA applied to Riemannian deformations of smoothed diurnal PA curves, then feeds these derived quantities into separate linear mixed models to test associations with physical function. No equation or step incorporates the PF outcome into the definition or fitting of the deformations, momenta, or principal components. No self-citations are invoked as load-bearing uniqueness theorems or ansatzes for the core pipeline. The reported p-values (PC1 with PF, DE×period interaction) arise from independent regression steps rather than any fitted-input-called-prediction or self-definitional loop. The chain therefore does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from functional data analysis and Riemannian geometry for curves, plus a data-driven choice for the number of components; no new entities are postulated.

free parameters (1)
  • Number of principal components retained
    Selected to explain at least 90% of variability in the deformations for both periods
axioms (2)
  • domain assumption Diurnal physical activity can be represented as smooth functional curves suitable for Riemannian analysis
    Invoked to enable the deformation modeling and MFPCA steps
  • domain assumption Riemannian deformations provide a geometrically valid way to quantify changes in timing and amplitude
    Core modeling premise stated in the methods description

pith-pipeline@v0.9.0 · 5708 in / 1424 out tokens · 67953 ms · 2026-05-08T13:03:55.261669+00:00 · methodology

discussion (0)

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Reference graph

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