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

Purpose: To investigate whether longitudinal changes in timing and magnitude of PA are associated with physical function (PF) in older women.

Methods: Women from OPACH study with accelerometry at baseline and WHISH study W1 and W2 were included.

Minute-level PA counts were averaged and smoothed as diurnal PA curves.

Consecutive-visit change was modeled within periods (baseline--W1 and W1--W2) as a Riemannian deformation from earlier to later curves, with two-dimensional initial momenta characterizing timing and magnitude shifts.

Multivariate functional principal component analysis (MFPCA) summarized coupled timing-magnitude patterns, and principal component (PC) scores and deformation energy were derived for each participant-period.

Linear mixed-effects models related these features to RAND-36 PF, adjusting for baseline PF and covariates.

Results: Mean PA deformation in both periods showed downward shifts in PA magnitude and temporal redistribution after 10:00.

Top 15 PCs explained at least 90% of variability in both periods.

PC1 captured diurnal PA increase/decrease, explaining 22.4% of variability for baseline--W1 and 20.8% for W1--W2.

Among participants with complete PF scores and baseline covariates (N=1,157), higher PC1 scores, reflecting relative increase/maintenance of PA across day, were positively associated with PF (P<0.0001).

Deformation energy, a metric for overall diurnal pattern change between visits, showed a significant interaction with period for PF (P=0.003), with a larger positive association during W1--W2 than during baseline--W1.

Conclusions: In older women, longitudinal changes in diurnal PA accumulation were associated with PF.

Riemannian deformation analysis identified clinically interpretable markers of PA pattern change that may capture functional-aging information not represented by conventional PA summaries.