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Align

Align illustration: two phase-shifted curves warped onto a shared template

Register and align curves to separate amplitude from phase variability.

Functional observations often exhibit two fundamentally different sources of variation: amplitude (how tall or deep the features are) and phase (when those features occur). Standard statistical methods conflate the two, leading to washed-out means and inflated variance estimates. The Align module provides elastic alignment tools built on the Fisher-Rao metric and the Square Root Slope Function (SRSF) framework to cleanly decompose these sources of variability.

Elastic Alignment
SRSF registration, Karcher mean, and amplitude/phase separation.
Advanced Elastic Alignment
Closed, constrained, penalized, and multi-resolution alignment.
Landmark Registration
Align curves by matching landmark locations with monotone warps.
TSRVF
Linearized elastic analysis in a transported tangent space.
Comparing Methods
No alignment vs elastic vs landmark, side by side.
Shape Analysis
Shape-preserving registration and geodesic computations.

Scope & limitations

fdars alignment operates on real-valued curves sampled on a single shared grid (argvals) common to every observation — the Fdata constructor and stack enforce this. Keep these boundaries in mind:

  • Sparse or irregular per-curve sampling is not supported. Pre-smooth each curve onto the shared grid first. The one exception is elastic_partial_match, which accepts two mismatched grids but is strictly pairwise.
  • The elastic machinery is 1-D (SRSF / Fisher–Rao). For ordinary real-valued curves in Euclidean space, TSRVF collapses to SRVF. Genuinely manifold-valued trajectories (sphere, SPD/covariance-over-time, shapes, rotations) are not supported — there are no exp/log maps, geodesic mean, or PGA.
  • Constrained-range or density-valued curves are not supported. Transform them to an unconstrained space before aligning.
  • Supported beyond the basics: closed/periodic curves (elastic_align_pair_closed, karcher_mean_closed) and landmark-constrained alignment (elastic_align_pair_constrained).