Functional inference on rotational curves under sample‐specific group actions and identification of human gait

Abstract Inspired by the problem of gait reproducibility (reidentifying individuals across doctor's visits) we develop two‐sample permutation tests under a sample‐specific group action on Lie groups with a bi‐invariant Riemannian metric. These tests rely on consistent estimators and pairwise curve alignment. To this end, we propose Gaussian perturbation models and for the special case of curves on the group of 3D rotations we provide asymptotic consistency and, employing a quaternion point of view, fast spatial alignment of pointwise extrinsic mean curves. In our application to rotations of the tibia versus the femur at the knee joint under the spatial action of marker placement and the temporal action of different walking speeds, obtained from an experiment, we solve the problem of gait reproducibility.