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We extend Gaussian perturbation models in classical functional data analysisto the three-dimensional rotational group where a zero-mean Gaussian process inthe Lie algebra under the Lie exponential spreads multiplicatively around acentral curve. As an estimator, we introduce point-wise extrinsic mean curveswhich feature strong perturbation consistency, and which are asymptoticallya.s. unique and differentiable, if the model is so. Further, we consider thegroup action of time warping and that of spatial isometries that are connectedto the identity. The latter can be asymptotically consistently estimated iflifted to the unit quaternions. Introducing a generic loss for Lie groups, theformer can be estimated, and based on curve length, due to asymptoticdifferentiability, we propose two-sample permutation tests involving variouscombinations of the group actions. This methodology allows inference on gaitpatterns due to the rotational motion of the lower leg with respect to theupper leg. This was previously not possible because, among others, the usualanalysis of separate Euler angles is not independent of marker placement, evenif performed by trained specialists.

Functional Inference on Rotational Curves and Identification of Human Gait at the Knee Joint