Probabilistic size-and-shape functional mixed models

The reliable recovery and uncertainty quantification of a fixed effect function in a functional mixed model, for modelling population- and object-level variability in noisily observed functional data, is a notoriously challenging task: variations along the and axes are confounded with additive measurement error, and cannot in general be disentangled. The question then as to what properties of may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable under a Bayesian functional mixed model. The size-and-shape of is a geometric property invariant to a family of space-time unitary transformations, viewed as rotations of the Hilbert space, that jointly transform the and axes. A random object-level unitary transformation then captures size-and-shape \emph{preserving} deviations of from an individual function, while a random linear term and measurement error capture size-and-shape \emph{altering} deviations. The model is regularized by appropriate priors on the unitary transformations, posterior summaries of which may then be suitably interpreted as optimal data-driven rotations of a fixed orthonormal basis for the Hilbert space. Our numerical experiments demonstrate utility of the proposed model, and superiority over the current state-of-the-art.
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