Probabilistic size-and-shape functional mixed models

Fangyi Wang, Karthik Bharath, Oksana Chkrebtii, Sebastian Kurtek

Keywords: statistical shape analysis, size-and-shape perturbation model, Bayesian random effects model, norm-preserving transformation, phase function

TL;DR: We develop a Bayesian mixed effects model for inference on the functional size-and-shape mean via a combination of size-and-shape preserving and altering random effects.

Abstract: The reliable recovery and uncertainty quantification of a fixed effect function $\mu$ in a functional mixed model, for modeling population- and object-level variability in noisily observed functional data, is a notoriously challenging task: variations along the $x$ and $y$ axes are confounded with additive measurement error, and cannot in general be disentangled. The question then as to what properties of $\mu$ may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable $\mu$ under a Bayesian functional mixed model. The size-and-shape of $\mu$ is a geometric property invariant to a family of space-time unitary transformations, viewed as rotations of the Hilbert space, that jointly transform the $x$ and $y$ axes. A random object-level unitary transformation then captures size-and-shape preserving deviations of $\mu$ from an individual function, while a random linear term and measurement error capture size-and-shape 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.

Primary Area: Probabilistic methods (for example: variational inference, Gaussian processes)

Submission Number: 17964