Probabilistic size-and-shape functional mixed models

Wang, Fangyi; Bharath, Karthik; Chkrebtii, Oksana; Kurtek, Sebastian

Abstract

The reliable recovery and uncertainty quantification of a fixed effect function µ 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 x and y axes are confounded with additive measurement error, and cannot in general be disentangled. The question then as to what properties of µ may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable µ under a Bayesian functional mixed model. The size-and-shape of µ 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 µ 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.

Citation

Wang, F., Bharath, K., Chkrebtii, O., & Kurtek, S. (2024, December). Probabilistic size-and-shape functional mixed models. Presented at Thirty-Eighth Annual Conference on Neural Information Processing Systems, Vancouver, Canada

• Presentation Conference Type: Conference Paper (published)
• Conference Name: Thirty-Eighth Annual Conference on Neural Information Processing Systems
• Start Date: Dec 10, 2024
• End Date: Dec 15, 2024
• Acceptance Date: Oct 30, 2024
• Publication Date: Dec 10, 2024
• Deposit Date: Jan 29, 2025
• Publicly Available Date: Jan 31, 2025
• Peer Reviewed: Peer Reviewed
• Public URL: https://nottingham-repository.worktribe.com/output/44690325
• External URL: https://neurips.cc/Conferences/2024