Probabilistic size-and-shape functional mixed models

• URL: http://arxiv.org/abs/2411.18416v1
• Date: Wed, 27 Nov 2024 15:00:33 GMT
• Title: Probabilistic size-and-shape functional mixed models
• Authors: Fangyi Wang, Karthik Bharath, Oksana Chkrebtii, Sebastian Kurtek,
• Abstract summary: We show that it is possible to recover the size-and-shape of a square-integrable $mu$ under a functional mixed model. A random object-level unitary transformation then captures size-and-shape emphpreserving deviations of $mu$ from an individual function.
• Score: 6.424762079392286
• License:
• Abstract: The reliable recovery and uncertainty quantification of a fixed effect function $\mu$ 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 $\mu$ 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 \emph{preserving} deviations of $\mu$ 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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