Probabilistic size-and-shape functional mixed models
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909406499700736 |
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| author | Wang, Fangyi Bharath, Karthik Chkrebtii, Oksana Kurtek, Sebastian |
| author_facet | Wang, Fangyi Bharath, Karthik Chkrebtii, Oksana Kurtek, Sebastian |
| contents | 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 \emph{preserving} deviations of $μ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Probabilistic size-and-shape functional mixed models Wang, Fangyi Bharath, Karthik Chkrebtii, Oksana Kurtek, Sebastian Methodology Computation Machine Learning 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 \emph{preserving} deviations of $μ$ 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. |
| title | Probabilistic size-and-shape functional mixed models |
| topic | Methodology Computation Machine Learning |
| url | https://arxiv.org/abs/2411.18416 |