Optimal Designs for Regression on Lie Groups
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arXiv
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| Format: | Preprint |
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2024
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| author | Chakraborty, Somnath Dette, Holger Kroll, Martin |
| author_facet | Chakraborty, Somnath Dette, Holger Kroll, Martin |
| contents | We consider a linear regression model with complex-valued response and predictors from a compact and connected Lie group. The regression model is formulated in terms of eigenfunctions of the Laplace-Beltrami operator on the Lie group. We show that the normalized Haar measure is an approximate optimal design with respect to all Kiefer's $Φ_p$-criteria. Inspired by the concept of $t$-designs in the field of algebraic combinatorics, we then consider so-called $λ$-designs in order to construct exact $Φ_p$-optimal designs for fixed sample sizes in the considered regression problem. In particular, we explicitly construct $Φ_p$-optimal designs for regression models with predictors in the Lie groups $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$, the groups of $2\times 2$ unitary matrices and $3\times 3$ orthogonal matrices with determinant equal to $1$, respectively. We also discuss the advantages of the derived theoretical results in a concrete biological application. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Designs for Regression on Lie Groups Chakraborty, Somnath Dette, Holger Kroll, Martin Statistics Theory 62K05 We consider a linear regression model with complex-valued response and predictors from a compact and connected Lie group. The regression model is formulated in terms of eigenfunctions of the Laplace-Beltrami operator on the Lie group. We show that the normalized Haar measure is an approximate optimal design with respect to all Kiefer's $Φ_p$-criteria. Inspired by the concept of $t$-designs in the field of algebraic combinatorics, we then consider so-called $λ$-designs in order to construct exact $Φ_p$-optimal designs for fixed sample sizes in the considered regression problem. In particular, we explicitly construct $Φ_p$-optimal designs for regression models with predictors in the Lie groups $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$, the groups of $2\times 2$ unitary matrices and $3\times 3$ orthogonal matrices with determinant equal to $1$, respectively. We also discuss the advantages of the derived theoretical results in a concrete biological application. |
| title | Optimal Designs for Regression on Lie Groups |
| topic | Statistics Theory 62K05 |
| url | https://arxiv.org/abs/2410.00429 |