Measure Selection for Functional Linear Model
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915472308436992 |
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| author | Iao, Su I Müller, Hans-Georg |
| author_facet | Iao, Su I Müller, Hans-Georg |
| contents | Advancements in modern science have led to an increased prevalence of functional data, which are usually viewed as elements of the space of square-integrable functions $L^2$. Core methods in functional data analysis, such as functional principal component analysis, are typically grounded in the Hilbert structure of $L^2$ and rely on inner products based on integrals with respect to the Lebesgue measure over a fixed domain. A more flexible framework is proposed, where the measure can be arbitrary, allowing natural extensions to unbounded domains and prompting the question of optimal measure choice. Specifically, a novel functional linear model is introduced that incorporates a data-adaptive choice of the measure that defines the space, alongside an enhanced function principal component analysis. Selecting a good measure can improve the model's predictive performance, especially when the underlying processes are not well-represented when adopting the default Lebesgue measure. Simulations, as well as applications to COVID-19 data and the National Health and Nutrition Examination Survey data, show that the proposed approach consistently outperforms the conventional functional linear model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measure Selection for Functional Linear Model Iao, Su I Müller, Hans-Georg Methodology Advancements in modern science have led to an increased prevalence of functional data, which are usually viewed as elements of the space of square-integrable functions $L^2$. Core methods in functional data analysis, such as functional principal component analysis, are typically grounded in the Hilbert structure of $L^2$ and rely on inner products based on integrals with respect to the Lebesgue measure over a fixed domain. A more flexible framework is proposed, where the measure can be arbitrary, allowing natural extensions to unbounded domains and prompting the question of optimal measure choice. Specifically, a novel functional linear model is introduced that incorporates a data-adaptive choice of the measure that defines the space, alongside an enhanced function principal component analysis. Selecting a good measure can improve the model's predictive performance, especially when the underlying processes are not well-represented when adopting the default Lebesgue measure. Simulations, as well as applications to COVID-19 data and the National Health and Nutrition Examination Survey data, show that the proposed approach consistently outperforms the conventional functional linear model. |
| title | Measure Selection for Functional Linear Model |
| topic | Methodology |
| url | https://arxiv.org/abs/2509.00583 |