On the use of the Gram matrix for multivariate functional principal components analysis
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909227378802688 |
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| author | Golovkine, Steven Gunning, Edward Simpkin, Andrew J. Bargary, Norma |
| author_facet | Golovkine, Steven Gunning, Edward Simpkin, Andrew J. Bargary, Norma |
| contents | Dimension reduction is crucial in functional data analysis (FDA). The key tool to reduce the dimension of the data is functional principal component analysis. Existing approaches for functional principal component analysis usually involve the diagonalization of the covariance operator. With the increasing size and complexity of functional datasets, estimating the covariance operator has become more challenging. Therefore, there is a growing need for efficient methodologies to estimate the eigencomponents. Using the duality of the space of observations and the space of functional features, we propose to use the inner-product between the curves to estimate the eigenelements of multivariate and multidimensional functional datasets. The relationship between the eigenelements of the covariance operator and those of the inner-product matrix is established. We explore the application of these methodologies in several FDA settings and provide general guidance on their usability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12949 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the use of the Gram matrix for multivariate functional principal components analysis Golovkine, Steven Gunning, Edward Simpkin, Andrew J. Bargary, Norma Methodology Machine Learning 62R10 Dimension reduction is crucial in functional data analysis (FDA). The key tool to reduce the dimension of the data is functional principal component analysis. Existing approaches for functional principal component analysis usually involve the diagonalization of the covariance operator. With the increasing size and complexity of functional datasets, estimating the covariance operator has become more challenging. Therefore, there is a growing need for efficient methodologies to estimate the eigencomponents. Using the duality of the space of observations and the space of functional features, we propose to use the inner-product between the curves to estimate the eigenelements of multivariate and multidimensional functional datasets. The relationship between the eigenelements of the covariance operator and those of the inner-product matrix is established. We explore the application of these methodologies in several FDA settings and provide general guidance on their usability. |
| title | On the use of the Gram matrix for multivariate functional principal components analysis |
| topic | Methodology Machine Learning 62R10 |
| url | https://arxiv.org/abs/2306.12949 |