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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2412.17971 |
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| _version_ | 1866915424293093376 |
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| author | Vidal, Marc Leman, Marc Aguilera, Ana M. |
| author_facet | Vidal, Marc Leman, Marc Aguilera, Ana M. |
| contents | We develop a theory for functional independent component analysis in an infinite-dimensional framework using Sobolev spaces that accommodate smoother functions. The notion of penalized kurtosis is introduced motivated by Silverman's method for smoothing principal components. This approach allows for a classical definition of independent components obtained via projection onto the eigenfunctions of a smoothed kurtosis operator mapping a whitened functional random variable. We discuss the theoretical properties of this operator in relation to a generalized Fisher discriminant function and the relationship it entails with the Feldman-Hájek dichotomy for Gaussian measures, both of which are critical to the principles of functional classification. The proposed estimators are a particularly competitive alternative in binary classification of functional data and can eventually achieve the so-called near-perfect classification, which is a genuine phenomenon of high-dimensional data. Our methods are illustrated through simulations, various real datasets, and used to model electroencephalographic biomarkers for the diagnosis of depressive disorder. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17971 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functional independent component analysis by choice of norm: a framework for near-perfect classification Vidal, Marc Leman, Marc Aguilera, Ana M. Statistics Theory 46N30, 47A52, 60G15 We develop a theory for functional independent component analysis in an infinite-dimensional framework using Sobolev spaces that accommodate smoother functions. The notion of penalized kurtosis is introduced motivated by Silverman's method for smoothing principal components. This approach allows for a classical definition of independent components obtained via projection onto the eigenfunctions of a smoothed kurtosis operator mapping a whitened functional random variable. We discuss the theoretical properties of this operator in relation to a generalized Fisher discriminant function and the relationship it entails with the Feldman-Hájek dichotomy for Gaussian measures, both of which are critical to the principles of functional classification. The proposed estimators are a particularly competitive alternative in binary classification of functional data and can eventually achieve the so-called near-perfect classification, which is a genuine phenomenon of high-dimensional data. Our methods are illustrated through simulations, various real datasets, and used to model electroencephalographic biomarkers for the diagnosis of depressive disorder. |
| title | Functional independent component analysis by choice of norm: a framework for near-perfect classification |
| topic | Statistics Theory 46N30, 47A52, 60G15 |
| url | https://arxiv.org/abs/2412.17971 |