Statistical Inference for Bures-Wasserstein Flows
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909229755924480 |
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| author | Santoro, Leonardo V. Panaretos, Victor M. |
| author_facet | Santoro, Leonardo V. Panaretos, Victor M. |
| contents | We develop a statistical framework for conducting inference on collections of time-varying covariance operators (covariance flows) over a general, possibly infinite dimensional, Hilbert space. We model the intrinsically non-linear structure of covariances by means of the Bures-Wasserstein metric geometry. We make use of the Riemmanian-like structure induced by this metric to define a notion of mean and covariance of a random flow, and develop an associated Karhunen-Loève expansion. We then treat the problem of estimation and construction of functional principal components from a finite collection of covariance flows, observed fully or irregularly.
Our theoretical results are motivated by modern problems in functional data analysis, where one observes operator-valued random processes -- for instance when analysing dynamic functional connectivity and fMRI data, or when analysing multiple functional time series in the frequency domain. Nevertheless, our framework is also novel in the finite-dimensions (matrix case), and we demonstrate what simplifications can be afforded then. We illustrate our methodology by means of simulations and data analyses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13764 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Statistical Inference for Bures-Wasserstein Flows Santoro, Leonardo V. Panaretos, Victor M. Methodology Statistics Theory 62R10, 62R20, 62R30, 62G05, 60G57 We develop a statistical framework for conducting inference on collections of time-varying covariance operators (covariance flows) over a general, possibly infinite dimensional, Hilbert space. We model the intrinsically non-linear structure of covariances by means of the Bures-Wasserstein metric geometry. We make use of the Riemmanian-like structure induced by this metric to define a notion of mean and covariance of a random flow, and develop an associated Karhunen-Loève expansion. We then treat the problem of estimation and construction of functional principal components from a finite collection of covariance flows, observed fully or irregularly. Our theoretical results are motivated by modern problems in functional data analysis, where one observes operator-valued random processes -- for instance when analysing dynamic functional connectivity and fMRI data, or when analysing multiple functional time series in the frequency domain. Nevertheless, our framework is also novel in the finite-dimensions (matrix case), and we demonstrate what simplifications can be afforded then. We illustrate our methodology by means of simulations and data analyses. |
| title | Statistical Inference for Bures-Wasserstein Flows |
| topic | Methodology Statistics Theory 62R10, 62R20, 62R30, 62G05, 60G57 |
| url | https://arxiv.org/abs/2310.13764 |