On the two-step estimation of the cross--power spectrum for dynamical inverse problems
| Autor: | Vallarino, Elisabetta, Sommariva, Sara, Piana, Michele, Sorrentino, Alberto |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1361-6420/ab67dc |
| Popis: | We consider the problem of reconstructing the cross--power spectrum of an unobservable multivariate stochatic process from indirect measurements of a second multivariate stochastic process, related to the first one through a linear operator. In the two--step approach, one would first compute a regularized reconstruction of the unobservable signal, and then compute an estimate of its cross--power spectrum from the regularized solution. We investigate whether the optimal regularization parameter for reconstruction of the signal also gives the best estimate of the cross--power spectrum. We show that the answer depends on the regularization method, and specifically we prove that, under a white Gaussian assumption: (i) when regularizing with truncated SVD the optimal parameter is the same; (ii) when regularizing with the Tikhonov method, the optimal parameter for the cross--power spectrum is lower than half the optimal parameter for the signal. We also provide evidence that a one--step approach would likely have better mathematical properties of the two--step approach. Our results apply particularly to the brain connectivity estimation from magneto/electro-encephalographic recordings and provide a formal interpretation of recent empirical results. Comment: 16 pages, 3 figures |
| Databáze: | arXiv |
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