On the two-step estimation of the cross--power spectrum for dynamical inverse problems

TL;DR

This paper analyzes the two-step method for estimating the cross-power spectrum of an unobservable process from indirect measurements, revealing how regularization choices affect the optimal parameters for reconstruction and spectral estimation.

Contribution

It provides theoretical insights into the relationship between regularization parameters for signal reconstruction and spectral estimation in inverse problems.

Findings

Optimal regularization parameter for signal reconstruction matches that for cross-power spectrum with truncated SVD.

For Tikhonov regularization, the optimal spectral parameter is lower than half the signal's.

One-step approach may have better mathematical properties than the two-step method.

Abstract

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.