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

We consider the problem of reconstructing the cross-power spectrum of an unobservable multivariate stochastic 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 than 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.