  We propose a data-driven way to reduce the noise of covariance matrices of
nonstationary systems. In the case of stationary systems, asymptotic approaches
were proved to converge to the optimal solutions. Such methods produce
eigenvalues that are highly dependent on the inputs, as common sense would
suggest. Our approach proposes instead to use a set of eigenvalues totally
independent from the inputs and that encode the long-term averaging of the
influence of the future on present eigenvalues. Such an influence can be the
predominant factor in nonstationary systems. Using real and synthetic data, we
show that our data-driven method outperforms optimal methods designed for
stationary systems for the filtering of both covariance matrix and its inverse,
as illustrated by financial portfolio variance minimization, which makes out
method generically relevant to many problems of multivariate inference.
