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.

翻译：我们提出一种数据驱动方法，用于降低非平稳系统协方差矩阵中的噪声。对于平稳系统，渐进方法已被证明能够收敛至最优解。按照常理，此类方法产生的特征值高度依赖于输入信息。相反，我们的方法采用一组完全独立于输入的特征值，这些特征值编码了未来对当前特征值影响的长期平均效应——而这种影响可能正是非平稳系统中的主导因素。通过真实与合成数据实验表明：我们提出的数据驱动方法在协方差矩阵及其逆矩阵的滤波效果上均优于专为平稳系统设计的最优方法（金融投资组合方差最小化案例为此提供了实证）。这使得我们的方法具有普适性，可广泛应用于多变量推断的各类问题。