Computing linear response statistics using orthogonal polynomial based estimators: An RKHS formulation

We study the problem of estimating linear response statistics under external perturbations using time series of unperturbed dynamics. A standard approach to this estimation problem is to employ the Fluctuation-Dissipation Theory, which requires the knowledge of the functional form of the underlying unperturbed density that is not available in general. To overcome this issue, we consider a nonparametric density estimator formulated by the kernel embedding of distributions. To avoid the computational expense associated with using radial type kernels, we consider the "Mercer-type" kernels constructed based on the classical orthogonal bases defined on non-compact domains. While the resulting representation is analogous to Polynomial Chaos Expansion(PCE), by studying in the reproducing kernel Hilbert space(RKHS) setting, we establish the uniform convergence of the estimator. More importantly, the RKHS formulation allows one to systematically address a practical question of identifying the PCE basis for a consistent estimation through the decay property of the target functions that can be quantified using the available data. In terms of the linear response estimation, our study provides practical conditions for the well-posedness of not only the estimator but also the well-posedness of the underlying response statistics. We provide a theoretical guarantee for the convergence of the estimator to the underlying linear response statistics. Finally, we offer an error bound for the density estimation that accounts for the Monte-Carlo averaging over non-i.i.d time series and the biases due to truncation. This error bound helps understand the feasibility as well as limitation of the kernel embedding with Mercer-type kernels. Numerically, we verify the effectiveness of the kernel embedding linear response estimator on two stochastic dynamics with known, yet, non-trivial equilibrium densities.