  Observable operator models (OOMs) and related models are one of the most
important and powerful tools for modeling and analyzing stochastic systems.
They exactly describe dynamics of finite-rank systems and can be efficiently
and consistently estimated through spectral learning under the assumption of
identically distributed data. In this paper, we investigate the properties of
spectral learning without this assumption due to the requirements of analyzing
large-time scale systems, and show that the equilibrium dynamics of a system
can be extracted from nonequilibrium observation data by imposing an
equilibrium constraint. In addition, we propose a binless extension of spectral
learning for continuous data. In comparison with the other continuous-valued
spectral algorithms, the binless algorithm can achieve consistent estimation of
equilibrium dynamics with only linear complexity.
