  We propose a novel data-driven linear inverse model, called Colored-LIM, to
extract the linear dynamics and diffusion matrix that define a linear
stochastic process driven by an Ornstein-Uhlenbeck colored-noise. The
Colored-LIM is a new variant of the classical linear inverse model (LIM) which
relies on the white noise assumption. Similar to LIM, the Colored-LIM
approximates the linear dynamics from a finite realization of a stochastic
process and then solves the diffusion matrix based on, for instance, a
generalized fluctuation-dissipation relation, which can be done by solving a
system of linear equations. The main difficulty is that in practice, the
colored-noise process can be hardly observed while it is correlated to the
stochastic process of interest. Nevertheless, we show that the local behavior
of the correlation function of the observable encodes the dynamics of the
stochastic process and the diffusive behavior of the colored-noise.
  In this article, we review the classical LIM and develop Colored-LIM with a
mathematical background and rigorous derivations. In the numerical experiments,
we examine the performance of both LIM and Colored-LIM. Finally, we discuss
some false attempts to build a linear inverse model for colored-noise driven
processes, and investigate the potential misuse and its consequence of LIM in
the appendices.
