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.

翻译：我们提出一种新型数据驱动线性逆模型——Colored-LIM，用于提取由奥恩斯坦-乌伦贝克有色噪声驱动的线性随机过程所定义的线性动力学与扩散矩阵。Colored-LIM是经典线性逆模型（LIM）的新变体，后者基于白噪声假设。与LIM类似，Colored-LIM通过随机过程的有限实现近似线性动力学，进而基于广义涨落-耗散关系等途径求解扩散矩阵，这可通过求解线性方程组实现。主要难点在于实际中有色噪声过程难以直接观测，且其与目标随机过程存在相关性。然而研究表明，观测变量相关函数的局部行为蕴含随机过程的动力学信息以及有色噪声的扩散特性。本文在回顾经典LIM的基础上，从数学背景和严格推导出发构建Colored-LIM。通过数值实验对比LIM与Colored-LIM的性能，最后讨论构建有色噪声驱动过程线性逆模型的若干错误尝试，并在附录中探讨LIM的潜在误用及其后果。