The recurrence plot was originally proposed for visualizing time series data. As recurrence plots have mainly been used for analyzing time series generated from nonlinear deterministic systems, it is not well known whether they can be applied to gain insight into analyzing time series generated from a random dynamical system, in which stochastic components play a central role. In this study, we demonstrate that a recurrence plot can provide new viewpoints for the stochasticity in the underlying dynamics. In particular, we present three theorems: the first theorem demonstrates that a recurrence plot can eventually establish one-to-one correspondence with a joint set of initial conditions and a series of stochastic inputs if the underlying dynamics is expansive and topologically transitive; the second theorem distinguishes deterministic and stochastic systems; and the third theorem enables the second theorem to be used for a shorter time series. Moreover, we propose a stochasticity test based on a recurrence plot. The theorems and stochasticity test are verified by numerical examples as well as real datasets.

最初提出了递归图用于可视化时间序列数据。由于递归图主要用于分析由非线性确定性系统生成的时间序列，因此人们尚不知道是否可以将它们用于深入分析由随机分量起主要作用的随机动力系统生成的时间序列。在这项研究中，我们证明了递归图可以为基础动力学中的随机性提供新的观点。特别是，我们提出了三个定理：第一个定理证明，如果基础动力学是扩展的并且在拓扑上是可传递的，则递归图最终可以与一组联合初始条件和一系列随机输入建立一对一的对应关系。第二定理区分确定性系统和随机系统；第三定理使第二定理可以用于更短的时间序列。此外，我们提出了基于递归图的随机检验。通过数值例子和真实数据集验证了定理和随机检验。