Given a sequence of empirical distribution data (e.g. a movie of a spatiotemporal process such as a fluid flow), this work develops an ensemble data assimilation method to estimate the transition probability that represents a finite approximation of the Frobenius–Perron operator. This allows a dynamical systems knowledge to be incorporated into a prior ensemble, which provides sensible estimates in instances of limited observation. We demonstrate improved estimates over a constrained optimization approach based on a quadratic programming problem. The estimated transition probability then enables several probabilistic analysis of dynamical systems. We focus only on the identification of coherent patterns from the estimated Markov transition to demonstrate its application as a proof-of-concept. To the best of our knowledge, there have not been many works on data-driven methods to identify coherent patterns from this type of data. While here the results are presented only in the context of dynamical systems applications, this work we present here has the potential to make a contribution in wider application areas that require the estimation of transition probabilities from a time-ordered spatio-temporal distribution data.

给定一系列经验分布数据（例如，时空过程的电影，例如流体流动），这项工作开发了一种整体数据同化方法，以估计表示Frobenius-Perron算子的有限近似的跃迁概率。这允许将动态系统知识合并到先前的合奏中，从而在有限观察的情况下提供合理的估计。我们展示了基于二次规划问题的约束优化方法的改进估计。然后，估计的过渡概率可以对动力系统进行几项概率分析。我们仅着重于从估计的马尔可夫转变中识别相干模式，以证明其作为概念证明的应用。据我们所知，关于从这种类型的数据识别一致模式的数据驱动方法的工作还很少。虽然此处仅在动力学系统应用程序的上下文中显示结果，但是我们在此处提供的这项工作可能会在更广泛的应用程序领域做出贡献，这些应用程序需要根据时间顺序的时空分布数据估算转换概率。