A review of symbolic dynamics and symbolic reconstruction of dynamical systems

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications.

Automated summary

Discretization of nonlinear time series can be achieved through partitioning, permutations, recurrence plots, and horizontal visibility graphs. This review summarizes and compares these methods from the perspective of symbolic dynamics, which offers a suitable framework for studying symbolic representation and reconstruction of nonlinear time series.