Estimation of entropies and dimensions by nonlinear symbolic time series analysis

摘要

Symbolic nonlinear time series analysis is useful for time series in which the data are known to a very low degree of precision, for example series obtained with inexpensive sensors or sensors that must be very robust and simple, and for time series with a large amount of noise. In addition, symbolic analysis may be useful for cases in which efficiency of the analysis is important. Two main tasks for nonlinear time series analysis are the computations of entropies (or more accurately entropy rates) and dimensions, specifically the metric entropy and the information dimension. Entropy rates are best motivated by an analogy with Markov presses, which we discuss in this article. The metric entropy for a symbolic time series obtained from a map, with the sampling time equal to the map period, have been studied previously with and without noise. Here we propose and explore new methods of computing the metric entropy for periodically driven flows and for autonomous flows, i.e., flows driven independently of time. We also investigate methods of computing the information dimension from a symbolic non-linear time series. We show that the usual definition of information dimension based on the diameters of the partition sets is not adequate, and propose a definition based on several possible measures of the size of partition sets. We test these methods on several simple one- and two-dimensional maps.