A mutual information statistic for assessing state space partitions of dynamical systems

We propose a mutual information statistic to quantify the information encoded by a partition of the state space of a dynamical system. We measure the mutual information between each point's symbolic trajectory history under a coarse partition (one with few unique symbols) and its partition assignment under a fine partition (one with many unique symbols). When applied to a set of test cases, this statistic demonstrates predictable and consistent behavior. Empirical results and the statistic's formulation suggest that partitions based on trajectory history, such as the ordinal partition, perform best. As an application, we introduce the weighted ordinal partition, an extension of the popular ordinal partition with parameters that can be optimized using the mutual information statistic, and demonstrate improvements over the ordinal partition in time series analysis. We also demonstrate the weighted ordinal partition's applicability to real experimental datasets.