A State-Transition Information Space for Time-Series Dynamics: Theory and Application
Characterizing dynamical organization in time series requires distinguishing the diversity of accessible states from uncertainty in their temporal transitions. Here we introduce a state-transition information space based on two normalized entropy measures derived from ordinal patterns: K_q, quantifying ordinal-state diversity, and K_t, quantifying transition uncertainty. Their joint K_t-K_q representation provides a two-dimensional framework in which dynamical regimes and their temporal evolution can be examined. Theoretical properties establish the bounded relation 0 <= K_t <= K_q <= 1, while canonical time series identify distinct empirical domains ranging from ordered dynamics to near-maximal randomness. Sliding-window analysis further shows that systems can exhibit temporal trajectories through the K_t-K_q plane rather than remaining at a fixed dynamical state. Application to 1,879 monthly naturalized streamflow records from the U.S. rivers shows that river dynamics occupy a distinct intermediate region between ordered and highly disordered regimes. Moreover, river positions shift systematically toward higher K_t and K_q with increasing drainage area, revealing scale-dependent organization of streamflow dynamics. The framework therefore provides a compact means of comparing state diversity, transition uncertainty, and their temporal and spatial organization across time-series systems.