Predictability Exponent of Stochastic Dynamical Systems

Predicting the trajectory of stochastic dynamical systems (SDSs) is an intriguing problem in numerous fields, where characterizing the predictability of SDSs is of fundamental importance. Prior works have tackled this issue by indirectly investigating the uncertainty of distribution in each prediction. How accurately the trajectory of SDSs can be directly predicted still remains open. This paper proposes a new metric, namely predictability exponent, to characterize the decaying rate of probability that the prediction error never exceeds arbitrary $\epsilon$. To evaluate predictability exponent, we begin with providing a complete framework for model-known cases. Then, we bring to light the explicit relationship between predictability exponent and entropy by discrete approximation techniques. The definition and evaluation on predictability exponent are further extended to model-unknown cases by optimizing over model spaces, which build a bridge between the accuracy of trajectory predictions and popular entropy-based uncertainty measures. Examples of unpredictable trajectory design are presented to elaborate the applicability of the proposed predictability metric. Simulations are conducted to illustrate the efficiency of the obtained results.