Estimating predictability limit from processes with characteristic timescale, Part I: AR(1) process

Inferring intrinsic predictability (IP) or predictability limit (PL) from time series plays a crucial role in understanding complex systems and guiding predictions. Though PL is often considered to depend on the characteristic timescale (CT) of an underlying process, the quantitative relation between IP, PL and CT has not been well studied. As the simplest process with an adjustable CT, the Auto-Regression of order one, i.e. AR(1), is taken as a representative process to explore this quantitative relation, then this relation is leveraged to estimate PL. Our results show that directly estimating the PL highly relies on the CT of a specific AR(1) process, and the uncertainties and bias of PL estimations dramatically increase with the enhanced CT, which indicates that more data points and computational cost are required for reliably estimating PL from the process with a large CT value, and it is unrealizable to directly estimate PL from most of real-world series with limited length. To solve this problem, an IP metric, i.e. the time series predictability defined by the weighted permutation entropy (WPE), is proposed to indirectly estimate PL reliably with much lower uncertainties without biases for short series. The findings in this study can greatly improve the accuracy of PL estimation and in-depth understandings on the predictability studies.