Entropy of hydrological systems under small samples: Uncertainty and variability

• Two innovative estimators for entropy estimation for small samples are introduced.
• Multi-scale moving entropy-based analysis is employed to assess the uncertainty.
• Intrinsic property of entropy in hydrological uncertainty analyses is discussed.
• Correlations between entropy and other common statistics in hydrology are illustrated.

SummaryEntropy theory has been increasingly applied in hydrology in both descriptive and inferential ways. However, little attention has been given to the small-sample condition widespread in hydrological practice, where either hydrological measurements are limited or are even nonexistent. Accordingly, entropy estimated under this condition may incur considerable bias. In this study, small-sample condition is considered and two innovative entropy estimators, the Chao–Shen (CS) estimator and the James–Stein-type shrinkage (JSS) estimator, are introduced. Simulation tests are conducted with common distributions in hydrology, that lead to the best-performing JSS estimator. Then, multi-scale moving entropy-based hydrological analyses (MM-EHA) are applied to indicate the changing patterns of uncertainty of streamflow data collected from the Yangtze River and the Yellow River, China. For further investigation into the intrinsic property of entropy applied in hydrological uncertainty analyses, correlations of entropy and other statistics at different time-scales are also calculated, which show connections between the concept of uncertainty and variability.