On the Mathematical Relationship Between RMSE and NSE Across Evaluation Scenarios

Model evaluation metrics play a crucial role in hydrology, where accurate prediction of continuous variables such as streamflow and rainfall–runoff is essential for sustainable water resources management and climate resilience. Among these metrics, the Nash–Sutcliffe efficiency (NSE) is the most widely adopted, while the Root Mean Squared Error (RMSE) is also widely used because of its clear and intuitive interpretation. Although both metrics are functions of the sum of squared errors (SSE), their different normalizations can lead to subtle differences in model ranking under certain conditions. This study systematically investigates three scenarios: (i) both models are evaluated on the same dataset, (ii) both models are re-evaluated after adding new data, and (iii) one model is evaluated on the original dataset while the other is evaluated on the expanded dataset. For each case, we derive the mathematical conditions that determine whether one model can have simultaneously lower RMSE and NSE, or vice versa. The results demonstrate that RMSE and NSE always align on the same dataset in terms of model performance, but discrepancies can emerge when dataset expansion changes the total variance and models are compared across unequal evaluation bases. These findings clarify the interpretation of NSE and RMSE in hydrological model assessment and provide guidance for their use in broader machine learning applications. An important implication is that NSE can be artificially inflated: a model may appear to perform better simply by appending new data with high variance, even if its prediction errors remain large. This insight supports more reliable evaluation of hydrological models, contributing to better water management decisions under SDG 6 (Clean Water and Sanitation) and SDG 13 (Climate Action).