Efficiency & Error Metric

The aim of this section is to present all the efficiency & error metrics that can be used to calibrate the model and evaluate its performance in simulating discharges.

Denote \(Q\) and \(Q^*\) the simulated and observed discharge, respectively, with \(t\in]0 .. T]\) representing a time step for each.

NSE

The Nash-Sutcliffe Efficiency

\[j_{nse} = 1 - \frac{\sum_{t=1}^{T}\left(Q(t) - Q^*(t)\right)^2}{\sum_{t=1}^{T}\left(Q^*(t) - \mu_{Q^*}\right)^2}\]

with \(\mu_{Q^*}\) the mean of the observed discharge.

NNSE

The Normalized Nash-Sutcliffe Efficiency

\[j_{nnse} = \frac{1}{2 - j_{nse}}\]

KGE

The Kling-Gupta Efficiency

\[j_{kge} = 1 - \sqrt{(r - 1)^2 + (\alpha - 1)^2 + (\beta - 1)^2}\]

with \(r\) the Pearson correlation coefficient, \(\alpha\) the variability of prediction errors, and \(\beta\) the bias term. They are defined as follows:

\begin{eqnarray} &r& &=& &\frac{\text{cov}(Q, Q^*)}{\sigma_Q \sigma_{Q^*}}\\ &\alpha& &=& &\frac{\sigma_Q}{\sigma_{Q^*}}\\ &\beta& &=& &\frac{\mu_Q}{\mu_{Q^*}} \end{eqnarray}

with \(\text{cov}(Q, Q^*)\) the covariance between \(Q\) and \(Q^*\), \(\mu_{Q}\) and \(\mu_{Q^*}\) the mean of the simulated and observed discharge, respectively, and \(\sigma_{Q}\) and \(\sigma_{Q^*}\) the standard deviation of the simulated and observed discharge, respectively.

MAE

The Mean Absolute Error

\[j_{mae} = \frac{1}{T} \sum_{t=1}^T \lvert Q(t) - Q^*(t) \rvert\]

MAPE

The Mean Absolute Percentage Error

\[j_{mape} = \frac{1}{T} \sum_{t=1}^T \lvert \frac{Q(t) - Q^*(t)}{Q^*(t)} \rvert\]

MSE

The Mean Squared Error

\[j_{mse} = \frac{1}{T} \sum_{t=1}^T \left(Q(t) - Q^*(t)\right)^2\]

RMSE

The Root Mean Squared Error

\[j_{rmse} = \sqrt{j_{mse}}\]

LGRM

The Logarithmic Error

\[j_{lgrm} = \sum_{t=1}^T Q^*(t) \ln\left(\frac{Q(t)}{Q^*(t)}\right)^2\]