Table 3 Performance Metrics and nominated criteria for ANN_WQ simulation scenarios.

Correlation Coefficient (R²)

\({R}^{2}{=\left(\frac{\sum_{i=1}^{N}\left({Y}_{i}^{obs}-{{Y}_{mean}}^{obs}\right)\left({Y}_{i}^{sim}-{{Y}_{mean}}^{sim}\right)}{\sqrt{\sum_{i=1}^{N}{\left({Y}_{i}^{obs}-{{Y}_{mean}}^{obs}\right)}^{2}}\sqrt{\sum_{i=1}^{N}{\left({Y}_{i}^{sim}-{{Y}_{mean}}^{sim}\right)}^{2}}}\right)}^{2}\)

Equation 2⁶⁹

 > 0.5

Nash–Sutcliffe coefficient (NSE)

\(NSE=1- \left\lceil\frac{\sum_{i=1}^{N}{\left({Y}_{i}^{obs}-{Y}_{i}^{sim}\right)}^{2}}{\sum_{i=1}^{N}{\left({Y}_{i}^{obs}-{Y}_{i}^{mean}\right)}^{2}}\right\rceil\)

Equation 3^54,70

 > 0.5

Willmotts index (d)

\(d=1- \left\lceil\frac{\sum_{i=1}^{N}{\left({Y}_{i}^{obs}-{Y}_{i}^{sim}\right)}^{2}}{\sum_{i=1}^{N}{\left(\left|{Y}_{i}^{sim}-\right.\left.{{Y}_{mean}}^{obs}\right|+\left|\left|{Y}_{i}^{obs}-\right.\left.{{Y}_{mean}}^{sim}\right|\right.\right)}^{2}}\right\rceil\)

Equation 4⁷¹

 > 0.5

Root mean square error (RMSE)

\(RMSE= \sqrt{\frac{1}{N}\sum_{i=1}^{N}{\left({Y}_{i}^{sim}-{Y}_{i}^{obs}\right)}^{2}}\)

Equation 5^69,72

Lowest

Peak percentage deviation (pde)

\(100\sum_{i=1}^{N}\frac{1-{Y}_{max}^{sim}}{{Y}_{max}^{obs}}\)

Equation 6⁷³

 <  ± 25

Mean absolute error

\(MAE= \frac{1}{N}\sum_{i=1}^{N}{|Y}_{i}^{sim}-{Y}_{i}^{obs}|\)

Equation 7⁷²

Lowest