Table 3 Evaluation metrics.

Mean Squared Error (MSE)

\(\text {MSE} = \frac{1}{n} \sum _{i=1}^n \left( y_i - \hat{y}_i \right) ^2\)

Measures the average squared difference between predicted values (\(\hat{y}_i\)) and actual values (\(y_i\)). It penalizes larger errors more heavily, making it sensitive to outliers. Lower MSE values indicate better model performance.

Root Mean Squared Error (RMSE)

\(\text {RMSE} = \sqrt{\frac{1}{n} \sum _{i=1}^n \left( y_i - \hat{y}_i \right) ^2}\)

The square root of MSE, providing a more interpretable measure of error in the same units as the target variable. It emphasizes larger discrepancies.

Mean Absolute Error (MAE)

\(\text {MAE} = \frac{1}{n} \sum _{i=1}^n \left| y_i - \hat{y}_i \right|\)

Measures the average absolute difference between predicted and actual values, providing a straightforward measure of model accuracy. It is less sensitive to outliers than MSE.

Mean Bias Error (MBE)

\(\text {MBE} = \frac{1}{n} \sum _{i=1}^n \left( \hat{y}_i - y_i \right)\)

Captures the average bias in model predictions, indicating whether the model tends to systematically overestimate or underestimate actual values. Positive MBE indicates overestimation, while negative MBE indicates underestimation.

Pearson’s Correlation Coefficient (r)

\(r = \frac{\sum _{i=1}^n \left( y_i - \bar{y} \right) \left( \hat{y}_i - \bar{\hat{y}} \right) }{\sqrt{\sum _{i=1}^n \left( y_i - \bar{y} \right) ^2 \sum _{i=1}^n \left( \hat{y}_i - \bar{\hat{y}} \right) ^2}}\)

Measures the linear relationship between predicted and actual values, reflecting the strength and direction of their association. High r values indicate strong predictive performance.

Coefficient of Determination (R²)

\(R^2 = 1 - \frac{\sum _{i=1}^n \left( y_i - \hat{y}_i \right) ^2}{\sum _{i=1}^n \left( y_i - \bar{y} \right) ^2}\)

Quantifies the proportion of variance in actual values that is predictable from the model. An \(R^2\) value of 1 indicates perfect predictive accuracy.

Nash-Sutcliffe Efficiency (NSE)

\(NSE = 1 - \frac{\sum _{i=1}^n (y_i - \hat{y}_i)^2}{\sum _{i=1}^n (y_i - \bar{y})^2}\)

Evaluates predictive power by comparing observed data variance to residual variance. Ranges from -\(\infty\) to 1, with 1 indicating perfect prediction.

Willmott Index (WI)

\(WI = 1 - \frac{\sum _{i=1}^n |y_i - \hat{y}_i|}{\sum _{i=1}^n (|y_i - \bar{y}| + |\hat{y}_i - \bar{y}|)}\)

Quantifies the agreement between predicted and actual values, ranging from 0 to 1, with values closer to 1 indicating better performance.