Table 6 Performance Measuring Parameters.
From: Advanced air quality prediction using multimodal data and dynamic modeling techniques
Metric | Formula | Description |
|---|---|---|
Mean Squared Error (MSE) | \(\text{MSE}=\frac{1}{\text{n}}\sum_{\text{i}=1}^{\text{n}}{({\text{y}}_{\text{i}}-\widehat{{\text{y}}_{\text{i}}})}^{2}\) (25) | Measures the average squared difference between predicted and actual values |
Mean Absolute Error (MAE) | \(\text{MAE}=\frac{1}{\text{n}}\sum_{\text{i}=1}^{\text{n}}{(|{\text{y}}_{\text{i}}-\widehat{{\text{y}}_{\text{i}}}|)}\) (26) | Measures the average absolute difference between predicted and actual values |
Root Mean Squared Error (RMSE) | \(\text{RMSE}=\sqrt[ ]{\frac{1}{\text{n}}\sum_{\text{i}=1}^{\text{n}}{({\text{y}}_{\text{i}}-\widehat{{\text{y}}_{\text{i}}})}^{2}}\) (27) | The square root of MSE gives error in the same units as the target variable |
R-squared (R2) | \({\text{R}}^{2}=[1-\frac{\frac{1}{\text{n}}\sum_{\text{i}=1}^{\text{n}}{({\text{y}}_{\text{i}}-\widehat{{\text{y}}_{\text{i}}})}^{2} }{\frac{1}{\text{n}}\sum_{\text{i}=1}^{\text{n}}{({\text{y}}_{\text{i}}-\widehat{{\text{y}}_{ }})}^{2} }]\) (28) | Indicates how well the model explains the variance in the target variable |
Explained Variance Score (EVS) | \(\text{EVS}=[1-\frac{\text{Var }{({\text{y}}_{\text{i}}-\widehat{{\text{y}}_{\text{i}}})}^{2} }{\text{Var}(\text{y}) }]\) (29) | Measures how much of the variance in the data is explained by the model |
Mean Absolute Percentage Error (MAPE) | \(\text{MAPE}=\frac{1}{\text{n}}\sum_{\text{i}=1}^{\text{n}}{(|\frac{{\text{y}}_{\text{i}}-\widehat{{\text{y}}_{\text{i}}}}{{\text{y}}_{\text{i}}}|)}\) (30) | Measures prediction accuracy as a percentage |
Precision (P) | \(\text{P}=\frac{\text{TP}}{(\text{TP}+\text{FP})}\) (31) | Fraction of relevant instances retrieved |
Recall (RC) | \(\text{RC}=\frac{\text{TP}}{(\text{TP}+\text{FN})}\) (32) | Fraction of relevant instances that were retrieved |
F1-Score (FS) | \(FS=2\times \frac{P\times RC}{(P+RC)}\) (33) | The harmonic mean of precision and recall balances both metrics |
Area Under the Curve—Receiver Operating Characteristic AUC-ROC | AUC-ROC is computed as the area under the ROC curve (TPR vs. FPR) | Measures the model’s ability to distinguish between classes |
