Table 2 Performance metrics for regression analysis.
Error | Mathematical equations | Definition |
|---|---|---|
Mean Squared Logarithmic Error (MSLE) | \(\:MSLE=\:\frac{1}{n}\:{\sum\:}_{i=1}^{n}{\left({log}\left(1+{y}_{i}\right)+{log}\left(1+{\widehat{y}}_{i}\right)\right)}^{2}\) | Quantifies prediction accuracy by calculating the average squared difference between the natural logarithm of predicted values and the natural logarithm of actual values. |
Root Mean Squared Logarithmic Error (RMSLE) | \(\:RMSLE=\:\sqrt{\frac{1}{n}\:{\sum\:}_{i=1}^{n}{\left({log}\left(1+{y}_{i}\right)+{log}\left(1+{\widehat{y}}_{i}\right)\right)}^{2}}\) | Computes the square root of the average squared logarithmic differences between predicted and observed values. This metric converts logarithmic-scale discrepancies into the original units of the target variable, preserving interpretability while emphasizing proportional accuracy. |
Variance Score (R-squared) | \(\:{R}^{2}\:=1\:-\:\frac{{\sum\:}_{i=1}^{N}{({y}_{i}-{\widehat{y}}_{i})}^{2}}{{\sum\:}_{i=1}^{N}{({y}_{i}-\:\overline{{y}_{i}})}^{2}}\) | Quantifies the proportion of variance in the target variable captured by the regression model, serving as a statistical measure of predictive accuracy. Values range from 0 (indicating no explanatory power) to 1 (representing perfect alignment between predictions and observed data), with higher scores reflecting stronger model performance. |
Mean Absolute Error (MAE) | \(\:MAE=\:\frac{1}{n}\:{\sum\:}_{i=1}^{n}\left|{y}_{i}-{\widehat{y}}_{i}\right|\) | Computes the average absolute difference between predicted and true values, employing a linear scoring mechanism that assigns equal weight to all errors irrespective of their size. |
Mean Absolute Percentage Error (MAPE) | \(\:MAPE=\:\frac{1}{n}\:{\sum\:}_{i=1}^{n}\left|\frac{{y}_{i}-{\widehat{y}}_{i}}{{y}_{i}}\right|\times\:100\%\) | This metric calculates prediction errors as percentage deviations relative to actual values, providing scale-independent comparisons that facilitate intuitive evaluation across datasets with differing magnitudes or measurement units. |
Root Mean Squared Error (RMSE) | \(\:RMSE=\:\sqrt{\frac{1}{n}\sum\:_{i=1}^{n}{({P}_{i}\:-\:{O}_{i})}^{2}}\) | This metric quantifies the average magnitude of prediction errors by squaring deviations before averaging, thereby disproportionately penalizing larger inaccuracies compared to minor discrepancies. |
Mean Squared Error (MSE) | \(\:MSE=\:\frac{1}{n}\:{\sum\:}_{i=1}^{n}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}\) | This metric calculates the average squared difference between a model’s predicted outputs and the corresponding true values across a dataset. |
