🔬 R² (R-squared)

R² (R-squared)#

The share of variance in the target explained by a regression model.

Important

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What it is#

R², the coefficient of determination, is the proportion of variance in the target that the model explains. It compares the model’s residual error to the variance around the mean:

\[R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}, \qquad SS_{\text{res}} = \sum_i (y_i - \hat{y}_i)^2, \quad SS_{\text{tot}} = \sum_i (y_i - \bar{y})^2.\]

How to read it#

It usually runs 0 to 1 — 1 is a perfect fit, 0 means the model does no better than predicting the mean (and it can go negative for a model worse than that). Being dimensionless, it complements MAE / RMSE, which report error in the target’s units; in simple regression it equals \(r^2\), the squared Pearson correlation.

Caveats#

R² never decreases when predictors are added (even noise), so use adjusted R² to compare models of different size; it is sensitive to outliers, assumes the modeled relationship, and a high value implies neither causation nor good out-of-sample performance.

Theme: Model Evaluation & Uncertainty · All terminology

Hint

Mind map — connected ideas

Forecast Error · Average Absolute Error (AAE) · Relative accuracy · Regression Coefficient · Target Variable · Point Forecasts

Hint

More in Model Evaluation & Uncertainty

Average Absolute Error (AAE) · Baseline Heuristics · Bootstrap · Bootstrap Confidence Intervals (CIs) · Coverage · Cramér’s V · DeLong’s Test · KS Statistic (Kolmogorov–Smirnov Statistic) · Likelihood Ratio (LR) · Mann–Whitney U Test (also called the Wilcoxon rank-sum test) · MASE (Mean Absolute Scaled Error) · Mean Absolute Error (MAE) · Mean Absolute Percentage Error (MAPE) · Mean Squared Error (MSE)