Explained variance

1. Definition #

For variance of the observations \(\mathrm{Var}(y)\) and variance of the residuals \(\mathrm{Var}(y - \hat{y})\):

$$ \mathrm{Explained\ Variance} = 1 - \frac{\mathrm{Var}(y - \hat{y})}{\mathrm{Var}(y)} $$

• Values close to 1 indicate that most variability is captured.
• 0 equals a baseline “predict the mean” model.
• Negative values imply the predictions are worse than using the mean.

2. Computing in Python #

from sklearn.metrics import explained_variance_score

ev = explained_variance_score(y_test, y_pred)
print(f"Explained Variance: {ev:.3f}")

explained_variance_score supports multi-output regression via multioutput="raw_values" if needed.

3. Explained variance vs. R² #

• Sensitivity to bias: R² reacts to bias (systematic error) because it considers mean squared error, while explained variance ignores bias and focuses on variance.
• Use cases: when you care about capturing fluctuation rather than matching levels exactly.
• Reporting: pair the score with R² to provide both variance-explanation and overall fit insights.

4. Practical applications #

• Risk/volatility forecasting: check whether the model captures the variability around a target level.
• Multi-output regression: compare explained variance per output to identify which targets are harder.
• Bias correction needed: high explained variance with large MAE indicates the model matches volatility but is biased; adjust intercepts accordingly.

5. Pairing with other metrics #

Summary #

• Explained variance complements R² by isolating how well the model captures variability.
• Easily computed via explained_variance_score, including multi-output cases.
• Always interpret it together with bias-focused metrics (MAE, MBE) to gain a balanced view.