Evaluation Metrics for Regression

Regression metrics all try to say "how close were the predictions" in a single number, but they disagree about what "close" means — and that disagreement matters exactly when outliers are present or when comparing models. Knowing which metric lies to you in which situation is the actual skill.

Key idea

RMSE and MAE disagree exactly when outliers matter, and R² can look fine on a model that is useless.

MSE and RMSE

$$\text{MSE} = \frac{1}{n}\sum_i (y_i - \hat y_i)^2, \qquad \text{RMSE} = \sqrt{\text{MSE}}$$

RMSE is in the same units as the target (unlike MSE), which makes it more interpretable. Both square the error, so large deviations are penalised disproportionately more than small ones.

MAE

$$\text{MAE} = \frac{1}{n}\sum_i |y_i - \hat y_i|$$

Penalises every error linearly — a single huge miss doesn't dominate the score the way it does for RMSE.

When they disagree, worked numerically

Consider errors $[1, 1, 1, 1, 10]$: $\text{MAE} = 2.8$, $\text{RMSE} = \sqrt{(4 \times 1 + 100)/5} \approx 4.56$. The single outlier of 10 barely moves MAE but nearly doubles RMSE relative to what four small errors alone would produce — RMSE is far more sensitive to that one outlier.

MAPE and its problems

$$\text{MAPE} = \frac{100\%}{n}\sum_i \left|\frac{y_i - \hat y_i}{y_i}\right|$$

Expresses error as a percentage, which sounds appealingly scale-free — but it's undefined when $y_i = 0$, and it's asymmetric: it penalises over-predictions more heavily than under-predictions of the same absolute size (an over-prediction can produce an arbitrarily large percentage error, while an under-prediction is capped at 100%).

SMAPE

$$\text{SMAPE} = \frac{100\%}{n}\sum_i \frac{|y_i - \hat y_i|}{(|y_i| + |\hat y_i|)/2}$$

A symmetric variant addressing MAPE's over/under-prediction asymmetry, though it introduces its own quirks near zero.

R² and adjusted R²

$$R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}} = 1 - \frac{\sum_i (y_i - \hat y_i)^2}{\sum_i (y_i - \bar y)^2}$$

$R^2$ answers "what fraction of the target's variance does the model explain, relative to just predicting the mean?" $R^2 = 1$ is a perfect fit, $R^2 = 0$ matches predicting the mean, and $R^2$ can go negative if the model is worse than that baseline. Adjusted $R^2$ penalises adding features that don't improve the fit, correcting for the fact that plain $R^2$ never decreases when you add more predictors, even useless ones.

Symbol	Meaning
$SS_{\text{res}}$	residual sum of squares — the model's remaining error
$SS_{\text{tot}}$	total sum of squares — variance around the mean, the "naive baseline" error
$\bar y$	the mean of the true targets

The ways R² misleads

$R^2$ can be high on a curved relationship fit with a linear model that is systematically biased in a pattern the residuals reveal — high $R^2$ says nothing about whether the shape of the fit is right. And $R^2$ values are not comparable across different datasets — an $R^2$ of 0.6 on a noisy dataset can represent a far better model than an $R^2$ of 0.9 on a nearly deterministic one.

Residual analysis as the real diagnostic

A single R² cannot distinguish these three. Curvature means a missing non-linear term; a fan means the variance depends on the prediction, which breaks the constant-variance assumption behind most confidence intervals.

Plotting residuals ($y_i - \hat y_i$) against predicted values or against each feature reveals problems no single number does: a curved pattern means the model is missing non-linearity; a funnel shape (residual spread growing with prediction) means heteroscedasticity, violating the constant-noise assumption behind squared-error losses.

Quantile loss for prediction intervals

Instead of predicting a single number, quantile loss trains a model to predict a specific quantile (e.g. the 90th percentile) of the target distribution — the basis of prediction intervals rather than point estimates:

$$L_\tau(y, \hat y) = \max\big(\tau(y - \hat y), (\tau - 1)(y - \hat y)\big)$$

Selection table

Situation	Reach for
Outliers should dominate the score	RMSE
Outliers should not dominate the score	MAE
Need a scale-free percentage, no zero targets	SMAPE over MAPE
Want "fraction of variance explained"	R² (with residual plots as backup)
Need an uncertainty range, not a point estimate	quantile loss

Code: metrics from scratch, a rank-flip case, residual plots

regression_metrics_demo.py

import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import mean_squared_error, mean_absolute_error, r2_score

def mse(y, yhat): return np.mean((y - yhat) ** 2)
def rmse(y, yhat): return np.sqrt(mse(y, yhat))
def mae(y, yhat): return np.mean(np.abs(y - yhat))

rng = np.random.default_rng(0)
y = rng.normal(size=100)

# Model A: many small errors. Model B: mostly perfect, one huge miss.
model_a_preds = y + rng.normal(scale=0.5, size=100)
model_b_preds = y.copy()
model_b_preds[0] += 10  # one huge miss

print("Model A: RMSE=%.3f MAE=%.3f" % (rmse(y, model_a_preds), mae(y, model_a_preds)))
print("Model B: RMSE=%.3f MAE=%.3f" % (rmse(y, model_b_preds), mae(y, model_b_preds)))
print("-> RMSE ranks B worse (outlier-sensitive); MAE may rank B better (mostly perfect)")

# Verify against sklearn
print("sklearn check:", mean_squared_error(y, model_a_preds, squared=False),
      mean_absolute_error(y, model_a_preds), r2_score(y, model_a_preds))

# --- Residual plot: good fit vs heteroscedastic fit ---
x = rng.uniform(0, 10, 200)
y_good = 2 * x + rng.normal(scale=1.0, size=200)
y_hetero = 2 * x + rng.normal(scale=0.2 * x + 0.1, size=200)  # noise grows with x

fig, axes = plt.subplots(1, 2, figsize=(10, 4))
axes[0].scatter(x, y_good - (2 * x)); axes[0].set_title("healthy residuals")
axes[1].scatter(x, y_hetero - (2 * x)); axes[1].set_title("heteroscedastic (funnel shape)")
plt.savefig("residual_plots.png")

See also

• Evaluation Metrics for Classification — the classification-side equivalent.
• Loss Functions — MSE and MAE as training objectives, not just reported metrics.