Evaluating Regression Models

Regression models are used to predict continuous numeric values. To evaluate how well they perform, we rely on specific metrics that measure prediction accuracy and model fit.

The two most commonly used are:

• Mean Squared Error (MSE): The average of the squared differences between predicted and actual values (the closer to 0, the better)
• Coefficient of Determination (R²): Measures how well the model explains the variance in the target variable (the closer to 1.0, the better)

Mean Squared Error (MSE) is calculated as the average of the squared differences between predicted and actual values:

$$MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y_i})^2$$

The Coefficient of Determination (R²) represents how well the model explains the variance of the target variable:

$$R² = 1 - \frac{MSE}{Var(y)}$$

The following example demonstrates how to evaluate a regression model using the R² score:

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from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score
from sklearn.model_selection import train_test_split

# Generate synthetic regression data
import numpy as np
rng = np.random.RandomState(0)
X_reg = 2 * rng.rand(50, 1)
y_reg = 4 + 3 * X_reg.ravel() + rng.randn(50)

# Split into training and test sets
X_train, X_test, y_train, y_test = train_test_split(X_reg, y_reg, test_size=0.2, random_state=42)

# Train the model
reg = LinearRegression()
reg.fit(X_train, y_train)

# Make predictions
y_pred = reg.predict(X_test)

# Evaluate the model
r2 = r2_score(y_test, y_pred)
print(f"R² score: {r2:.3f}")

Possible values of R² are:

• 1.0: Perfect prediction
• 0: No improvement over predicting the mean
• Negative: Worse than predicting the mean

• Use classification metrics for categorical outputs and regression metrics for continuous outputs.
• The most common regression metrics are Mean Squared Error (MSE) and Coefficient of Determination (R²).
• Lower MSE and higher R² values generally indicate better performance.