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Practice

Evaluating Regression Models

Regression models are machine learning models used to predict continuous numerical values. The most commonly used metrics for evaluating their performance 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)

Formula for Mean Squared Error​

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

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

Formula for Coefficient of Determination (R²)​

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

R2=1−MSEVar(y)R² = 1 - \frac{MSE}{Var(y)}

Regression Example: R² Score​

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

R² Score Example
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

Key Takeaways​

  • Use classification metrics for categorical outputs and regression metrics for continuous outputs.
  • Regression models are typically evaluated using Mean Squared Error (MSE) and Coefficient of Determination (R²).

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