Regression evaluation (MSE, RMSE, R²) in ML Python

Introduction

We use regression evaluation to check how well a model predicts numbers. It helps us see if the model's guesses are close to the real answers.

When predicting house prices based on features like size and location.

When estimating the temperature for the next day using past weather data.

When forecasting sales numbers for a store based on past sales.

When predicting a student's test score from study hours and attendance.

Syntax

ML Python

from sklearn.metrics import mean_squared_error, r2_score

mse = mean_squared_error(true_values, predicted_values)
rmse = mean_squared_error(true_values, predicted_values, squared=False)
r2 = r2_score(true_values, predicted_values)

mean_squared_error calculates the average squared difference between real and predicted values.

Setting squared=False in mean_squared_error gives the root mean squared error (RMSE), which is easier to understand because it is in the same units as the target.

Examples

Calculate MSE for a small set of true and predicted values.

ML Python

mse = mean_squared_error([3, -0.5, 2, 7], [2.5, 0.0, 2, 8])

Calculate RMSE by setting squared=False.

ML Python

rmse = mean_squared_error([3, -0.5, 2, 7], [2.5, 0.0, 2, 8], squared=False)

Calculate R² score to see how well predictions fit the true values.

ML Python

r2 = r2_score([3, -0.5, 2, 7], [2.5, 0.0, 2, 8])

Sample Model

This program shows how to calculate three common regression evaluation metrics using simple lists of true and predicted values.

ML Python

from sklearn.metrics import mean_squared_error, r2_score

# True values (real answers)
true_values = [3, -0.5, 2, 7]
# Predicted values from a model
predicted_values = [2.5, 0.0, 2, 8]

# Calculate Mean Squared Error
mse = mean_squared_error(true_values, predicted_values)
# Calculate Root Mean Squared Error
rmse = mean_squared_error(true_values, predicted_values, squared=False)
# Calculate R-squared score
r2 = r2_score(true_values, predicted_values)

print(f"Mean Squared Error (MSE): {mse:.3f}")
print(f"Root Mean Squared Error (RMSE): {rmse:.3f}")
print(f"R-squared (R²): {r2:.3f}")

Important Notes

MSE and RMSE measure error size; smaller values mean better predictions.

R² score shows how well the model explains the data; values closer to 1 mean better fit.

RMSE is easier to interpret than MSE because it is in the same units as the target variable.

Summary

Regression evaluation helps check prediction accuracy.

MSE and RMSE measure average error size; RMSE is the square root of MSE.

R² score shows how well predictions match real values, with 1 being perfect.

Practice

1. What does the Mean Squared Error (MSE) measure in regression evaluation?

easy

A. The total number of data points

B. The average of the squared differences between predicted and actual values

C. The correlation between features

D. The percentage of correct predictions

Solution

1. Step 1: Understand MSE definition

   MSE calculates the average of squared differences between predicted and actual values to measure error size.

2. Step 2: Compare options to definition

   Only The average of the squared differences between predicted and actual values correctly describes MSE as average squared error; others describe unrelated concepts.

3. Final Answer:

   The average of the squared differences between predicted and actual values -> Option B

4. Quick Check:

   MSE = average squared error [OK]

Hint: MSE = average squared error between prediction and truth [OK]

Common Mistakes:

• Confusing MSE with accuracy
• Thinking MSE measures correlation
• Mixing MSE with count of data points

2. Which of the following Python code snippets correctly calculates the Root Mean Squared Error (RMSE) given true values y_true and predicted values y_pred?

easy

A. rmse = np.sum((y_true - y_pred)**2)

B. rmse = np.mean(np.abs(y_true - y_pred))

C. import numpy as np
rmse = np.sqrt(np.mean((y_true - y_pred)**2))

D. rmse = np.sqrt(np.sum(y_true - y_pred))

Solution

1. Step 1: Recall RMSE formula

   RMSE is the square root of the mean of squared differences between true and predicted values.

2. Step 2: Match code to formula

   import numpy as np rmse = np.sqrt(np.mean((y_true - y_pred)**2)) correctly computes squared differences, averages them, then takes square root. Others miss steps or use wrong operations.

3. Final Answer:

   import numpy as np rmse = np.sqrt(np.mean((y_true - y_pred)**2)) -> Option C

4. Quick Check:

   RMSE = sqrt(mean squared error) [OK]

Hint: RMSE = sqrt of mean squared error, use np.sqrt(np.mean(...)) [OK]

Common Mistakes:

• Using sum instead of mean
• Taking sqrt before mean
• Using absolute difference instead of squared

3. Given the true values y_true = [3, -0.5, 2, 7] and predicted values y_pred = [2.5, 0.0, 2, 8], what is the R² score?

medium

A. 1.0

B. 0.865

C. 0.75

D. 0.948

Solution

1. Step 1: Calculate R² score formula

   R² = 1 - (sum of squared residuals / total sum of squares). Calculate residuals and totals:

2. Step 2: Compute sums

   Residuals squared: (3-2.5)^2 + (-0.5-0)^2 + (2-2)^2 + (7-8)^2 = 0.25 + 0.25 + 0 + 1 = 1.5
   Total sum squares: sum((y_true - mean(y_true))^2) with mean= (3-0.5+2+7)/4=2.875
   Totals: (3-2.875)^2 + (-0.5-2.875)^2 + (2-2.875)^2 + (7-2.875)^2 = 0.015625 + 11.390625 + 0.765625 + 17.015625 = 29.1875

3. Step 3: Calculate R²

   R² = 1 - (1.5 / 29.1875) ≈ 1 - 0.0514 = 0.9486

4. Final Answer:

   0.948 -> Option D

5. Quick Check:

   R² ≈ 0.95 means good fit [OK]

Hint: R² = 1 - (SS_res / SS_tot), calculate sums carefully [OK]

Common Mistakes:

• Using sum of residuals instead of squared residuals
• Forgetting to subtract mean in total sum squares
• Mixing up numerator and denominator

4. You have this Python code to compute MSE but it gives wrong results:

def mse(y_true, y_pred):
    return sum(y_true - y_pred)**2 / len(y_true)

What is the error?

medium

A. The subtraction is done element-wise but sum is outside, causing wrong squaring

B. The function should use absolute values instead of squares

C. The length should be squared, not the sum

D. The function is missing import statements

Solution

1. Step 1: Analyze the code operation order

   The code does sum(y_true - y_pred) first, then squares the total sum, which is incorrect for MSE.

2. Step 2: Correct MSE calculation

   MSE requires squaring each difference first, then averaging. The code should square element-wise before summing.

3. Final Answer:

   The subtraction is done element-wise but sum is outside, causing wrong squaring -> Option A

4. Quick Check:

   Square differences before sum for MSE [OK]

Hint: Square each difference before summing for MSE [OK]

Common Mistakes:

• Squaring sum instead of sum of squares
• Using absolute difference instead of squared
• Ignoring element-wise operations

5. You trained two regression models on different datasets. Model A has MSE=4 and R²=0.8, Model B has MSE=2 and R²=0.6. Which model is better and why?

hard

A. Model B is better because it has lower MSE, indicating smaller average error

B. Model A is better because it has higher R², showing better fit to data

C. Both models are equally good because MSE and R² contradict

D. Cannot decide without RMSE values

Solution

1. Step 1: Understand MSE and R² meanings

   MSE measures average squared error (lower is better). R² measures proportion of variance explained (higher is better).

2. Step 2: Compare models on error and fit

   Model B has lower MSE (2 < 4), so predictions are closer on average. Model A has higher R² but larger error, meaning it fits variance better but with bigger errors.

3. Step 3: Prioritize error for prediction quality

   Lower error (MSE) usually means better prediction accuracy, so Model B is better despite lower R².

4. Final Answer:

   Model B is better because it has lower MSE, indicating smaller average error -> Option A

5. Quick Check:

   Lower MSE means better prediction accuracy [OK]

Hint: Lower MSE beats higher R² for prediction accuracy [OK]

Common Mistakes:

• Choosing model with higher R² ignoring error size
• Thinking MSE and R² always agree
• Ignoring what metric matters for task