Regression evaluation (MSE, RMSE, R²) in ML Python - ML Experiment: Train & Evaluate

Experiment - Regression evaluation (MSE, RMSE, R²)

Problem:You have trained a regression model to predict house prices. The model's performance is measured by Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and R-squared (R²). Currently, the model shows a high MSE and low R² on the test data, indicating poor prediction accuracy.

Current Metrics:MSE: 250000000, RMSE: 15811.39, R²: 0.45

Issue:The model predictions are not accurate enough. The high MSE and low R² show the model does not explain the data well.

Your Task

Improve the regression model evaluation by calculating and interpreting MSE, RMSE, and R² correctly on test data. Aim to reduce MSE and RMSE and increase R² compared to current metrics.

Do not change the model architecture or training process.

Only modify the evaluation code to correctly compute metrics.

Use sklearn metrics functions.

Hint 1

Hint 2

Hint 3

Hint 4

Solution

ML Python

from sklearn.datasets import load_boston
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
import numpy as np

# Load dataset
boston = load_boston()
X = boston.data
y = boston.target

# Split data
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

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

# Predict
y_pred = model.predict(X_test)

# Calculate metrics
mse = mean_squared_error(y_test, y_pred)
rmse = np.sqrt(mse)
r2 = r2_score(y_test, y_pred)

print(f"MSE: {mse:.2f}")
print(f"RMSE: {rmse:.2f}")
print(f"R²: {r2:.2f}")

Added correct calculation of MSE using mean_squared_error function.

Computed RMSE by taking square root of MSE.

Calculated R² using r2_score function.

Ensured predictions and true values are correctly matched.

Results Interpretation

Before: MSE = 250000000, RMSE = 15811.39, R² = 0.45

After: MSE = 24.29, RMSE = 4.93, R² = 0.67

Correctly calculating and interpreting regression metrics helps understand model performance better. Lower MSE and RMSE mean predictions are closer to true values. Higher R² means the model explains more variance in the data.

Bonus Experiment

Try evaluating the model using Mean Absolute Error (MAE) and compare it with MSE and RMSE.

💡 Hint

Use mean_absolute_error from sklearn.metrics and observe how MAE differs in sensitivity to outliers compared to MSE.

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