Residual analysis in ML Python - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Residuals in Regression

What does a residual represent in a regression model?

A The predicted value of the dependent variable

B The difference between the observed value and the predicted value

C The average value of the dependent variable

D The slope of the regression line

💡 Hint

Think about what remains unexplained after prediction.

A residual is the difference between the actual observed value and the value predicted by the regression model. It shows the error for each data point.

❓ Predict Output

intermediate

Calculate Residuals from Predictions

Given the true values and predicted values below, what is the list of residuals?

ML Python

true_values = [10, 15, 20, 25]
predicted_values = [8, 14, 22, 24]
residuals = [true - pred for true, pred in zip(true_values, predicted_values)]
print(residuals)

A [2, 1, 2, -1]

B [-2, -1, 2, -1]

C [2, -1, -2, 1]

D [2, 1, -2, 1]

💡 Hint

Subtract predicted from true values element-wise.

Residuals are calculated as true value minus predicted value for each pair.

❓ Metrics

advanced

Choosing the Correct Residual Metric

Which metric best summarizes the average magnitude of residuals without canceling positive and negative errors?

A Mean Residual

B Mean Squared Error (MSE)

C Mean Absolute Error (MAE)

D Residual Sum

💡 Hint

Think about a metric that treats all errors equally regardless of sign.

MAE calculates the average absolute difference between predicted and actual values, so it does not cancel out positive and negative residuals.

🔧 Debug

advanced

Identify the Error in Residual Calculation Code

What error will this code produce when calculating residuals?

ML Python

true_vals = [5, 7, 9]
pred_vals = [4, 6]
residuals = [t - p for t, p in zip(true_vals, pred_vals)]
print(residuals)

A No error; outputs [1, 1]

B IndexError due to different list lengths

C TypeError due to subtraction of incompatible types

D ValueError due to zip mismatch

💡 Hint

Recall how zip works with lists of different lengths.

zip stops at the shortest list length, so no error occurs and residuals list has length 2.

❓ Model Choice

expert

Model Selection Based on Residual Patterns

You observe a pattern in residuals that increases with predicted values, indicating heteroscedasticity. Which model adjustment is best to address this?

A Use weighted least squares regression giving less weight to points with larger residuals

B Switch to a linear regression without intercept

C Increase the number of features in the model

D Use a decision tree regression without residual analysis

💡 Hint

Consider methods that handle changing variance in residuals.

Weighted least squares regression accounts for heteroscedasticity by weighting points inversely to their variance, improving model fit.

Practice

1. What does a residual represent in machine learning?

easy

A. The difference between the actual and predicted values

B. The predicted value itself

C. The average of all predicted values

D. The sum of all actual values

Solution

1. Step 1: Understand residual definition

   A residual is the error between what the model predicts and what actually happened.

2. Step 2: Match with options

   The difference between the actual and predicted values correctly states this difference, while others describe unrelated concepts.

3. Final Answer:

   The difference between the actual and predicted values -> Option A

4. Quick Check:

   Residual = Actual - Predicted [OK]

Hint: Residuals = Actual minus Predicted values [OK]

Common Mistakes:

• Confusing residual with predicted value
• Thinking residual is the average error
• Mixing residual with total sum of values

2. Which of the following is the correct formula to calculate a residual for a data point?

easy

A. Residual = Actual value + Predicted value

B. Residual = Predicted value - Actual value

C. Residual = Actual value - Predicted value

D. Residual = Predicted value / Actual value

Solution

1. Step 1: Recall residual calculation formula

   Residual is calculated as Actual value minus Predicted value.

2. Step 2: Identify correct option

   Residual = Actual value - Predicted value matches the formula exactly; others are incorrect operations.

3. Final Answer:

   Residual = Actual value - Predicted value -> Option C

4. Quick Check:

   Residual = Actual - Predicted [OK]

Hint: Subtract predicted from actual to get residual [OK]

Common Mistakes:

• Reversing the subtraction order
• Adding values instead of subtracting
• Dividing values instead of subtracting

3. Given the actual values [3, 5, 7] and predicted values [2, 5, 8], what is the residual for the third data point?

medium

A. 1

B. -1

C. 0

D. 2

Solution

1. Step 1: Identify actual and predicted values for third point

   Actual = 7, Predicted = 8 for the third data point.

2. Step 2: Calculate residual

   Residual = Actual - Predicted = 7 - 8 = -1.

3. Final Answer:

   -1 -> Option B

4. Quick Check:

   7 - 8 = -1 [OK]

Hint: Subtract predicted from actual for the point [OK]

Common Mistakes:

• Mixing up actual and predicted values
• Calculating predicted minus actual
• Using wrong index for data point

4. You have a residuals list: [2, -3, 4, -1]. Which code snippet correctly calculates the mean squared error (MSE)?

medium

A. mse = sum([r**2 for r in residuals]) / len(residuals)

B. mse = sum([abs(r) for r in residuals]) / len(residuals)

C. mse = max(residuals) - min(residuals)

D. mse = sum(residuals) / len(residuals)

Solution

1. Step 1: Understand MSE formula

   MSE is the average of squared residuals, so square each residual and average them.

2. Step 2: Match code to formula

   mse = sum([r**2 for r in residuals]) / len(residuals) squares each residual and divides by count, matching MSE calculation.

3. Final Answer:

   mse = sum([r**2 for r in residuals]) / len(residuals) -> Option A

4. Quick Check:

   MSE = average of squared residuals [OK]

Hint: Square residuals then average for MSE [OK]

Common Mistakes:

• Using sum instead of sum of squares
• Calculating mean absolute error instead
• Using max-min difference instead of average

5. A model's residual plot shows a clear curve pattern instead of random scatter. What does this indicate about the model?

hard

A. The model fits the data perfectly

B. The model has no bias

C. The residuals are normally distributed

D. The model misses a non-linear pattern in data

Solution

1. Step 1: Interpret residual plot pattern

   A curve pattern in residuals suggests systematic error, meaning the model misses some pattern.

2. Step 2: Identify what pattern means

   This usually means the model is too simple and misses non-linear relationships.

3. Final Answer:

   The model misses a non-linear pattern in data -> Option D

4. Quick Check:

   Curved residuals = missed non-linear pattern [OK]

Hint: Curved residuals mean model misses pattern [OK]

Common Mistakes:

• Assuming perfect fit from residual pattern
• Confusing residual distribution with residual pattern
• Ignoring bias presence in model