Multiple linear regression in ML Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is multiple linear regression?

Multiple linear regression is a method to predict a number by using two or more input features. It finds a straight line (or plane) that best fits the data points in multiple dimensions.

beginner

Write the general formula for multiple linear regression.

The formula is: y = b0 + b1*x1 + b2*x2 + ... + bn*xn + e, where y is the predicted value, b0 is the intercept, b1 to bn are coefficients, x1 to xn are input features, and e is the error term.

beginner

Why do we use multiple linear regression instead of simple linear regression?

Because multiple linear regression can use many features to make better predictions, while simple linear regression uses only one feature. This helps capture more information from the data.

intermediate

What does the coefficient (like b1) represent in multiple linear regression?

Each coefficient shows how much the predicted value changes when that feature increases by one unit, keeping other features constant.

intermediate

How do we measure how well a multiple linear regression model fits the data?

We use metrics like R-squared, which tells us the percentage of variation in the target explained by the features, and Mean Squared Error (MSE), which measures average prediction error.

What does the intercept (b0) in multiple linear regression represent?

A The predicted value when all features are zero

B The slope of the first feature

C The error term

D The average of all features

The intercept is the predicted output when all input features are zero.

Which metric tells how much of the target's variation is explained by the model?

A Mean Squared Error

B Precision

C Accuracy

D R-squared

R-squared shows the proportion of variance in the target explained by the features.

If a coefficient b2 is negative, what does it mean?

A Feature 2 has no effect

B As feature 2 increases, the predicted value decreases

C Feature 2 is the most important

D The model is wrong

A negative coefficient means the predicted value goes down when that feature goes up.

Multiple linear regression can be used when:

A There are multiple input features

B There is only one input feature

C The target is categorical

D The data is images

Multiple linear regression uses two or more input features to predict a continuous target.

What is the error term (e) in the regression formula?

A The sum of all features

B The predicted value

C The difference between actual and predicted values

D The intercept

The error term represents the difference between the actual and predicted values.

Explain how multiple linear regression uses features to make predictions.

Think about how each feature contributes to the final predicted value.

Describe two ways to evaluate the performance of a multiple linear regression model.

One metric shows explained variance, the other shows average error.

Practice

1. What does multiple linear regression do in simple terms?

easy

A. It predicts a result using several input features combined.

B. It groups data into clusters based on similarity.

C. It reduces the number of features by combining them.

D. It finds the shortest path between two points.

Solution

1. Step 1: Understand the purpose of multiple linear regression

   Multiple linear regression uses multiple inputs to predict a single output by finding the best weights for each input.

2. Step 2: Compare options with this understanding

   Only It predicts a result using several input features combined. correctly describes predicting a result using several inputs combined.

3. Final Answer:

   It predicts a result using several input features combined. -> Option A

4. Quick Check:

   Multiple inputs predict output = A [OK]

Hint: Remember: multiple inputs, one output prediction [OK]

Common Mistakes:

• Confusing regression with clustering
• Thinking it reduces features instead of weighting them
• Mixing it up with pathfinding algorithms

2. Which of the following is the correct formula for multiple linear regression prediction?

easy

A. y = w0 + w1*x1 + w2*x2 + ... + wn*xn

B. y = w0 * x1 + w1 * x2 + ... + wn * xn

C. y = w0 / x1 + w1 / x2 + ... + wn / xn

D. y = w0 - w1*x1 - w2*x2 - ... - wn*xn

Solution

1. Step 1: Recall the multiple linear regression formula

   The prediction is a sum of a bias term plus each input multiplied by its weight: y = w0 + w1*x1 + w2*x2 + ... + wn*xn.

2. Step 2: Check each option against the formula

   Only y = w0 + w1*x1 + w2*x2 + ... + wn*xn matches the correct formula with addition and multiplication.

3. Final Answer:

   y = w0 + w1*x1 + w2*x2 + ... + wn*xn -> Option A

4. Quick Check:

   Sum of weighted inputs plus bias = B [OK]

Hint: Sum weighted inputs plus bias for prediction [OK]

Common Mistakes:

• Using multiplication instead of addition between terms
• Dividing weights by inputs
• Subtracting weights instead of adding

3. Given weights w0=2, w1=3, w2=-1 and inputs x1=4, x2=5, what is the predicted value y using multiple linear regression?

medium

A. 9.5

B. 9

C. 5

D. 10

Solution

1. Step 1: Apply the formula y = w0 + w1*x1 + w2*x2

   Substitute values: y = 2 + 3*4 + (-1)*5.

2. Step 2: Calculate the prediction

   Calculate: 3*4 = 12, (-1)*5 = -5, sum = 2 + 12 - 5 = 9.

3. Step 3: Recheck calculation

   Sum is 9, but 10 says 10, so re-check: 2 + 12 - 5 = 9, so correct answer is 9.

4. Final Answer:

   9 -> Option B

5. Quick Check:

   2 + 12 - 5 = 9 [OK]

Hint: Multiply weights by inputs, then add bias [OK]

Common Mistakes:

• Adding weights instead of multiplying by inputs
• Forgetting to add bias term w0
• Mixing up signs of weights

4. Identify the error in this Python code for multiple linear regression prediction:

def predict(x1, x2, w0, w1, w2):
    return w0 + w1 * x1 + w2 + x2

print(predict(3, 4, 1, 2, 3))

medium

A. The print statement syntax is incorrect.

B. The function is missing a return statement.

C. The function parameters are in wrong order.

D. The term w2 + x2 should be w2 * x2 for correct multiplication.

Solution

1. Step 1: Analyze the formula inside the function

   The formula should multiply each input by its weight: w0 + w1*x1 + w2*x2.

2. Step 2: Identify the mistake in the code

   The code uses w2 + x2 instead of w2 * x2, which adds instead of multiplies.

3. Final Answer:

   The term w2 + x2 should be w2 * x2 for correct multiplication. -> Option D

4. Quick Check:

   Multiplication needed, not addition = A [OK]

Hint: Check all input-weight pairs use multiplication [OK]

Common Mistakes:

• Using + instead of * between weight and input
• Forgetting return statement (not here though)
• Mixing parameter order
• Incorrect print syntax

5. You have a dataset with features: hours studied (x1), hours slept (x2), and exam score (y). You fit a multiple linear regression model and get weights: w0=5, w1=4, w2=2. What does the weight w1=4 tell you about hours studied?

hard

A. Hours slept affects exam score more than hours studied.

B. Hours studied has no effect on exam score.

C. For each extra hour studied, exam score increases by 4 points, holding sleep constant.

D. Exam score decreases by 4 points for each hour studied.

Solution

1. Step 1: Understand the meaning of weights in multiple linear regression

   Each weight shows how much the output changes when that input increases by one unit, keeping others fixed.

2. Step 2: Interpret w1=4 for hours studied

   A weight of 4 means exam score increases by 4 points for each additional hour studied, assuming hours slept stays the same.

3. Final Answer:

   For each extra hour studied, exam score increases by 4 points, holding sleep constant. -> Option C

4. Quick Check:

   Weight shows effect per unit input = D [OK]

Hint: Weight = change in output per unit input [OK]

Common Mistakes:

• Thinking weight means no effect
• Confusing positive and negative effect
• Mixing effects of different features