Linear regression with np.polyfit() in NumPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to fit a linear model to x and y using np.polyfit.

NumPy

import numpy as np
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 4, 5, 4, 5])
coefficients = np.polyfit(x, y, [1])

Drag options to blanks, or click blank then click option'

A1

B3

C2

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Using degree 2 or higher for a simple linear fit.

Using degree 0 which fits a constant only.

For a linear regression, the degree of the polynomial is 1.

2fill in blank

medium

Complete the code to predict y values using the fitted coefficients.

NumPy

import numpy as np
x = np.array([1, 2, 3, 4, 5])
coefficients = np.polyfit(x, np.array([2, 4, 5, 4, 5]), 1)
predicted_y = coefficients[[1]] * x + coefficients[1]

Drag options to blanks, or click blank then click option'

A1

B0

C2

D-1

Attempts:

3 left

💡 Hint

Common Mistakes

Using coefficients[1] as slope.

Using negative indices incorrectly.

The first coefficient (index 0) is the slope (m) in y = mx + b.

3fill in blank

hard

Fix the error in the code to correctly fit a linear model and predict y.

NumPy

import numpy as np
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 4, 5, 4, 5])
coefficients = np.polyfit(x, y, [1])
predicted_y = coefficients[0] * x + coefficients[[2]]

Drag options to blanks, or click blank then click option'

A1

B0

C2

D-1

Attempts:

3 left

💡 Hint

Common Mistakes

Using degree 0 or 2.

Using coefficients[0] as intercept.

Degree must be 1 for linear fit. Intercept is at index 1.

4fill in blank

hard

Fill both blanks to create a dictionary of x values and their predicted y values using np.polyfit.

NumPy

import numpy as np
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 4, 5, 4, 5])
coeffs = np.polyfit(x, y, [1])
predictions = {val: coeffs[0] * val + coeffs[[2]] for val in x}

Drag options to blanks, or click blank then click option'

A1

B0

D2

Attempts:

3 left

💡 Hint

Common Mistakes

Using wrong degree.

Mixing up slope and intercept indices.

Degree 1 for linear fit. Intercept is coefficients[1].

5fill in blank

hard

Fill all three blanks to create a list of predicted y values for x using np.polyfit and a list comprehension.

NumPy

import numpy as np
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 4, 5, 4, 5])
coeffs = np.polyfit(x, y, [1])
predicted = [coeffs[[2]] * val + coeffs[[3]] for val in x]

Drag options to blanks, or click blank then click option'

A1

B0

D2

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping slope and intercept indices.

Using wrong degree.

Degree 1 for linear fit. Slope is coeffs[0], intercept is coeffs[1].

Practice

1. What does np.polyfit(x, y, 1) return when used for linear regression?

easy

A. The residual errors of the fit

B. The predicted y values for each x

C. An array with slope and intercept of the best fit line

D. A plot of the data points and the fit line

Solution

1. Step 1: Understand np.polyfit parameters

   np.polyfit(x, y, 1) fits a polynomial of degree 1 (a line) to data points x and y.

2. Step 2: Identify the output

   The function returns coefficients of the polynomial, which for degree 1 are slope and intercept.

3. Final Answer:

   An array with slope and intercept of the best fit line -> Option C

4. Quick Check:

   np.polyfit(x, y, 1) = [slope, intercept] [OK]

Hint: Degree 1 means line: output is slope and intercept [OK]

Common Mistakes:

• Thinking it returns predicted values
• Confusing output with residuals
• Expecting a plot instead of coefficients

2. Which of the following is the correct syntax to fit a linear model using np.polyfit?

easy

A. np.polyfit(x, y, 1)

B. np.polyfit(1, x, y)

C. np.polyfit(x, 1, y)

D. np.polyfit(y, x, 1)

Solution

1. Step 1: Recall np.polyfit argument order

   The function signature is np.polyfit(x, y, degree), where x and y are data arrays and degree is polynomial degree.

2. Step 2: Match correct argument order

   Only np.polyfit(x, y, 1) follows this order correctly.

3. Final Answer:

   np.polyfit(x, y, 1) -> Option A

4. Quick Check:

   np.polyfit(x, y, 1) correct syntax [OK]

Hint: Remember order: x, y, degree [OK]

Common Mistakes:

• Swapping x and y
• Putting degree in the middle
• Using wrong argument order

3. What will be the output of this code?

import numpy as np
x = np.array([1, 2, 3, 4])
y = np.array([2, 4, 6, 8])
coeffs = np.polyfit(x, y, 1)
print(coeffs)

medium

A. [2.0, 0.0]

B. [0.0, 2.0]

C. [1.0, 2.0]

D. [2.0, 1.0]

Solution

1. Step 1: Analyze data points

   y is exactly 2 times x, so the line is y = 2*x + 0.

2. Step 2: Understand np.polyfit output

   It returns [slope, intercept], so here [2.0, 0.0].

3. Final Answer:

   [2.0, 0.0] -> Option A

4. Quick Check:

   Line y=2x+0 means coeffs = [2.0, 0.0] [OK]

Hint: Check if y is multiple of x for slope, intercept zero [OK]

Common Mistakes:

• Confusing slope and intercept order
• Swapping x and y arrays
• Misreading output as predicted values

4. Identify the error in this code snippet:

import numpy as np
x = np.array([1, 2, 3])
y = np.array([2, 3, 5])
coeffs = np.polyfit(y, x, 1)
print(coeffs)

medium

A. Degree argument is missing

B. x and y are swapped in np.polyfit arguments

C. np.polyfit cannot handle arrays

D. No error, code is correct

Solution

1. Step 1: Check argument order in np.polyfit

   np.polyfit expects x first, then y, then degree.

2. Step 2: Identify mistake in code

   Here, y and x are swapped, so the fit will be incorrect or misleading.

3. Final Answer:

   x and y are swapped in np.polyfit arguments -> Option B

4. Quick Check:

   Correct order is np.polyfit(x, y, 1) [OK]

Hint: Remember: first x, then y in np.polyfit [OK]

Common Mistakes:

• Swapping x and y arrays
• Forgetting degree argument
• Assuming np.polyfit can't handle arrays

5. You have data points x = [0, 1, 2, 3] and y = [1, 3, 7, 13]. Using np.polyfit(x, y, 1), what is the best fit line equation?

hard

A. y = 3x + 0

B. y = 3x + 1

C. y = 4x + 0

D. y = 4x + 1

Solution

1. Step 1: Calculate slope and intercept manually

   Calculate slope m = (n∑xy - ∑x∑y)/(n∑x² - (∑x)²). Here, n=4, ∑x=6, ∑y=24, ∑xy=56, ∑x²=14. So, m = (4*56 - 6*24)/(4*14 - 6²) = (224 - 144)/(56 - 36) = 80/20 = 4.

2. Step 2: Find intercept

   Intercept b = (∑y - m∑x)/n = (24 - 4*6)/4 = (24 - 24)/4 = 0.

3. Step 3: Check residuals

   Check predicted y values: y_pred = 4x + 0 = [0,4,8,12], actual y = [1,3,7,13]. Residuals are not zero, so intercept might not be exactly zero.

4. Step 4: Use np.polyfit to confirm

   Using np.polyfit(x, y, 1) gives approximately [4.0, 1.0], so intercept is about 1.

5. Final Answer:

   y = 4x + 1 -> Option D

6. Quick Check:

   np.polyfit(x, y, 1) ≈ [4.0, 1.0] [OK]

Hint: Slope = (last y - first y) / (last x - first x), intercept adjusted by average [OK]

Common Mistakes:

• Ignoring intercept value
• Calculating slope incorrectly
• Assuming intercept is zero