Polynomial features in ML Python

Introduction

Polynomial features help us find patterns that are not straight lines by adding powers of the original data. This lets simple models learn curves and more complex shapes.

When you want to predict something that changes in a curve, like the growth of plants over time.

When your data points don't line up in a straight line but seem to follow a curve.

When you want to improve a simple model by giving it more information about combinations of features.

When you want to capture interactions between features without manually creating new columns.

Syntax

ML Python

from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(degree=2, include_bias=False)
X_poly = poly.fit_transform(X)

degree controls the highest power of features to include.

include_bias=False means we don't add a column of ones (intercept).

Examples

Adds original features, their squares, and pairwise products.

ML Python

poly = PolynomialFeatures(degree=2)
X_poly = poly.fit_transform(X)

Adds features up to cubes without the bias column.

ML Python

poly = PolynomialFeatures(degree=3, include_bias=False)
X_poly = poly.fit_transform(X)

Returns the original features plus a bias column (if include_bias=True).

ML Python

poly = PolynomialFeatures(degree=1)
X_poly = poly.fit_transform(X)

Sample Model

This example shows how to turn two features into polynomial features, train a simple model, and check how well it fits.

ML Python

from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
import numpy as np

# Sample data: hours studied and hours slept
X = np.array([[1, 2], [2, 3], [3, 4], [4, 5]])
y = np.array([5, 7, 9, 11])  # target: test score

# Create polynomial features of degree 2
poly = PolynomialFeatures(degree=2, include_bias=False)
X_poly = poly.fit_transform(X)

# Train linear regression on polynomial features
model = LinearRegression()
model.fit(X_poly, y)

# Predict on training data
y_pred = model.predict(X_poly)

# Calculate mean squared error
mse = mean_squared_error(y, y_pred)

print(f"Original features:\n{X}")
print(f"Polynomial features:\n{X_poly}")
print(f"Predictions: {y_pred}")
print(f"Mean Squared Error: {mse:.4f}")

Important Notes

Polynomial features can increase the number of columns quickly, so use small degrees for many features.

Always scale your data before polynomial transformation if features have very different scales.

Higher degree polynomials can cause overfitting, so test carefully.

Summary

Polynomial features add powers and combinations of original features to help models learn curves.

Use PolynomialFeatures from scikit-learn to create these new features easily.

Be careful with degree size to avoid too many features or overfitting.

Practice

1. What is the main purpose of using PolynomialFeatures in machine learning?

easy

A. To create new features by adding powers and combinations of existing features

B. To reduce the number of features in the dataset

C. To normalize the data between 0 and 1

D. To split the dataset into training and testing sets

Solution

1. Step 1: Understand the role of PolynomialFeatures

   PolynomialFeatures generates new features by raising existing features to powers and combining them, helping models learn curves.

2. Step 2: Compare with other options

   Feature reduction, normalization between 0 and 1, and splitting into training/testing sets describe different preprocessing steps, not feature creation with powers.

3. Final Answer:

   To create new features by adding powers and combinations of existing features -> Option A

4. Quick Check:

   PolynomialFeatures = create new polynomial features [OK]

Hint: PolynomialFeatures adds powers and combos of features [OK]

Common Mistakes:

• Confusing feature creation with normalization
• Thinking it reduces features instead of expanding
• Mixing it up with data splitting

2. Which of the following is the correct way to import and create polynomial features of degree 2 using scikit-learn?

easy

A. from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(degree=2)

B. from sklearn.linear_model import PolynomialFeatures
poly = PolynomialFeatures(2)

C. import PolynomialFeatures from sklearn.preprocessing
poly = PolynomialFeatures(degree=2)

D. from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(3)

Solution

1. Step 1: Check the correct import statement

   PolynomialFeatures is in sklearn.preprocessing, so 'from sklearn.preprocessing import PolynomialFeatures' is correct.

2. Step 2: Verify the degree parameter

   To create degree 2 features, use degree=2 in the constructor.

3. Final Answer:

   from sklearn.preprocessing import PolynomialFeatures poly = PolynomialFeatures(degree=2) -> Option A

4. Quick Check:

   Import from preprocessing and set degree=2 [OK]

Hint: Import from preprocessing and set degree=2 [OK]

Common Mistakes:

• Importing from wrong module
• Forgetting 'degree=' keyword
• Setting wrong degree value

3. Given the code below, what is the output of X_poly?

from sklearn.preprocessing import PolynomialFeatures
import numpy as np
X = np.array([[2, 3]])
poly = PolynomialFeatures(degree=2, include_bias=False)
X_poly = poly.fit_transform(X)
print(X_poly)

medium

A. [[2 3 5 6 9]]

B. [[1 2 3 4 6 9]]

C. [[2 3 4 6 9]]

D. [[2 3 4 5 6 9]]

Solution

1. Step 1: Understand PolynomialFeatures output with degree=2 and include_bias=False

   Features include original features, their squares, and pairwise products: [x1, x2, x1^2, x1*x2, x2^2].

2. Step 2: Calculate values for X = [2, 3]

   x1=2, x2=3; x1^2=4, x1*x2=6, x2^2=9; so output is [[2, 3, 4, 6, 9]].

3. Final Answer:

   [[2 3 4 6 9]] -> Option C

4. Quick Check:

   Polynomial features = original + squares + products [OK]

Hint: Output includes original, squares, and cross-products [OK]

Common Mistakes:

• Including bias term when include_bias=False
• Miscomputing squares or products
• Adding extra features not in degree 2

4. Identify the error in the following code snippet that uses PolynomialFeatures:

from sklearn.preprocessing import PolynomialFeatures
X = [[1, 2], [3, 4]]
poly = PolynomialFeatures(degree=3)
X_poly = poly.fit_transform(X)
print(X_poly)

medium

A. X should be a NumPy array, not a list of lists

B. No error; code runs correctly

C. Missing import for NumPy

D. Degree 3 is not supported by PolynomialFeatures

Solution

1. Step 1: Check input type compatibility

   PolynomialFeatures accepts lists or arrays, so X as list of lists is valid.

2. Step 2: Verify degree parameter and imports

   Degree 3 is supported; no NumPy import needed if not used explicitly.

3. Final Answer:

   No error; code runs correctly -> Option B

4. Quick Check:

   PolynomialFeatures accepts lists and degree 3 [OK]

Hint: PolynomialFeatures accepts lists; degree 3 is valid [OK]

Common Mistakes:

• Assuming input must be NumPy array
• Thinking degree 3 is invalid
• Expecting import errors without NumPy usage

5. You have a dataset with 3 features and want to add polynomial features up to degree 3. How many features will the transformed dataset have if include_bias=False?

hard

A. 10

B. 20

C. 16

D. 19

Solution

1. Step 1: Use formula for number of polynomial features

   Number of features = C(n + d, d) - 1 if include_bias=False, where n=3, d=3.

2. Step 2: Calculate combinations

   C(3+3, 3) = C(6, 3) = 20; subtract 1 for no bias gives 19 features.

3. Final Answer:

   19 -> Option D

4. Quick Check:

   Features = combinations(6,3)-1 = 19 [OK]

Hint: Use combinations(n+d, d) minus bias if excluded [OK]

Common Mistakes:

• Forgetting to subtract bias feature
• Using wrong combination formula
• Confusing degree with number of features