Polynomial features in Data Analysis Python - Step-by-Step Execution

Concept Flow - Polynomial features

Start with original features

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Choose polynomial degree

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Generate new features by multiplying original features

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Add interaction terms (products of different features)

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Combine all polynomial features into new dataset

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Use new features for modeling

We start with original data features, pick a polynomial degree, create new features by multiplying originals and their combinations, then use these new features for analysis.

Execution Sample

Data Analysis Python

from sklearn.preprocessing import PolynomialFeatures
import pandas as pd

X = pd.DataFrame({'x1': [1, 2], 'x2': [3, 4]})
poly = PolynomialFeatures(degree=2, include_bias=False)
X_poly = poly.fit_transform(X)

This code creates polynomial features of degree 2 from two original features x1 and x2.

Execution Table

Step	Input Features	Degree	Generated Features	Output Shape
1	{'x1': 1, 'x2': 3}	2	[x1=1, x2=3, x1^2=1, x1*x2=3, x2^2=9]	(1, 5)
2	{'x1': 2, 'x2': 4}	2	[x1=2, x2=4, x1^2=4, x1*x2=8, x2^2=16]	(1, 5)
3	All rows processed	2	Combined polynomial features for all samples	(2, 5)
4	End	-	-	Transformation complete

💡 All input rows processed and polynomial features generated for degree 2

Variable Tracker

Variable	Start	After 1	After 2	Final
X	{'x1': [1,2], 'x2': [3,4]}	{'x1': 1, 'x2': 3}	{'x1': 2, 'x2': 4}	{'x1': [1,2], 'x2': [3,4]}
X_poly	None	[1, 3, 1, 3, 9]	[2, 4, 4, 8, 16]	[[1,3,1,3,9],[2,4,4,8,16]]

Key Moments - 3 Insights

Why does the output have more features than the input?

Because polynomial features include original features, their squares, and interaction terms (products), increasing the total number of features as shown in execution_table rows 1 and 2.

What does include_bias=False mean in PolynomialFeatures?

It means the output does not include a constant feature of 1 (bias term). This is why the output features start with original features and their combinations, not a column of ones.

How does degree affect the number of generated features?

Higher degree means more combinations and powers of features, so the number of generated features grows quickly, as seen by the polynomial terms in execution_table rows 1 and 2.

Visual Quiz - 3 Questions

Look at the execution_table at Step 1, what is the value of the interaction term x1*x2?

A 1

B 9

C 3

D 0

💡 Hint

Check the Generated Features column at Step 1 in the execution_table.

At which step does the transformation process complete for all input rows?

A Step 3

B Step 1

C Step 2

D Step 4

💡 Hint

Look for the step mentioning 'All rows processed' in the execution_table.

If we set degree=1, how would the number of generated features change compared to degree=2?

A It would be the same number of features

B It would be fewer features, only original ones

C It would be more features, including squares

D It would include only interaction terms

💡 Hint

Refer to the concept_snapshot about degree and feature generation.

Concept Snapshot

PolynomialFeatures creates new features by raising original features to powers and multiplying them.
Syntax: PolynomialFeatures(degree=n, include_bias=False).
Degree controls max power and interaction terms.
Output shape grows with degree and number of original features.
Useful to capture nonlinear relationships in data.

Full Transcript

Polynomial features start from original data columns. We pick a degree, like 2, to create new features by multiplying and squaring originals. For example, with features x1 and x2, degree 2 creates x1, x2, x1 squared, x1 times x2, and x2 squared. This increases the number of features, helping models learn more complex patterns. The code example uses sklearn's PolynomialFeatures to do this transformation. The execution table shows step-by-step how each row's features expand. Key points include why output features increase, what include_bias means, and how degree affects feature count. The visual quiz checks understanding of interaction terms, process steps, and degree effects. The snapshot summarizes usage and behavior for quick reference.