Polynomial regression in ML Python - Model Pipeline Trace

Model Pipeline - Polynomial regression

Polynomial regression fits a curved line to data by adding powers of the input as new features. This helps capture relationships that are not straight lines.

Data Flow - 6 Stages

1Data in

100 rows x 1 column→Raw input feature (e.g., house size in 100 sq ft units)→100 rows x 1 column

[2.5, 3.0, 4.1, 5.5, ...]

↓

2Preprocessing

100 rows x 1 column→Scale input feature to range 0-1→100 rows x 1 column

[0.25, 0.30, 0.41, 0.55, ...]

↓

3Feature Engineering

100 rows x 1 column→Add polynomial features: x, x^2, x^3→100 rows x 3 columns

[[0.25, 0.0625, 0.0156], [0.30, 0.09, 0.027], [0.41, 0.1681, 0.0689], ...]

↓

4Model trains

100 rows x 3 columns→Fit linear regression on polynomial features→Model with 3 coefficients + intercept

Coefficients: [1.2, -0.5, 0.3], Intercept: 0.7

↓

5Metrics improve

Model + training data→Calculate loss and R^2 score→Loss scalar, R^2 scalar

Loss: 0.02, R^2: 0.95

↓

6Prediction

New input feature (1 column)→Transform input to polynomial features and predict output→Predicted output scalar

Input: 0.4, Output: 1.12

Training Trace - Epoch by Epoch

Loss
0.15 |****
0.08 |**
0.04 |*
0.025|*
0.02 |*
      +------------
       1 2 3 4 5 Epochs

Epoch	Loss ↓	Accuracy ↑	Observation
1	0.15	0.70	Initial fit with high error and moderate accuracy
2	0.08	0.85	Loss decreased, model capturing curve better
3	0.04	0.92	Further improvement, curve fits data well
4	0.025	0.95	Loss low, accuracy high, model converging
5	0.02	0.96	Minimal improvement, training stabilizing

Prediction Trace - 3 Layers

Layer 1: Input feature

Layer 2: Polynomial feature expansion

Layer 3: Linear model prediction

Model Quiz - 3 Questions

What does adding polynomial features do in this pipeline?

A It allows the model to fit curved relationships.

B It reduces the number of input features.

C It normalizes the output values.

D It removes noise from the data.

Key Insight

Polynomial regression extends simple linear regression by adding powers of the input feature. This lets the model learn curved patterns in data, improving fit when relationships are not straight lines.

Practice

1. What is the main purpose of polynomial regression in machine learning?

easy

A. To perform clustering on unlabeled data

B. To reduce the number of features in the dataset

C. To classify data into categories

D. To model non-linear relationships by adding powers of input features

Solution

1. Step 1: Understand the goal of polynomial regression

   Polynomial regression extends linear regression by adding powers of input features to capture curves in data.

2. Step 2: Compare options with this goal

   Only To model non-linear relationships by adding powers of input features describes modeling non-linear relationships by adding powers of features, which matches polynomial regression.

3. Final Answer:

   To model non-linear relationships by adding powers of input features -> Option D

4. Quick Check:

   Polynomial regression = model curves [OK]

Hint: Polynomial regression fits curves by adding powers of inputs [OK]

Common Mistakes:

• Confusing polynomial regression with classification
• Thinking it reduces features instead of expanding them
• Mixing it up with clustering methods

2. Which Python code snippet correctly creates polynomial features of degree 3 using scikit-learn?

easy

A. from sklearn.linear_model import PolynomialFeatures
poly = PolynomialFeatures(3)
X_poly = poly.transform(X)

B. from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(degree=3)
X_poly = poly.fit_transform(X)

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

D. from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(degree=3)
X_poly = poly.transform(X)

Solution

1. Step 1: Check import and class usage

   PolynomialFeatures is in sklearn.preprocessing, so import must be from there.

2. Step 2: Verify method calls and degree

   Degree=3 is correct. fit_transform(X) creates polynomial features properly. from sklearn.preprocessing import PolynomialFeatures poly = PolynomialFeatures(degree=3) X_poly = poly.fit_transform(X) matches all these.

3. Final Answer:

   from sklearn.preprocessing import PolynomialFeatures poly = PolynomialFeatures(degree=3) X_poly = poly.fit_transform(X) -> Option B

4. Quick Check:

   Correct import + degree + fit_transform = A [OK]

Hint: Use fit_transform with PolynomialFeatures from sklearn.preprocessing [OK]

Common Mistakes:

• Importing PolynomialFeatures from wrong module
• Using transform without fit first
• Setting wrong degree value

3. Given this code, what will be the shape of X_poly if X has shape (5, 1) and degree=2?

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

medium

A. (5, 1)

B. (5, 2)

C. (5, 3)

D. (5, 4)

Solution

1. Step 1: Understand PolynomialFeatures output shape

   For degree=2 and one feature, output includes [1, x, x^2], so 3 features total.

2. Step 2: Calculate shape based on input rows and new features

   Input has 5 rows, so output shape is (5, 3).

3. Final Answer:

   (5, 3) -> Option C

4. Quick Check:

   Degree 2 with 1 feature = 3 columns [OK]

Hint: Output columns = combinations with degree + 1 for bias [OK]

Common Mistakes:

• Forgetting the bias (constant 1) column
• Assuming output shape equals input shape
• Confusing degree with number of features

4. Identify the error in this polynomial regression code snippet:

from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
poly = PolynomialFeatures(degree=2)
X_poly = poly.transform(X)
model = LinearRegression()
model.fit(X_poly, y)

medium

A. Using transform before fitting PolynomialFeatures

B. LinearRegression cannot be used with polynomial features

C. Missing import for LinearRegression

D. Degree should be 1 for polynomial regression

Solution

1. Step 1: Check usage of PolynomialFeatures methods

   PolynomialFeatures must be fitted before transform; fit_transform fits and transforms in one step.

2. Step 2: Identify the error in code

   Code calls poly.transform(X) without fitting first, causing error. Correct is poly.fit_transform(X).

3. Final Answer:

   Using transform before fitting PolynomialFeatures -> Option A

4. Quick Check:

   fit_transform needed before transform [OK]

Hint: Always fit PolynomialFeatures before transform or use fit_transform [OK]

Common Mistakes:

• Calling transform without fit first
• Thinking LinearRegression can't handle polynomial features
• Ignoring method order in preprocessing

5. You have data with a clear curve pattern. You fit a polynomial regression model with degree 5 and get very low training error but high error on new data. What is the best next step?

hard

A. Reduce the polynomial degree to avoid overfitting

B. Increase the polynomial degree to 10 for better fit

C. Remove polynomial features and use linear regression only

D. Add more polynomial features of degree 6

Solution

1. Step 1: Understand the problem of low training but high test error

   This indicates overfitting: model fits training data too closely but fails to generalize.

2. Step 2: Choose a solution to reduce overfitting

   Reducing polynomial degree simplifies the model, helping it generalize better on new data.

3. Final Answer:

   Reduce the polynomial degree to avoid overfitting -> Option A

4. Quick Check:

   Overfitting? Lower degree polynomial [OK]

Hint: Lower degree to reduce overfitting when test error is high [OK]

Common Mistakes:

• Increasing degree worsens overfitting
• Removing polynomial features loses curve fit
• Adding higher degree features increases complexity