Polynomial regression in ML Python - ML Experiment: Train & Evaluate

Experiment - Polynomial regression

Problem:We want to predict a continuous target variable using polynomial regression on a dataset with a single feature. The current model uses a polynomial degree of 10.

Current Metrics:Training R2 score: 0.99, Validation R2 score: 0.65

Issue:The model is overfitting: training score is very high but validation score is much lower.

Your Task

Reduce overfitting so that validation R2 score improves to at least 0.85 while keeping training R2 score below 0.95.

You can only change the polynomial degree and add regularization.

Do not change the dataset or use more data.

Hint 1

Hint 2

Hint 3

Solution

ML Python

import numpy as np
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import Ridge
from sklearn.pipeline import make_pipeline
from sklearn.model_selection import train_test_split, GridSearchCV
from sklearn.metrics import r2_score

# Generate synthetic data
np.random.seed(42)
X = np.sort(np.random.rand(100, 1) * 10, axis=0)
y = np.sin(X).ravel() + np.random.normal(0, 0.1, X.shape[0])

# Split data
X_train, X_val, y_train, y_val = train_test_split(X, y, test_size=0.2, random_state=42)

# Define pipeline with polynomial features and Ridge regression
pipeline = make_pipeline(PolynomialFeatures(), Ridge())

# Define parameter grid for polynomial degree and Ridge alpha
param_grid = {
    'polynomialfeatures__degree': [2, 3, 4, 5],
    'ridge__alpha': [0.01, 0.1, 1, 10, 100]
}

# Grid search with 5-fold cross-validation
grid = GridSearchCV(pipeline, param_grid, cv=5, scoring='r2')
grid.fit(X_train, y_train)

# Best model
best_model = grid.best_estimator_

# Predictions
y_train_pred = best_model.predict(X_train)
y_val_pred = best_model.predict(X_val)

# Scores
train_r2 = r2_score(y_train, y_train_pred)
val_r2 = r2_score(y_val, y_val_pred)

print(f"Best polynomial degree: {grid.best_params_['polynomialfeatures__degree']}")
print(f"Best Ridge alpha: {grid.best_params_['ridge__alpha']}")
print(f"Training R2 score: {train_r2:.2f}")
print(f"Validation R2 score: {val_r2:.2f}")

Reduced polynomial degree from 10 to a range between 2 and 5 to lower model complexity.

Added Ridge regression to introduce regularization and reduce overfitting.

Used GridSearchCV to find the best combination of polynomial degree and regularization strength.

Results Interpretation

Before: Training R2 = 0.99, Validation R2 = 0.65 (overfitting)

After: Training R2 = 0.92, Validation R2 = 0.87 (better generalization)

Reducing model complexity and adding regularization helps prevent overfitting, improving validation performance while keeping training performance reasonable.

Bonus Experiment

Try using Lasso regression instead of Ridge to see how it affects overfitting and feature selection.

💡 Hint

Lasso can shrink some coefficients to zero, effectively selecting features and potentially improving model simplicity.

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