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As a data enthusiast, I often find myself diving into the world of predictive modeling and machine learning. Recently, I embarked on a project that involved creating and refining various regression models using Python. The journey not only enhanced my technical skills but also deepened my understanding of how different approaches to modeling can impact results. In this blog, I’ll share my experiences, insights, and the Python code I used to achieve these results.

Understanding the Data

The first step in any data science project is understanding the data at hand. For this project, I worked with a dataset that included various features of cars, such as year, mileage, tax, mpg, and engineSize. My goal was to predict the price of the cars based on these features.

Data Preparation

Before jumping into modeling, I needed to prepare my data. This involved cleaning, transforming, and augmenting it. Here’s how I approached this task in Python:

import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import Ridge
from sklearn.metrics import mean_squared_error, r2_score
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import GridSearchCV

# Load the dataset
df = pd.read_csv('car_data.csv')

# Split the data into features and target
features = ['year', 'mileage', 'tax', 'mpg', 'engineSize']
target = 'price'

X = df[features]
y = df[target]

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

In the code above, I used the train_test_split function to divide the dataset into training (80%) and testing (20%) sets. This is crucial for evaluating the model’s performance later.

Model Development

Base Model

I began my modeling journey with a simple linear regression model to establish a baseline.

from sklearn.linear_model import LinearRegression

# Create and fit the linear regression model
lin_reg = LinearRegression()
lin_reg.fit(X_train, y_train)

# Predict on the test set
y_pred = lin_reg.predict(X_test)

# Calculate R² and MSE
r2 = r2_score(y_test, y_pred)
mse = mean_squared_error(y_test, y_pred)

print(f'R²: {r2}, Mean Squared Error (MSE): {mse}')

Running this code produced the following results:

• R²: 0.6917
• Mean Squared Error (MSE): 6912744.91

These metrics indicated that the linear regression model did a decent job of predicting the car prices based on the features.

Polynomial Features

Next, I decided to explore the impact of polynomial features to see if I could enhance the model’s performance.

# Create a pipeline with polynomial features and linear regression
poly_pipeline = Pipeline([
    ('poly', PolynomialFeatures(degree=2)),
    ('scaler', StandardScaler()),
    ('regressor', LinearRegression())
])

# Fit the model
poly_pipeline.fit(X_train, y_train)

# Predict on the test set
y_poly_pred = poly_pipeline.predict(X_test)

# Calculate R² and MSE
poly_r2 = r2_score(y_test, y_poly_pred)
poly_mse = mean_squared_error(y_test, y_poly_pred)

print(f'Polynomial Model R²: {poly_r2}, Mean Squared Error (MSE): {poly_mse}')

The results showed a slight decrease in performance:

• R²: 0.7667
• MSE: 5234038.0655

While polynomial features added complexity to the model, they did not significantly improve predictive power.

Ridge Regression

Next, I ventured into regularization with Ridge regression, aiming to prevent overfitting.

# Create and fit a Ridge regression model
ridge_model = Ridge(alpha=0.1)
ridge_model.fit(X_train, y_train)

# Predict on the test set
ridge_pred = ridge_model.predict(X_test)

# Calculate R² and MSE
ridge_r2 = r2_score(y_test, ridge_pred)
ridge_mse = mean_squared_error(y_test, ridge_pred)

print(f'Ridge Regression R²: {ridge_r2}, Mean Squared Error (MSE): {ridge_mse}')

The Ridge regression yielded:

• R²: 0.6917
• MSE: 6912725.8010

Interestingly, the performance dropped significantly. This indicated that the regularization might have overly constrained the model.

Ridge Polynomial Regression

To combine the benefits of polynomial features and regularization, I implemented Ridge Polynomial Regression.

# Create a pipeline with polynomial features and Ridge regression
ridge_poly_pipeline = Pipeline([
    ('poly', PolynomialFeatures(degree=2)),
    ('scaler', StandardScaler()),
    ('regressor', Ridge(alpha=0.1))
])

# Fit the model
ridge_poly_pipeline.fit(X_train, y_train)

# Predict on the test set
y_ridge_poly_pred = ridge_poly_pipeline.predict(X_test)

# Calculate R² and MSE
ridge_poly_r2 = r2_score(y_test, y_ridge_poly_pred)
ridge_poly_mse = mean_squared_error(y_test, y_ridge_poly_pred)

print(f'Ridge Polynomial Model R²: {ridge_poly_r2}, Mean Squared Error (MSE): {ridge_poly_mse}')

This model gave the following results:

• R²: 0.6733
• MSE: 7326174.8781

The Ridge Polynomial Regression performed better than the simple Ridge regression, demonstrating the effectiveness of combining polynomial features with regularization.

Grid Search for Hyperparameter Tuning

To ensure that I was using the best regularization parameter, I employed Grid Search for tuning the alpha parameter.

# Define a grid of alpha values
alpha_values = [0.01, 0.1, 1, 10, 100]

# Create a Ridge regression model
ridge = Ridge()

# Set up Grid Search
grid_search = GridSearchCV(estimator=ridge, param_grid={'alpha': alpha_values}, scoring='neg_mean_squared_error', cv=4)
grid_search.fit(X_train, y_train)

# Get the best alpha value
best_alpha = grid_search.best_params_['alpha']
print(f'Best Alpha: {best_alpha}')

# Predict with the best model
best_ridge = Ridge(alpha=best_alpha)
best_ridge.fit(X_train, y_train)
best_ridge_pred = best_ridge.predict(X_test)

# Calculate R² and MSE
best_ridge_r2 = r2_score(y_test, best_ridge_pred)
best_ridge_mse = mean_squared_error(y_test, best_ridge_pred)

print(f'Grid Search Ridge R²: {best_ridge_r2}, Mean Squared Error (MSE): {best_ridge_mse}')

The results from the Grid Search revealed:

• Best Alpha: 0.01
• MSE: 13840985.99
• R²: 0.3827

Despite finding the optimal alpha, the Ridge regression still underperformed compared to the base model.

Visualizing the Results

To better understand and compare the results of the various models, I created visualizations.

import matplotlib.pyplot as plt

# Model performance data
models = ['Linear Regression', 'Polynomial Model', 'Ridge Regression', 'Ridge Polynomial', 'GridSearch']
r2_scores = [0.6917, 0.7667, 0.6917, 0.6733, 0.3827]
mse_scores = [6912744.91, 5234038.0655, 6912725.801, 7326174.8781, 13840985.99]

# Plot R² Scores
plt.figure(figsize=(12, 5))
plt.subplot(1, 2, 1)
sns.barplot(data=results_df, x=models, y=r2_scores, palette='coolwarm')
plt.title('R² Scores of Different Models')
plt.ylabel('R² Score')
plt.ylim(0, 1)
plt.xticks(rotation=15)

# Plot MSE Scores
plt.subplot(1, 2, 2)
sns.barplot(data=results_df, x=models, y=mse_scores, palette='viridis')
plt.title('Mean Squared Error (MSE) of Different Models')
plt.ylabel('MSE')
plt.ylim(0, max(mse_scores) + 1000000)
plt.xticks(rotation=15)

plt.tight_layout()
plt.show()

Conclusion on Model Performance

Reflecting on the results, several conclusions emerged:

1. Model Comparison:

• The base linear regression model provided a solid baseline with an R² of 0.6917. This indicated a reasonable fit to the data.
• The polynomial model introduced complexity but did not enhance predictive power, with a slight drop in performance (R²: 0.7667.
• Ridge regression, while designed to prevent overfitting, underperformed significantly (R²: 0.6917), indicating that regularization might have overly constrained the model’s ability to capture relationships in the data.
• The Ridge Polynomial Regression showed a notable improvement over simple Ridge regression (R²: 0.6732), suggesting that combining polynomial features with regularization can
• GridSearch Ridge regression, while designed to prevent overfitting, underperformed significantly (R²: 0.3827), indicating that regularization might have overly constrained the model’s ability to capture relationships in the data.