📈 ML Polynomial Regression: Unlocking the Power of Curved Relationships

ML Polynomial Regression

When dealing with real-world data in Machine Learning, we often encounter complex, non-linear patterns that simple linear models can’t accurately capture. That’s where Polynomial Regression steps in — a flexible and powerful extension of linear regression, specially crafted to handle such non-linearity.

In this blog post, we’ll explore what Polynomial Regression is, why it’s needed, how it differs from other regression techniques, and finally, how to implement it in Python using a practical example.

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🔍 What is Polynomial Regression?

Polynomial Regression is a type of regression analysis where the relationship between the independent variable (x) and dependent variable (y) is modeled as an nth-degree polynomial. The general form of the polynomial equation looks like this: y=b0+b1x+b2x2+b3x3+…+bnxny = b_0 + b_1x + b_2x^2 + b_3x^3 + \ldots + b_nx^n

Despite the “polynomial” name, it’s still considered a linear model — not because the curve is linear, but because the coefficients b0,b1,…,bnb_0, b_1, …, b_n are combined linearly.

✅ Polynomial Regression vs. Linear Regression

It’s helpful to compare it with:

• Simple Linear Regression: y=b0+b1xy = b_0 + b_1x
• Multiple Linear Regression: y=b0+b1x1+b2x2+…+bnxny = b_0 + b_1x_1 + b_2x_2 + \ldots + b_nx_n
• Polynomial Regression: y=b0+b1x+b2x2+b3x3+…+bnxny = b_0 + b_1x + b_2x^2 + b_3x^3 + \ldots + b_nx^n

All are linear in terms of parameters, but Polynomial Regression adds higher-degree features to model curved data trends.

🎯 Why Do We Need Polynomial Regression?

Linear models are great for linearly distributed data. But what happens when your data curves? Applying a linear model to non-linear data will lead to:

• High errors
• Poor predictions
• Increased loss function values

In such cases, Polynomial Regression can effectively model the curve and yield more accurate predictions. Imagine trying to predict salary based on experience — a CEO’s salary doesn’t increase linearly with years of service!

🧠 How Does It Work?

Polynomial Regression works by:

1. Transforming features into higher-degree polynomial terms.
2. Fitting a linear regression model on this transformed data.

In essence:

“We convert the original feature space into a polynomial space to fit complex curves using a linear algorithm.”

💼 Real-Life Use Case: Bluff Detection in Salary Prediction

Let’s dive into a real-world example.

Problem Statement:

A company is hiring a new candidate who claims a previous salary of $160K/year. The HR team wants to verify this claim using their salary dataset of top 10 positions (with levels and salaries). As the relationship between level and salary is non-linear, we’ll build a Polynomial Regression model to predict the truthfulness of the claim.

🛠️ Step-by-Step Implementation Using Python

Step 1: Import Libraries and Dataset

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# Load dataset
dataset = pd.read_csv('Position_Salaries.csv')

# Extract features and labels
X = dataset.iloc[:, 1:2].values  # Position levels
y = dataset.iloc[:, 2].values    # Salaries

We’re only using “Level” and “Salary” columns — position names are descriptive and not used for modeling.

Step 2: Build and Fit a Linear Regression Model

from sklearn.linear_model import LinearRegression

lin_reg = LinearRegression()
lin_reg.fit(X, y)

This model will act as our baseline.

Step 3: Build and Fit a Polynomial Regression Model

from sklearn.preprocessing import PolynomialFeatures

poly_features = PolynomialFeatures(degree=4)  # You can try degree=2, 3, 5, etc.
X_poly = poly_features.fit_transform(X)

poly_reg = LinearRegression()
poly_reg.fit(X_poly, y)

The PolynomialFeatures class expands our features to include powers of the original values.

📊 Visualizing Results

Visualize Linear Regression Predictions

plt.scatter(X, y, color='blue')
plt.plot(X, lin_reg.predict(X), color='red')
plt.title("Linear Regression - Bluff Detection")
plt.xlabel("Position Level")
plt.ylabel("Salary")
plt.show()

You’ll notice this straight line doesn’t fit the non-linear salary data well.

Visualize Polynomial Regression Predictions

plt.scatter(X, y, color='blue')
plt.plot(X, poly_reg.predict(poly_features.fit_transform(X)), color='green')
plt.title("Polynomial Regression - Bluff Detection")
plt.xlabel("Position Level")
plt.ylabel("Salary")
plt.show()

With degree 4, the curve fits almost perfectly, capturing the complexities of the dataset.

🔮 Making Predictions

Predict with Linear Regression

lin_pred = lin_reg.predict([[6.5]])
print("Linear Prediction:", lin_pred)

Output: [330378.78] — Overestimates the value significantly.

Predict with Polynomial Regression

poly_pred = poly_reg.predict(poly_features.fit_transform([[6.5]]))
print("Polynomial Prediction:", poly_pred)

Output: [158862.45] — Much closer to the candidate’s claimed salary.

🧾 Final Thoughts

Polynomial Regression is a fantastic tool in the Machine Learning toolbox when you’re working with non-linear data. It helps you build powerful models while sticking to simple linear algorithms under the hood.

✅ Quick Summary:

• Use Polynomial Regression when data shows a non-linear relationship.
• It transforms input features into polynomial features.
• It’s still a linear model — just in an expanded feature space.
• Increasing the polynomial degree improves accuracy (up to a point).
• It is ideal for small datasets where precision is crucial.

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📌 Bonus Tip:

Always experiment with different polynomial degrees (e.g., 2, 3, 4, 5) to find the optimal balance between underfitting and overfitting.

Whether you’re building a salary prediction system, analyzing sales trends, or modeling any complex data — Polynomial Regression is a go-to choice when linear models fall short.

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