Polynomial Regression

Data follows a straight line

But in many real-world problems:

Data follows a curve

Polynomial Regression helps model such curved relationships.

Real-Life Example

Suppose we want to predict:

• Salary based on experience

• Plant growth based on temperature

• Sales based on advertisement spending

Sometimes the output does not increase in a straight-line pattern.

Example Dataset

If we plot this data:

• The relationship appears curved

• A straight line cannot fit properly

Why Polynomial Regression is Needed

Linear Regression works well only when:

Input and output have a linear relationship

But real-world data often contains:

• Curves

• Nonlinear trends

• Complex patterns

Polynomial Regression converts linear models into nonlinear models by adding powers of input variables.

Polynomial Regression Equation

Degree 2 Polynomial

y = b0 + b1x + b2x²

Degree 3 Polynomial

y = b0 + b1x + b2x² + b3x³

Where:

• y → Predicted output

• x → Input feature

• b0 → Intercept

• b1, b2, b3 → Coefficients

Understanding Polynomial Features

Suppose:

x = 2

Then:

x² = 4
x³ = 8

These additional powers help the model capture curved relationships.

Mathematical Example

Suppose the model equation is:

y = 2 + 3x + x²

Predict output for:

x = 4

Substitute values:

y = 2 + 3(4) + (4²)

Calculate:

3(4) = 12
4² = 16

Final value:

y = 2 + 12 + 16

y = 30

Visual Difference

Linear Regression

Straight Line

Polynomial Regression

Curved Line

Practical Example Using Python

Step 1: Import Libraries

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

from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures

Step 2: Create Dataset

data = {
    "Experience": [1, 2, 3, 4, 5],
    "Salary": [15, 20, 28, 40, 60]
}

df = pd.DataFrame(data)

print(df)

Step 3: Define Features and Target

X = df[["Experience"]]
y = df["Salary"]

Step 4: Convert to Polynomial Features

poly = PolynomialFeatures(degree=2)

X_poly = poly.fit_transform(X)

This creates:

x and x² features

Step 5: Train Model

model = LinearRegression()

model.fit(X_poly, y)

Step 6: Predict New Value

Predict salary for:

Experience = 6

prediction = model.predict(poly.transform([[6]]))

print(prediction)

Expected Output

[80.]

Step 7: Visualize Polynomial Curve

plt.scatter(df["Experience"], df["Salary"])

x_range = np.linspace(1, 5, 100).reshape(-1,1)

plt.plot(
    x_range,
    model.predict(poly.transform(x_range)),
    color="red"
)

plt.xlabel("Experience")
plt.ylabel("Salary")

plt.title("Polynomial Regression")

plt.show()

Understanding the Graph

• Blue dots → Actual data points

• Red curve → Polynomial regression curve

The curve fits nonlinear data better than a straight line.

Degree of Polynomial

Important Point

Higher degree:

• Increases flexibility

• May cause overfitting

Choosing the correct degree is important.

Advantages

• Handles nonlinear data

• Better curve fitting

• Improves prediction accuracy

• Easy extension of Linear Regression

Limitations

• Higher degree may overfit

• Sensitive to outliers

• Complex interpretation

Real-World Applications

Important Interview Points

1. Polynomial Regression models nonlinear relationships.

2. Polynomial features add powers of input variables.

3. Degree controls curve complexity.

4. Higher polynomial degree may cause overfitting.

5. Polynomial Regression is built on Linear Regression.

Summary

Polynomial Regression is a supervised learning algorithm used to model nonlinear relationships between input and output variables. It extends Linear Regression by adding polynomial features, allowing the model to fit curved data patterns more effectively.