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Implementation of Polynomial Regression: Python Code & Data Set

Polynomial regression is a powerful machine learning technique used to model non-linear relationships. A perfect real-world example is the relationship between car speed and braking distance, which follows a quadratic regression pattern rather than a straight line. In this guide, we will explore how to calculate polynomial regression coefficients using the least squares method, solve the equation mathematically, and implement it in Python with a real dataset. We will also visualize the results to understand how braking distance increases with speed.

Scenario

Imagine you are analyzing how a car's speed affects the braking distance required to stop. The relationship is non-linear because as speed increases, the braking distance grows exponentially.

A linear model (Y=b0+b1X) won’t fit well, so we use polynomial regression (degree 2):

Braking Distance=b0+b1(Speed)+b2(Speed)2

 

Sample Dataset (Speed vs Braking Distance)

Speed (km/h) Braking Distance (m)
10 5
20 20
30 45
40 80
50 125
  • As speed increases, the braking distance grows non-linearly (quadratically).
  • We need to calculate b0,b1,b2 to fit a quadratic curve.

Step-by-Step Solution

  1. Create the Equation:
    Using Y=b0+b1X+b2X2, we set up equations for given data points.
  2. Convert into Matrix Form:

       3.Solve for b0,b1,b2​ using matrix algebra. 

             We solve the equation AX=B using matrix algebra:

                                                                                                             X=A−1B

             Solving the system, we get:

b0≈0, b1≈0, b2≈0.05

            Thus, the polynomial equation for braking distance is:

Braking Distance=0.05×Speed2

            This equation perfectly fits the given data

Python Code for Polynomial Regression: Car Speed vs Braking Distance

import numpy as np
import matplotlib.pyplot as plt

# Sample dataset: Speed (X) vs Braking Distance (Y)
speed = np.array([10, 20, 30, 40, 50])
braking_distance = np.array([5, 20, 45, 80, 125])

# Create the coefficient matrix A
A = np.vstack([np.ones_like(speed), speed, speed**2]).T

# Solve for coefficients b0, b1, b2
b0, b1, b2 = np.linalg.lstsq(A, braking_distance, rcond=None)[0]

# Generate smooth curve for visualization
speed_smooth = np.linspace(5, 55, 100)
braking_smooth = b0 + b1 * speed_smooth + b2 * speed_smooth**2

# Plot the data points
plt.scatter(speed, braking_distance, color="blue", label="Data Points")

# Plot the polynomial curve
plt.plot(speed_smooth, braking_smooth, color="red", label="Polynomial Fit (Degree 2)")

# Labels and title
plt.xlabel("Speed (km/h)")
plt.ylabel("Braking Distance (m)")
plt.title("Polynomial Regression: Speed vs Braking Distance")
plt.legend()
plt.grid(True)

# Show the plot
plt.show()

# Print the equation
print(f"Polynomial Equation: Braking Distance = {b0:.2f} + {b1:.2f}*Speed + {b2:.2f}*Speed^2")

 


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