Polynomial Regression Degree 3 Example Python using Data set
Introduction
Polynomial regression helps model non-linear relationships. In this example, we analyze how temperature affects ice cream sales using a cubic polynomial equation:
Y=b0+b1X+b2X2+b3X3
where:
- Y = Ice Cream Sales
- X = Temperature
A cubic equation provides a better fit when the sales pattern fluctuates based on temperature extremes.
Dataset: Temperature vs Ice Cream Sales
| Temperature (°C) | Ice Cream Sales |
|---|---|
| 10 | 5 |
| 15 | 20 |
| 20 | 50 |
| 25 | 80 |
| 30 | 100 |
| 35 | 90 |
| 40 | 60 |
This dataset shows that sales increase with temperature but start declining after a peak, making it a non-linear relationship.
Python implementation for Polynomial Regression (Degree 3) using the Temperature vs Ice Cream Sales dataset
import numpy as np
import matplotlib.pyplot as plt
# Step 1: Define the dataset (Temperature vs Ice Cream Sales)
temperature = np.array([10, 15, 20, 25, 30, 35, 40])
sales = np.array([5, 20, 50, 80, 100, 90, 60])
# Step 2: Create the polynomial feature matrix (Degree 3)
A = np.vstack([np.ones_like(temperature), temperature, temperature**2, temperature**3]).T
# Step 3: Solve for coefficients b0, b1, b2, b3
b0, b1, b2, b3 = np.linalg.lstsq(A, sales, rcond=None)[0]
# Print the polynomial equation
print(f"Polynomial Equation: Sales = {b0:.2f} + {b1:.2f}*Temp + {b2:.2f}*Temp^2 + {b3:.2f}*Temp^3")
# Step 4: Generate smooth predictions for plotting
temp_smooth = np.linspace(10, 40, 100)
sales_smooth = b0 + b1 * temp_smooth + b2 * temp_smooth**2 + b3 * temp_smooth**3
# Step 5: Plot the actual data points and polynomial regression curve
plt.scatter(temperature, sales, color="blue", label="Actual Sales")
plt.plot(temp_smooth, sales_smooth, color="red", label="Polynomial Fit (Degree 3)")
# Step 6: Labels, title, and legend
plt.xlabel("Temperature (°C)")
plt.ylabel("Ice Cream Sales")
plt.title("Polynomial Regression (Degree 3): Temperature vs Ice Cream Sales")
plt.legend()
plt.grid(True)
# Step 7: Show the plot
plt.show()