Polynomial regression is a technique we can use when the relationship between a predictor variable and a response variable is nonlinear. 

This type of regression takes the form:

Y = β₀ + β₁X + β₂X² + … + β_hX^h + ε

where h is  the “degree” of the polynomial.

The following step-by-step example shows how to perform polynomial regression in Python using sklearn.

Step 1: Create the Data

First, let’s create two NumPy arrays to hold the values for a predictor and response variable:

import matplotlib.pyplotas plt
import numpy as np

#define predictor and response variables
x = np.array([2, 3, 4, 5, 6, 7, 7, 8, 9, 11, 12])
y = np.array([18, 16, 15, 17, 20, 23, 25, 28, 31, 30, 29])

#create scatterplot to visualize relationship between x and y
plt.scatter(x, y)

From the scatterplot we can see that the relationship between x and y is not linear.

Thus, it’s a good idea to fit a polynomial regression model to the data to capture the non-linear relationship between the two variables.

Step 2: Fit the Polynomial Regression Model

The following code shows how to use functions from sklearn to fit a polynomial regression model with a degree of 3 to this dataset:

from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_modelimport LinearRegression

#specify degree of 3 for polynomial regression model
#include bias=False means don't force y-intercept to equal zero
poly = PolynomialFeatures(degree=3, include_bias=False)

#reshape data to work properly with sklearn
poly_features = poly.fit_transform(x.reshape(-1, 1))

#fit polynomial regression model
poly_reg_model = LinearRegression()
poly_reg_model.fit(poly_features, y)

#display model coefficients
print(poly_reg_model.intercept_, poly_reg_model.coef_)

33.62640037532282 [-11.83877127   2.25592957  -0.10889554]

Using the model coefficients displayed on the last line, we can write the fitted polynomial regression equation as:

y = -0.109x³ + 2.256x² – 11.839x + 33.626

This equation can be used to find the expected value for the response variable based on a given value for the predicted variable.

y = -0.109(4)³ + 2.256(4)² – 11.839(4) + 33.626= 15.39

Note: To fit a polynomial regression model with a different degree, simply change the value for the degree argument within the PolynomialFeatures() function.

Step 3: Visualize the Polynomial Regression Model

Lastly, we can create a simple plot to visualize the fitted polynomial regression model over the original data points:

#use model to make predictions on response variable
y_predicted = poly_reg_model.predict(poly_features)

#create scatterplot of x vs. y
plt.scatter(x, y)

#add line to show fitted polynomial regression model
plt.plot(x, y_predicted, color='purple')

From the plot we can see that the polynomial regression model seems to fit the data well without .

Note: You can find the complete documentation for the sklearn PolynomialFeatures() function .

The following tutorials explain how to perform other common tasks using sklearn: