Please note the similarities between multiple linear regression and polynomial regression. In multiple linear regression, the relationship between the target variable and the predictor variables is expressed using the following equation:

Y is the target variable, X₁, X₂, … X_n are the predictor variables or the features. And e is the error called the residual. β₀, β₁, … β_n are called the coefficients, and they are selected in such a way that e or the error is minimum.

Please note that in polynomial regression, x can be compared with X₁, x² can be compared with X₂, xⁿ can be compared with X_n and so on.

Polynomial Regression using Python

For polynomial regression, we will load the “mpg” dataset from the seaborn library. The dataset contains various information such as the miles per gallon or mpg of a car, model, horsepower, weight, acceleration, the origin of the car, etc. In this article, we will show an example and for that purpose, we are only interested in the horsepower of a car and the mpg of the car.

The mpg of a car is a dependent variable or the target variable here. And the horsepower is a predictor variable. We have already seen using a scatterplot how these two variables are related to each other.

Now, we can use the following Python code to predict the mpg of a car from its horsepower.

import pandas
import seaborn
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures
from sklearn.metrics import r2_score, mean_squared_error
from matplotlib import pyplot
 
df = seaborn.load_dataset("mpg")
print(df.head())
print(df.info())
df = df.dropna()
 
pyplot.scatter(df["horsepower"], df["mpg"])
pyplot.xlabel("Horsepower")
pyplot.ylabel("MPG")
pyplot.savefig("poly-regression-scatter.png")
pyplot.close()
 
X_train, X_test, y_train, y_test = train_test_split(df[["horsepower"]], df["mpg"],
                                                    train_size=0.8, shuffle=True, random_state=1)
polynomial_features = PolynomialFeatures(degree=2, include_bias=False)
X_train_transformed = polynomial_features.fit_transform(X_train)
 
polynomial_regression = LinearRegression()
polynomial_regression.fit(X_train_transformed, y_train)
 
X_test_transformed = polynomial_features.transform(X_test)
y_test_predicted = polynomial_regression.predict(X_test_transformed)
 
df2 = pandas.DataFrame(list(zip(X_test['horsepower'], y_test, y_test_predicted)), columns=['horsepower', 'actual_mpg', 'predicted_mpg'])
df2.sort_values(by=['horsepower'], inplace=True)
print(df2.head())
pyplot.scatter(df["horsepower"], df["mpg"])
pyplot.xlabel("Horsepower")
pyplot.ylabel("MPG")
pyplot.plot(df2['horsepower'], df2['predicted_mpg'], color="red")
pyplot.savefig("poly-regression-final.png")
pyplot.close()
 
r2 = r2_score(y_test, y_test_predicted)
rmse = mean_squared_error(y_test, y_test_predicted, squared=False)
 
print("R2: ", r2)
print("RMSE: ", rmse)

We are first loading the dataset and printing the first few lines of the dataset. We also print information about the dataset using df.info() function…