Exploring Nonlinear Relationships with Polynomial Regression: A Comprehensive Guide
Polynomial regression is a regression analysis in which the relationship between the independent variable (also called the predictor variable) and the dependent variable (sometimes called the response variable) is described as an nth-degree polynomial function. In other words, polynomial regression models may fit curved or nonlinear connections between the variables instead of just fitting a linear relationship between the variables as in basic regression models.
Polynomial regression might be utilized when a linear regression model fails to reflect the underlying relationship between the variables sufficiently or when the data seems to follow a nonlinear trend. For instance, a polynomial regression model containing a quadratic or cubic component may be more suitable than a basic linear regression model if the data points follow a parabolic or cubic pattern.
Many regression approaches, including conventional least squares regression, ridge regression, and lasso regression, may be used to conduct polynomial regression. However, it is essential to remember that higher-degree polynomial models might be prone to overfitting, leading to the subpar prediction of new data. Consequently, it is important to carefully determine the degree of the polynomial depending on the intricacy of the underlying relationship and the available data.
Polynomial regression using Python and the scikit-learn library:
import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures
import matplotlib.pyplot as plt
x = np.array([0, 1, 2, 3, 4, 5])
y = np.array([3, 5, 4, 6, 8, 10])
X = x.reshape((-1, 1))
poly = PolynomialFeatures(degree=2)
X_poly = poly.fit_transform(X)
model = LinearRegression().fit(X_poly, y)
x_new = np.linspace(0, 5, num=50)
X_new = x_new.reshape((-1, 1))
X_new_poly = poly.transform(X_new)
y_new = model.predict(X_new_poly)
plt.scatter(x, y)
plt.plot(x_new, y_new)
plt.show()Here we first generate some example data x and y. We then reshape x to a 2D array X and fit a polynomial regression model with degree 2 using the PolynomialFeatures and LinearRegression classes from scikit-learn. We then generate some new data x_new and make predictions using the predict method of the LinearRegression model. Finally, we plot the original data and the predicted values using. matplotlib.
