Polynomial Regression is a type of regression analysis that is used to model the relationship between a dependent variable and one or more independent variables. It is a special case of multiple linear regression, where the independent variables are not linearly related to the dependent variable.
In Polynomial Regression, the relationship between the dependent variable and the independent variables is modeled as an nth degree polynomial equation. The goal of Polynomial Regression is to find the coefficients of the polynomial equation that best fit the data.
This is commonly used in applications such as finance, where it is used to model the relationship between variables such as stock prices and economic indicators. It is also used in machine learning and data analysis, where it is used to model the relationship between variables in a nonlinear way.
Overall, Polynomial Regression is a powerful tool for modeling complex relationships between variables and is widely used in various fields.
# Import necessary libraries
import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Generate some example data
np.random.seed(42)
X = 6 * np.random.rand(100, 1) - 3
y = 0.5 * X**2 + X + 2 + np.random.randn(100, 1)
# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Apply polynomial features transformation
degree = 2 # Set the degree of the polynomial
poly_features = PolynomialFeatures(degree=degree, include_bias=False)
X_poly_train = poly_features.fit_transform(X_train)
X_poly_test = poly_features.transform(X_test)
# Create a Polynomial Regression model
poly_model = LinearRegression()
# Train the model on the polynomial features
poly_model.fit(X_poly_train, y_train)
# Make predictions on the test data
y_poly_pred = poly_model.predict(X_poly_test)
# Evaluate the model
mse = mean_squared_error(y_test, y_poly_pred)
rmse = np.sqrt(mse)
print(f"Root Mean Squared Error: {rmse}")
# Plot the original data and the polynomial regression curve
plt.scatter(X, y, label='Original Data')
X_plot = np.linspace(-3, 3, 100).reshape(-1, 1)
X_plot_poly = poly_features.transform(X_plot)
y_plot = poly_model.predict(X_plot_poly)
plt.plot(X_plot, y_plot, color='red', label=f'Polynomial Regression (Degree {degree})')
plt.xlabel('X-axis label')
plt.ylabel('Y-axis label')
plt.title('Polynomial Regression')
plt.legend()
plt.show()
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