Linear Regression with NumPy

Linear regression is a fundamental statistical and machine learning technique used for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation. NumPy, a powerful library for numerical computing in Python, provides essential tools for implementing linear regression models from scratch. We’ll explore the key concepts of linear regression and demonstrate how to perform linear regression using NumPy.

Understanding Linear Regression

Linear regression aims to find a linear relationship between a dependent variable (Y) and one or more independent variables (X). The model assumes that this relationship can be expressed as:

Y = β₀ + β₁X₁ + β₂X₂ + … + β_nX_n + ε

Where:

• Y is the dependent variable (the variable we want to predict).
• X₁, X₂, …, X_n are the independent variables (features).
• β₀ is the intercept (the value of Y when all X values are zero).
• β₁, β₂, …, β_n are the coefficients (weights) of the independent variables.
• ε represents the error term (the difference between the predicted and actual values).

Performing Linear Regression with NumPy

To perform linear regression using NumPy, follow these steps:

1. Import NumPy:

import numpy as np

2. Define your data: Prepare your dataset with the dependent variable (Y) and independent variable(s) (X).

# Example data
X = np.array([1, 2, 3, 4, 5])
Y = np.array([2, 4, 5, 4, 5])

3. Calculate the coefficients: Use NumPy functions to calculate the coefficients β₀ and β₁.

# Calculate the coefficients
mean_x = np.mean(X)
mean_y = np.mean(Y)
n = len(X)

# Calculate β1 (slope) and β0 (intercept)
beta_1 = np.sum((X - mean_x) * (Y - mean_y)) / np.sum((X - mean_x) ** 2)
beta_0 = mean_y - (beta_1 * mean_x)

4. Make predictions: Use the calculated coefficients to make predictions.

# Make predictions
Y_pred = beta_0 + (beta_1 * X)

5. Visualize the results: You can use libraries like Matplotlib to visualize your linear regression model and predictions.

import matplotlib.pyplot as plt

# Plot the data points
plt.scatter(X, Y)

# Plot the regression line
plt.plot(X, Y_pred, color='red')

# Show the plot
plt.show()

See also: Linear Regression in Excel