Multiple Regression with NumPy

NumPy offers indispensable tools for developing multiple regression models from the ground up. This guide will explore key concepts of multiple regression and show you how to implement it using NumPy.

Understanding Multiple Regression

Multiple regression aims to find a linear relationship between a dependent variable (Y) and two or more independent variables (X₁, X₂, …, X_n). 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 Multiple Regression with NumPy

To perform multiple 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 multiple independent variables (X₁, X₂, …, X_n).

X1 = np.array([1, 2, 3, 4, 5])
X2 = np.array([2, 3, 4, 5, 6])
Y = np.array([3, 5, 7, 8, 10])

3. Normalize Your Data: Normalize your data if the scales of your features differ significantly to improve the stability and performance of your regression model.

X1_normalized = (X1 - np.mean(X1)) / np.std(X1)
X2_normalized = (X2 - np.mean(X2)) / np.std(X2)

4. Calculate the Coefficients: Utilize NumPy’s linear algebra functions to compute the coefficients β₀, β₁, and β₂.

X = np.column_stack((np.ones_like(X1), X1, X2))
coefficients = np.linalg.inv(X.T @ X) @ X.T @ Y
beta_0 = coefficients[0]
beta_1 = coefficients[1]
beta_2 = coefficients[2]

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

Y_pred = beta_0 + (beta_1 * X1) + (beta_2 * X2)