Linear regression is a statistical method that models the relationship between a dependent variable (y) and one or more independent variables (x₁, x₂, ..., xₚ). It assumes a linear relationship, which can be represented by a straight line when plotted.
Developed in the early 19th century, linear regression has evolved through advancements in statistical theory and computing power, becoming a cornerstone of predictive modeling in data science and machine learning.
Linear regression is fundamental for understanding relationships between variables, making predictions, and interpreting data trends. It serves as a basis for more complex machine learning algorithms and statistical models.
Assumes a linear correlation between predictors and the response variable.
Includes linearity, independence of errors, homoscedasticity, and normality of residuals.
Simple uses one predictor, while multiple incorporates multiple predictors.
y = β₀ + β₁x₁ + β₂x₂ + ... + βₚxₚ + ε
Each β quantifies the effect of its corresponding x-variable on y.
Minimizes the sum of squared differences between observed and predicted values to estimate β.
R², adjusted R² measure how well the model fits the data.
Examines the difference between predicted and actual values for model validity.
Ensures residuals meet statistical assumptions.
Predicts y using one predictor.
Uses multiple predictors to predict y.
Models non-linear relationships using polynomial terms.
Gathers relevant data for analysis.
Cleans and prepares data for regression.
Implements using tools like Python or R.
Analyzes model output to make data-driven decisions.
Overview of tools like scikit-learn in Python and lm() function in R.
# Python example using scikit-learn
from sklearn.linear_model import LinearRegression
import numpy as np
# Sample data
X = np.array([[1, 1], [1, 2], [2, 2], [2, 3]])
y = np.dot(X, np.array([1, 2])) + 3
# Fit model
model = LinearRegression().fit(X, y)
# Model coefficients
print("Coefficients:", model.coef_)
print("Intercept:", model.intercept_)Address the risks of overfitting and underfitting in model training.
Handles the issue of predictors being highly correlated with each other.
Ensures model assumptions are not violated to maintain accuracy.
Provides real-world examples and case studies where linear regression has been successfully applied.
Compares linear regression with other methods like logistic regression, ridge regression, and lasso regression.
Step-by-step guide to implementing linear regression in Google Colab using Python.
Linear regression plays a crucial role in data science, offering a methodical approach to analyzing and predicting relationships between variables. Its applications range widely across industries, from technology and finance to marketing and project management.