Linear Regression
Introduction
Linear regression is a fundamental statistical technique used to model the relationship between a dependent variable and one or more independent variables. In this lecture, we will learn how to perform linear regression in R, including model building, evaluation, and interpretation.
Key Concepts
1. What is Linear Regression?
Linear regression aims to find the best-fitting straight line through the data points that predicts the dependent variable based on the independent variables. The equation of a simple linear regression line is:
[ y = _0 + _1 x + ]
where:
( y ) is the dependent variable.
( _0 ) is the intercept.
( _1 ) is the slope.
( x ) is the independent variable.
( ) is the error term.
2. Assumptions of Linear Regression
For linear regression to provide reliable results, the following assumptions must be met:
Linearity: The relationship between the dependent and independent variables is linear.
Independence: Observations are independent of each other.
Homoscedasticity: The residuals have constant variance at all levels of the independent variable.
Normality: The residuals are normally distributed.
Performing Linear Regression in R
1. Building the Model
You can build a linear regression model using the lm() function in R.
# Creating a sample dataset
set.seed(123)
data <- data.frame(
x = rnorm(100),
y = 3 * rnorm(100) + 2 * rnorm(100) + rnorm(100)
)
# Building the linear regression model
model <- lm(y ~ x, data = data)
summary(model)2. Evaluating the Model
You can evaluate the model’s performance using various metrics such as R-squared, adjusted R-squared, and residual standard error.
# Model summary
summary(model)3. Making Predictions
You can use the model to make predictions on new data.
# Creating new data for prediction
new_data <- data.frame(x = c(-1, 0, 1))
# Making predictions
predictions <- predict(model, newdata = new_data)
print(predictions)4. Plotting the Regression Line
You can visualize the regression line along with the data points using the ggplot2 package.
# Installing and loading ggplot2
install.packages("ggplot2")
library(ggplot2)
# Plotting the regression line
ggplot(data, aes(x = x, y = y)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE, col = "red") +
labs(title = "Linear Regression", x = "Independent Variable (x)", y = "Dependent Variable (y)")Example: Comprehensive Linear Regression Analysis
Here’s a comprehensive example of performing linear regression analysis in R.
# Creating a sample dataset
set.seed(123)
data <- data.frame(
x = rnorm(100),
y = 3 * rnorm(100) + 2 * rnorm(100) + rnorm(100)
)
# Splitting the data into training and testing sets
library(caret)
trainIndex <- createDataPartition(data$y, p = 0.7, list = FALSE)
train_data <- data[trainIndex, ]
test_data <- data[-trainIndex, ]
# Building the linear regression model
model <- lm(y ~ x, data = train_data)
# Evaluating the model
summary(model)
# Making predictions on the test set
predictions <- predict(model, newdata = test_data)
# Calculating Mean Squared Error
mse <- mean((test_data$y - predictions)^2)
print(paste("Mean Squared Error:", mse))
# Plotting the regression line
library(ggplot2)
ggplot(train_data, aes(x = x, y = y)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE, col = "red") +
labs(title = "Linear Regression", x = "Independent Variable (x)", y = "Dependent Variable (y)")Summary
In this lecture, we covered how to perform linear regression in R, including building the model, evaluating its performance, making predictions, and visualizing the results. Linear regression is a powerful tool for understanding relationships between variables and making predictions based on those relationships.
Further Reading
For more detailed information, consider exploring the following resources:
Call to Action
If you found this lecture helpful, make sure to check out the other lectures in the ML R series. Happy coding!