• Predicts a response variable using multiple explanatory variables
• Extends simple linear regression to include multiple predictors
• Models a (possible) relationship between variables
• Finds the best-fitting line by minimizing errors
• Used for forecasting and data analysis across various disciplines

\[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_n x_n + \epsilon \]

• \(y\) = response variable
• \(\beta_0\) = intercept
• \(\beta_1, \beta_2, \dots, \beta_n\) = regression coefficients
• \(x_1, x_2, \dots, x_n\) = explanatory variables
• \(\epsilon\) = error

Regression coefficients are calculated
using the normal equation:

\[ \boldsymbol{\beta} = (X^T X)^{-1} X^T \boldsymbol{y} \]

Where:

• \(\boldsymbol{\beta}\) = vector of regression coefficients

• \(X\) = matrix of explanatory variables

• \(\boldsymbol{y}\) = vector of observed values for response variable

• \(X^T\) = transpose of the matrix \(X\)

• \((X^T X)^{-1}\) = inverse of the matrix \(X^T X\)

• Swiss provinces (1888): Fertility and economic indicators
• Key variables: Fertility, Agriculture, Education, etc.
• Source: Built-in dataset in R
• Use: Commonly used in regression analysis, helpful in understanding multiple linear regression
• Application: Models relationship between socioeconomic factors and fertility

library(ggplot2)

ggplot(swiss, aes(x=Agriculture, y=Fertility)) +
  geom_point(color = "blue") + 
  geom_smooth(method = "lm", color="red", se=FALSE) +
  labs(title = "Fertility vs. Agriculture",
       x = "Agriculture (% of Men Employed in Agriculture)",
        y = "Fertility Rate") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5)) +
  xlim(0, 100) +
  ylim(0, 100)

https://www.investopedia.com/terms/m/mlr.asp#

https://statisticsbyjim.com/regression/linear-regression/

https://statquest.org/statquest-multiple-regression-glms-part-1-5-clearly-explained/