Here we have 3 data points. Miles per gallon, Engine Displacement and Vehicle weight. The linear regression plane is created by taking the relationship of (Engine displacement + Vehicle weight) and comparing the duo to overall MPG.

This relationship from the previous slide can be mathmatically shown using the equation: \(Y=b_0 +b_1X_1 +b_2X_2\) where \(Y\) = MPG, \(X_1\)=engine Displacement and \(X_2\)= Vehicle Weight. \(b_0\) represents the y intercept, and \(b_n\) is change in Y for one-unit change in \(X_n\).

\(b_n\) is also know as a regression coefficient for some \(X_n\)

The code below is used to generate the chart in the following slide. Its pulling data from the mtcars dataset. Note that the standard error has been removed (se=F) and we’re using a simple formula to generate the linear regression line (formula = y~x). Mathmatically this formula can be represented as \(Y=a+bx\) where \(Y\) is Horsepower(Predicted value), \(a\) is the y-intercept, \(b\) is the slope, \(x\) is Engine Displacement(Independent Variable).

ggplot(mtcars, aes(x=disp, y = hp))+geom_point()+
  labs(
    title = "Displacement vs Horsepower using simple linear regression",
    x = 'Engine Displacement (Cubic inches)',
    y = 'Horsepower'
  ) +stat_smooth(method = 'lm', se=F, formula = y~x, color='blue')

As opposed to slide 4 showing a linear regression plane, here we have a linear regression line. This line is showing the relationship between engine displacement and horsepower produced by said engine. The confidence interval has been turned off for this chart.

The code used to generate this plot is similar to the previous slide. I use data from the Faithful dataset. The main difference is the inclusion of the standard error (shaded region). Standard error is calculated by the formula:

\(\sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n - 2}}\)