• We will be going over Simple Linear Regression
• It is a technique used to model between two variables
• We will use mtcars dataset and visualizations to understand how it works

• A method to model the relationship between a dependent variable \(y\) and one independent variable \(x\)

\[ y = \beta_0 + \beta_1x + \epsilon \]

Where:

• \(y\) is the dependent variable
• \(x\) is the independent variable
• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope
• \(\epsilon\) is the error term

\[ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}, \quad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x} \]

• These formulas give us the slope and intercept of the best fitting line using the least squares method
• Least squares estimators minimize the total squared vertical distance between the observed points and the regression line
• They are the basis for drawing the regression line shown in the next slide and interpreting the model

• This scatter plot shows the relationship between car weight and miles per gallon (MPG)
• The downward-sloping regression line indicates that as car weight increases, MPG tends to decrease

• Residuals are the differences between observed and predicted values
• This plot helps check if the residuals are randomly scattered, indicating a good linear fit

• This code will set up the 3D Plot to showcase Simple Linear Regression.

fig = plot_ly(data = mtcars, x =~ wt, y =~ mpg, z =~ hp,
               type = 'scatter3d', mode = 'markers',
               marker = list(size = 5, color =~ hp, 
                             colorscale = 'Viridis'))
fig

• This 3D plot visualizes the relationship between weight, MPG, and horsepower
• It helps us see how a third variable like horsepower may influence the relationship between weight and MPG

• Simple linear regression is a powerful tool in statistics
• It helps us understand and predict relationships between two variables
• Make sure to always check assumptions and use diagnostic plots to validate your model