Welcome to this presentation on Linear Regression.

In this talk, we will: - Introduce the fundamentals of linear regression, - Explain its mathematical foundation using key formulas, - Demonstrate practical examples using the built-in mtcars dataset, - Showcase visualizations with both ggplot2 and an interactive 3D plotly plot, - Discuss model diagnostics and conclude with insights for future analysis.

This project is part of the DAT 301 Homework 3 assignment. By the end of this presentation, you will have a clear understanding of how linear regression is implemented in R and how its results can be visualized and interpreted.

Linear regression is a statistical method used to model the relationship between a dependent variable (response) and one or more independent variables (predictors). In simple linear regression, the relationship is modeled using a straight line.

The model is represented by the equation: \[ y = \beta_0 + \beta_1 x + \epsilon \]

where: - \(y\) is the dependent variable, - \(x\) is the independent variable, - \(\beta_0\) is the intercept, - \(\beta_1\) is the slope, and - \(\epsilon\) is the error term that accounts for variability not explained by the model.

This technique is widely used across various fields to predict outcomes and analyze trends.

The goal of linear regression is to minimize the sum of squared residuals (errors) between the observed and predicted values. This is achieved by finding the best-fitting line.

The objective is: \[ \min_{\beta_0, \beta_1} \sum_{i=1}^{n} \left(y_i - \beta_0 - \beta_1 x_i\right)^2 \]

The formulas to calculate the coefficients are: \[ \hat{\beta_1} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} \] \[ \hat{\beta_0} = \bar{y} - \hat{\beta_1}\bar{x} \]

These equations form the foundation of simple linear regression.

We will use the built-in mtcars dataset to illustrate linear regression.
Below is a summary of the dataset:

summary(mtcars)

##       mpg             cyl             disp             hp       
##  Min.   :10.40   Min.   :4.000   Min.   : 71.1   Min.   : 52.0  
##  1st Qu.:15.43   1st Qu.:4.000   1st Qu.:120.8   1st Qu.: 96.5  
##  Median :19.20   Median :6.000   Median :196.3   Median :123.0  
##  Mean   :20.09   Mean   :6.188   Mean   :230.7   Mean   :146.7  
##  3rd Qu.:22.80   3rd Qu.:8.000   3rd Qu.:326.0   3rd Qu.:180.0  
##  Max.   :33.90   Max.   :8.000   Max.   :472.0   Max.   :335.0  
##       drat             wt             qsec             vs        
##  Min.   :2.760   Min.   :1.513   Min.   :14.50   Min.   :0.0000  
##  1st Qu.:3.080   1st Qu.:2.581   1st Qu.:16.89   1st Qu.:0.0000  
##  Median :3.695   Median :3.325   Median :17.71   Median :0.0000  
##  Mean   :3.597   Mean   :3.217   Mean   :17.85   Mean   :0.4375  
##  3rd Qu.:3.920   3rd Qu.:3.610   3rd Qu.:18.90   3rd Qu.:1.0000  
##  Max.   :4.930   Max.   :5.424   Max.   :22.90   Max.   :1.0000  
##        am              gear            carb      
##  Min.   :0.0000   Min.   :3.000   Min.   :1.000  
##  1st Qu.:0.0000   1st Qu.:3.000   1st Qu.:2.000  
##  Median :0.0000   Median :4.000   Median :2.000  
##  Mean   :0.4062   Mean   :3.688   Mean   :2.812  
##  3rd Qu.:1.0000   3rd Qu.:4.000   3rd Qu.:4.000  
##  Max.   :1.0000   Max.   :5.000   Max.   :8.000

Let’s visualize the relationship between car weight (wt) and miles per gallon (mpg), and then examine the model’s diagnostics.

Scatterplot with Regression Line (ggplot2):

library(ggplot2)
ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE, color = "blue") +
  labs(title = "MPG vs Weight with Regression Line",
       x = "Weight (1000 lbs)",
       y = "Miles per Gallon")

Residuals vs Fitted Plot (ggplot2):

# Fit the linear model
model <- lm(mpg ~ wt, data = mtcars)
# Create a data frame with fitted values and residuals
model_data <- data.frame(Fitted = fitted(model), Residuals = residuals(model))
ggplot(model_data, aes(x = Fitted, y = Residuals)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed", color = "red") +
  labs(title = "Residuals vs Fitted (ggplot2)", x = "Fitted Values", y = "Residuals")

For an interactive exploration, view this 3D scatter plot that shows the relationship between weight, mpg, and horsepower.

library(plotly)
p <- plot_ly(data = mtcars, x = ~wt, y = ~mpg, z = ~hp,
             type = "scatter3d", mode = "markers") %>%
     layout(title = "3D Scatter Plot: Weight, MPG, and Horsepower")
p

In summary, we have reviewed the fundamentals of linear regression, including its mathematical basis and practical application.

Recall the regression model: \[ y = \beta_0 + \beta_1 x + \epsilon \]

Thank you for your attention!

Feel free to ask any questions you might have.

Future Directions:

• Extending the analysis to multiple linear regression.
• Investigating non-linear relationships.
• Using other datasets to validate the model in different contexts.

Your feedback and questions are welcome!