• Technique to predict values based off of existing data points
• Uses a best-fit line to make predictions
• Predicts dependent variables using independent variables
• Independent variable is on the X-axis
• Dependent variable is on the Y-axis
• Common Use Cases:
  • Forecasting
  • Trend Analysis
  • Predictive Modelling

head(cars)

##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

• Cars contains 2 variables (one independent, one dependent)
• Ideal for linear regression

carsplot = plot_ly(cars, x=~speed, y=~dist, 
                   type="scatter", mode='markers') %>%
  layout(xaxis=list(title="Speed"), 
         yaxis=list(title="Stopping Distance")) %>%
  layout(title = "Speed vs Stopping Distance")

This code will plot the cars dataset.

Key Points

• Independent Variable: Speed
• Dependent Variable: Stopping Distance

To use linear regression in R, we must first create the model.

model = lm(dist ~ speed, data = cars)
model

## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## Coefficients:
## (Intercept)        speed  
##     -17.579        3.932

But before that, can we use math?

The linear regression equation is \(y = \beta_0 + \beta_1x\)

From the previous slide, we know that: \[ \beta_0 = -17.579\newline \beta_1 = 3.932 \] Therefore, our linear regression equation for cars is: \[y = -17.579 + 3.932x\] For a speed of 20 (\(x=20\)), we get a distance of: \[ y = -17.579 + 3.932(20) = 61.061 \]

To use R to do the same prediction, we use the model we made earlier.

Using this snippet of R code:

predicted_distance = predict(model, newdata = data.frame(speed = 20))

We get the output, which is 61.0690803 feet.

Using ggplot2, we can show the line of best fit.

lm() uses this formula to calculate the slope: \[ \beta_1 = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})} {\sum_{i=1}^{n}(x_i-\bar{x})^2} \] To calculate the line using R:

x = cars$speed; y=cars$dist
x_bar = mean(x); y_bar = mean(y)
slope = sum((x-x_bar) * (y-y_bar))/sum((x-x_bar)^2)

This gives us a slope of 3.932 feet/mph.

Now that we have the slope, we can calculate the intercept using: \[ \beta_0 = \bar{y} - \beta_1\bar{x} \] Substituting values that we already know:

\[ \beta_0 = 42.98 - 3.9324088\times15.4\newline \beta_0 \approx -17.579 \]

Using ggplot2 once again, we can plot the line of best fit that we calculated.