• Goal: Model how one variable changes with another
• Example: Predict height from age, sales from advertising

• One predictor x, one outcome y
• We assume a straight line pattern with some noise
• We will:
  • Choose a line
  • Measure errors
  • Pick the line with the smallest total error

\[ y = \beta_0 + \beta_1 x + \varepsilon \]

• \(y\): response (what we want to predict)
• \(x\): predictor (what we use to predict)
• \(\beta_0\): intercept (value of \(y\) when \(x = 0\))
• \(\beta_1\): slope (change in \(y\) when \(x\) increases by 1)
• \(\varepsilon\): random noise

We draw a line to get predicted values:

\[ \hat y_i = \hat\beta_0 + \hat\beta_1 x_i \]

The residual for point \(i\) is:

\[ \text{residual}_i = y_i - \hat y_i \]

Least squares chooses \(\hat\beta_0, \hat\beta_1\) to make

\[ \sum_{i=1}^n \text{residual}_i^2 \]

as small as possible.

ggplot(data, aes(x = x, y = y)) +
  geom_point(color = "blue", size = 3) +
  geom_smooth(method = "lm", se = FALSE, color = "red") +
  labs(x = "X",
       y = "Y") +
  theme_minimal()

Explanation:

geom_smooth(        # This add a fitted curve to the plot
  method = "lm",    # "lm" = Linear Model, on x and y variables
  se = FALSE,       # "se" = enables or disables confidence band
  color = "red"     # This makes the line red
)

• Not everything can be modeled with a straight line
• Outliers can throw off your equation if not accounted for

Linear regressions can work with multiple dimensions…

Look forwards to next lesson where you learn to work with 3D data!