data(Orange)
head(Orange)

  Tree  age circumference
1    1  118            30
2    1  484            58
3    1  664            87
4    1 1004           115
5    1 1231           120
6    1 1372           142

x <- Orange$age
y <- Orange$circumference
model <- lm(y~x, data = Orange)


orangefig <- plot_ly(Orange, x = ~age, y = ~circumference, z = ~Tree) %>%
  add_markers(color = ~Tree)

orangefig

model: \(\sum_{i=1}^n(y_i - \bar{y})^2 = \sum_{i=1}^n(\hat{y}_i - \bar{y})^2 + \sum_{i=1}^n(y_i - \hat{y}_i)^2\)

og <- ggplot(Orange, aes(x = age, y = circumference)) + geom_point(aes(color = Tree))
og + geom_smooth(method="lm", se=FALSE, color="coral")

\(circumference = \beta_0 + \beta_1\cdot age + \varepsilon\), where \(\varepsilon \sim\mathcal{N}(\mu=0; \,\,\sigma^2)\)

\(\displaystyle MSE = {SSE \over n-2} = {1\over n-2} \sum_{i=1}^n (y_i - \hat{y}_i)^2\)