Fit the linear SVM to olive oils, using a training split of 2/3, using only regions 2, 3, and the predictors linoleic and arachidic. Report the training and test error, list the support vectors, the coefficients for the support vectors and the equation for the separating hyperplane, and
$$???\times\text{linoleic}+???\times\text{arachidic}+??? > 0$$ and make a plot of the boundary.

```{r}
notsouth <- olive %>%
  filter(region != 1) %>%
  select(region, linoleic, arachidic) %>%
  mutate(region = factor(region)) %>%
  mutate(across(where(is.numeric), ~ (.x - mean(.x)) / sd(.x)))
```

```{r}
set.seed(2021)
notsouth_split <- initial_split(notsouth, prop = 2/3, strata = region)
notsouth_tr <- training(notsouth_split)
notsouth_ts <- testing(notsouth_split)

library(kernlab)

svm_mod <-
  svm_rbf(cost = 10) %>%
  set_mode("classification") %>%
  set_engine("kernlab", 
             kernel = "vanilladot", # linear kernel, see ?kernlab::ksvm()
             scaled = FALSE)

notsouth_svm <- svm_mod %>%
  fit(region ~ ., data = notsouth_tr)
```

```{r}
notsouth_p <- as_tibble(expand_grid(linoleic = seq(-2.2, 2.2, 0.1), 
                                    arachidic = seq(-2, 2, 0.1))) 

notsouth_p  <- notsouth_p  %>%
  mutate(region_svm = predict(notsouth_svm, notsouth_p )$.pred_class)

ggplot() +
  # Predicted values
  geom_point(data = notsouth_p, 
             aes(x = linoleic, 
                 y = arachidic,
                 color = region_svm), 
             alpha = 0.1) +
  # Overlay with actual data
  geom_point(data = notsouth, 
             aes(x = linoleic, 
                 y = arachidic, 
                 color = region, 
                 shape = region)) +
  # Circle the support vectors
  geom_point(data = notsouth_tr %>% slice(notsouth_svm$fit@SVindex), # Extract support vectors
             aes(x = linoleic, y = arachidic), 
             shape = 1, 
             size = 3,
             colour = "black") +
  scale_color_brewer("", palette="Dark2") +
  theme_bw() +
  theme(aspect.ratio = 1,
        legend.position = "none") +
  ggtitle("SVM") +
  geom_abline(intercept = 1.45396, slope = -3.478113)
```

The $\alpha$'s, indexes of support vectors and $\beta_0$, and the observations that are the support vectors are:

```{r}
notsouth_svm$fit@coef # \alpha * y_i
notsouth_svm$fit@SVindex # Indexes (row numbers) of support vectors
notsouth_svm$fit@b # Negative intercept term -b_0

# Extract support vectors using indexes
notsouth_tr[notsouth_svm$fit@SVindex, ] # Support vectors; notsouth_tr %>% slice(6,102,132)
notsouth_tr$region[notsouth_svm$fit@SVindex] # Response variable of support vectors
```