library(tidyverse) ## data manipulation
library(tidymodels) ## models
library(knitr) ## tables

## reproducible
set.seed(445)

## Data Preparation
# Run the following code to create two datasets.

n1 <- 20
n2 <- 200
p <- 2

## training data sets
x_small <- matrix(rnorm(n1 * p), ncol = p)
x_large <- matrix(rnorm(n2 * p), ncol = p)
y_small <- c(rep(-1, n1/2), rep(1, n1/2))
y_large <- c(rep(1, n2/4*3), rep(2, n2/4))

## shift data farther apart
x_small[y_small == 1,] <- x_small[y_small == 1,] + 1
x_large[1:100,] <- x_large[1:100,] + 2
x_large[101:150,] <- x_large[101:150,] - 2

## put data into dataframes
df_small <- data.frame(x_small, y = as.factor(y_small))
df_large <- data.frame(x_large, y = as.factor(y_large))

# 1. Make two scatterplots to inspect the small and large training data sets. Describe what you see.

## Support Vector Classifier
# 1. Fit a support vector classifier on the small data with $C = 10$ (use `scaled = FALSE` to indicated your data has not been scaled.)
# 2. How many support vectors were used to fit your classifier?
# 3. Predict a grid of $\boldsymbol X$ values between the range of $X_1$ and $X_2$. Plot these predictions using `geom_tile()` to visualize the decision boundary and add a scatteplot of training data on top, colored by training label. Describe what you see.
# 4. Alter your plot from 2 to change the shape of the support vectors.
# 5. Perform CV on the cost parameter. Which value of $C$ would you choose?
# 6. Repeat 3. and 4. using your chosen $C$ value. Describe what you see.

## Support Vector Machines
# 1. Split the large data frame into 50% training and 50% test.
# 2. Fit a linear SVM, radial SVM with $\gamma = 1$, and polynomial SVM with $d = 3$ using CV to choose the appropriate cost for each model.
# 3. Predict a grid of $\boldsymbol X$ values between the range of $X_1$ and $X_2$. Plot these predictions using `geom_tile()` to visualize the decision boundary and add a scatteplot of training data on top, colored by training label. Describe what you see.
# 4. Predict your test data with your three models. Which model would you choose?