Problems

1. (10 points) Textbook problem 10.2. You don't have to calculate anything; just make approximate markings.
2. (10 points) Textbook problem 10.5(a) and 10.5(c). You don't have to plot the points as characters; you can use different colors/shapes as long as you include a legend.
3. (10 points) Textbook problem 10.6
4. (10 points) Suppose you have a dataset $\{ \mathbf{x}\} = \{ ( x^{(1)}, x^{(2)} ) \}$ consisting of 2-dimensional vectors. You observe that $\text{var}( \{ x^{(1)} \} ) = 9$ and $\text{var}( \{ x^{(2)} \} ) = 4$ and also that $\text{cov}( \{ ( x^{(1)}, x^{(2)} ) \} ) = 6$.
   1. What is $\text{Covmat}(\{ \mathbf{x}\})$?
   2. Find the eigenvalues of $\text{Covmat}(\{ \mathbf{x}\})$.
   3. What do the eigenvalues say about the shape of the blob of the dataset $\{ \mathbf{x}\}$?
5. (10 points) Consider another dataset $\{ \mathbf{x}\}$ consisting of 4-dimensional vectors. Given below are $\text{mean}(\{ \mathbf{x} \})$ and the normalized eigenvectors $\mathbf{u}_i$ of $\text{Covmat}(\{ \mathbf{x}\})$, arranged in order of decreasing eigenvalue. Also given is an item $\mathbf{x}_1$ from the dataset. $$\text{mean}(\{ \mathbf{x} \})=\begin{bmatrix}1 \\ 1 \\ 1 \\ 1\end{bmatrix} \quad \mathbf{u}_1 =\begin{bmatrix}+0.5 \\ +0.5 \\ +0.5 \\ +0.5\end{bmatrix} \quad \mathbf{u}_2 =\begin{bmatrix}+0.5 \\ -0.5 \\ -0.5 \\ +0.5\end{bmatrix} \quad \mathbf{u}_3 =\begin{bmatrix}+0.5 \\ +0.5 \\ -0.5 \\ -0.5\end{bmatrix} \quad \mathbf{u}_4 =\begin{bmatrix}+0.5 \\ -0.5 \\ +0.5 \\ -0.5\end{bmatrix} \quad \mathbf{x}_1 =\begin{bmatrix}3 \\ 3 \\ 5 \\ 7\end{bmatrix}$$ Suppose you project the dataset onto its first two principal components and plot the resulting projection on a pair of axes. What are the coordinates of the projected point that represents $\mathbf{x}_1$?
6. (Extra 3 points) Textbook problem 10.3.