Homework 9

Due Mon. Nov. 8 at 11:59pm

Homework policies and submission instructions

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.