Question

Principal Component Analysis (25 points): Consider the observations representing $X_i \in \mathbb{R}^2$, $\begin{bmatrix} 2 & 3 \\ 5 & 4 \\ 1 & 0 \\ 4 & 1 \end{bmatrix}$ (a) Compute the unit length principal components of X. (b) Use PCA to approximate the data using one dimension. Derive the equation for a new, $X_i \in \mathbb{R}^2$, observation and its approximation. (c) Plot the 4 samples in X. Draw the directions of two eigenvectors and the projections of all the sample points on the both eigenvectors. Label all samples precisely. (d) Calculate the variance of the projections on the principle component direction (i.e., the eigenvector with the largest eigenvalue) as well as the projections on the second eigenvector. Which variance is larger? Justify your answer.

          Principal Component Analysis (25 points): Consider the observations
representing $X_i \in \mathbb{R}^2$,
$\begin{bmatrix} 2 & 3 \\ 5 & 4 \\ 1 & 0 \\ 4 & 1 \end{bmatrix}$
(a) Compute the unit length principal components of X.
(b) Use PCA to approximate the data using one dimension. Derive the
equation for a new, $X_i \in \mathbb{R}^2$, observation and its approximation.
(c) Plot the 4 samples in X. Draw the directions of two eigenvectors
and the projections of all the sample points on the both eigenvectors.
Label all samples precisely.
(d) Calculate the variance of the projections on the principle component
direction (i.e., the eigenvector with the largest eigenvalue) as well
as the projections on the second eigenvector. Which variance is
larger? Justify your answer.

Principal Component Analysis (25 points): Consider the observations
representing Xi ∈ℝ^2,
< b m a t r i x >
(a) Compute the unit length principal components of X.
(b) Use PCA to approximate the data using one dimension. Derive the
equation for a new, Xi ∈ℝ^2, observation and its approximation.
(c) Plot the 4 samples in X. Draw the directions of two eigenvectors
and the projections of all the sample points on the both eigenvectors.
Label all samples precisely.
(d) Calculate the variance of the projections on the principle component
direction (i.e., the eigenvector with the largest eigenvalue) as well
as the projections on the second eigenvector. Which variance is
larger? Justify your answer.

Added by Sean M.

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