Problems for Part I

**Principal Component Analysis**

1.Show part 2 of Proposition 2.1.

2.For *n* = 100, 200, 500 and 1,000, simulate data from the true distribution referred to in Example 2.3. Separately for each *n* carry out parts (a)–(c).

(a)Calculate the sample eigenvalues and eigenvectors.

(b)Calculate the two-dimensional PC data, and display them.

(c)Compare your results with those of Example 2.3 and comment.

(d)For *n* = 100, repeat the simulation 100 times, and save the eigenvalues you obtained from each simulation. Show a histogram of the eigenvalues (separately for each eigenvalue), and calculate the sample means of the eigenvalues. Compare the means to the eigenvalues of the true covariance matrix and comment.

3.Let **X** satisfy the assumptions ...

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