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 ...