Use the information in Exercise 12.2 to a. Compute 95% confidence intervals for the eigenvalues, assuming large samples and multinormal data. b. Examine the eigenvalue separation using the North et...

Use the information in Exercise 12.2 to

a. Compute 95% confidence intervals for the eigenvalues, assuming large samples and multinormal data.

b. Examine the eigenvalue separation using the North et al. rule of thumb.

Using the information in Exercise 12.2, calculate the eigenvector matrix [E] to be orthogonally rotated if

a. The resulting rotated eigenvectors are to be orthogonal.

b. The resulting principal components are to be uncorrelated.

Exercise 12.2

A principal component analysis of the data in Table A.3 yields the three eigenvectors e₁
^T
= [.593, .552, –.587], e₂
^T
= [ .332, –.831, –.446], and e₃
^T
= [ .734, –.069, .676], where the three elements in each vector pertain to the temperature, precipitation, and pressure data, respectively. The corresponding three eigenvalues are

₁
= 2.476, l ¼=0.356, and l = 0.169.

a. Was this analysis done using the covariance matrix or the correlation matrix? How can you tell?

b. How many principal components should be retained according to Kaiser’s rule, Jolliffe’s modification, and the broken stick model?

c. Reconstruct the data for 1951, using a synthesis truncated after the first two principal components.