Exercise Solution 9.8

1. Using a spreadsheet, we calculate the determinant of as 0.000015.
2. based upon the result of part (a), is conditionally multicollinear.
3. We define ν as the matrix whose columns comprise the eigenvectors of :

   [s1]

   

   Then

   [s2]

   

   where ¹D is the eight-dimensional random vector of principal components of . It has mean 0 and diagonal covariance matrix whose diagonal elements are the eigenvalues of :

   [s3]

   

   To construct our principal component remapping, we observe that the variances of the last two principal components of are small compared to the variances of the rest. We discard those two principal components, defining as the six-dimensional random vector comprising the first six principal components of . Then [s2] becomes

   [s4]

   

   where is a matrix comprising the first six columns of ν:

   [s5]

   
4. The conditional mean vector and conditional covariance matrix of are obtained directly from the conditional mean vector and covariance matrix of ¹D. Specifically, = 0, and

   [s6]

   
5. With the addition of the principal component remapping, Schematic [9.54] becomes

   [s7]