1. Show that the sum of the variances of β + (g)j is equal to the sum of the variances of j . That is, show that tr{Var[β + (g) ]} = tr{Var[ (g) ]}. 2. Show that the length of the β + (g) vector is...

1. Show that the sum of the variances of β⁺
_(g)j
is equal to the sum of the variances of

_j
. That is, show that tr{Var[β⁺
_(g)]} = tr{Var[
_(g)]}.

2. Show that the length of the β⁺
_(g)
vector is the same as the length of

_(g)

3. Use the logarithms of the nine independent variables in the peak flow runoff.

(a) Center and scale the independent variables to obtain Z and Z Z, the correlation matrix.

(b) Do the singular value decomposition on Z and construct the biplot for the first and second principal component dimensions. What proportion of the dispersion in the X-space is accounted for by these first two dimensions?

(c) Use the correlation matrix and the biplot to describe the correlational structure of the independent variables.