In using the PCA, suppose that the data set contains two variables, and the normalized versions are labeled as 1 and 2 . Also, after calculating the SVD, one obtains the matrix                 In this...

In using the PCA, suppose that the data set contains two variables, and the normalized versions are labeled as

₁
and

₂. Also, after calculating the SVD, one obtains the matrix

In this problem the PCA assumption is that
p
=

₁
v
₁.

(a) Assuming

₁₁
 0, show that the assumption
p
= α₁
v
₁
can be written as

₂
=

₁, where
 =
₂₁/
₁₁.

(b) Letting

₁
and

₂
be the original, unnormalized, data variables, show that the fitted line can be written as

where

₁
and

₂
are the means of

₁
and

₂, respectively. Also,
 =

₂/
₁, where

₁
and

₂
are the scale factors used for the respective variables.