(Solved): Consider the covariance matrix \sum =([4,-7],[-7,15]) of two variables, the age (x_(1)) of cars, mea ...
Consider the covariance matrix
\sum =([4,-7],[-7,15])of two variables, the age
(x_(1))of cars, measured in years, and its selling price
(x_(2)), measured in thousand rupees. (a) (i) Find the eigenvalues
\lambda _(1)and
\lambda _(2)of the covariance matrix
\sum . (ii) Determine the population principal components
Y_(1)and
Y_(2)for the covariance matrix
\sum . (iii) Show that
Var(Y_(1))=\lambda _(1)and
Cov(Y_(1),Y_(2))=0. (iv) Calculate the proportion of the total population variance explained by the first principal component. (v) Compute the correlation
\rho (Y_(1),x_(1)).
(2+4+4+2+3 Marks )(b) (i) Convert the covariance matrix to a correlation matrix
\rho . (ii) Determine the principal components
Y_(1)and
Y_(2)from correlation matrix
\rho and compute the proportion of total population variance explained by the first principal component. (iii) Compare the components calculated in part (a)(ii) with those obtained in part (b)(ii) and explain if they should be the same or not?
