Determine the population principal components Y_1 and Y_2 for the covariance mat
ID: 3226075 • Letter: D
Question
Determine the population principal components Y_1 and Y_2 for the covariance matrix sigma = [5 2 2 2]. Also, calculate the proportion of the total population variance explained by the first principal component. Convert the covariance matrix in Problem 1 into a correlation matrix rho. (a) Determine the principal components Y_1 and Y_2 from rho and compute the proportion of the total population variance explained by Y_1. (b) Compare the components calculated in Part a with those obtained in Problem 1. Are they the same? Should they be? Comment in detail. (c) Compute the correlations rho Y_1, Z_1, rho Y_1, Z_2, and rho Y_1, Z_1.Explanation / Answer
The analysis is performed using R statistical software.
1. We enter the covariance matrix and run princomp on it to get the principal components:
> covmat <- matrix(c(5,2,2,2),nrow=2)
> summary(princomp(covmat=covmat))
Importance of components:
Comp.1 Comp.2
Standard deviation 2.4494897 1.0000000
Proportion of Variance 0.8571429 0.1428571
Cumulative Proportion 0.8571429 1.0000000
Thus, 85.71% of the population variance is explained by the first principal component.
2. The covariance matrix is converted into a correlation matrix and the principal component is run again:
> cormat <- cov2cor(covmat)
> cormat
[,1] [,2]
[1,] 1.0000000 0.6324555
[2,] 0.6324555 1.0000000
> summary(princomp(covmat=cormat))
Importance of components:
Comp.1 Comp.2
Standard deviation 1.2776758 0.6062545
Proportion of Variance 0.8162278 0.1837722
Cumulative Proportion 0.8162278 1.0000000
(a) The proportion of the total population variance explained is 81.62%.
(b) The proportions explained by 1 and (a) above are different.
(c) Since Z1 and Z2 are not clearly given in the question, the answer can't be provided here.
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