(a) The output from names includes the following — sdev loadings center scores — Describe each these components mathematical terms answer this part question you can

(a) The output from names includes the following:

[1] "sdev" "loadings" "center" "scores"

Describe each of these components in mathematical terms. To answer this part of the question, you can print and plot the components to see what they contain and use R's help for further information.

(b) What are the first two eigenvalues of the covariance matrix?

(c) What is the eigenvector corresponding to the largest eigenvalue?

(d) Suppose you wish to “explain” at least 95% of the variation in the changes in the yield curves. Then how many principal components should you use?

SOLUTION

Part (a): The variable sdev holds the standard deviations of each principal components, these are also the square root of the eigenvalues of the covariance matrix. The variable loadings hold the eigenvectors of the covariance matrix. The variable center hold the means that were subtracted in each feature dimension in computing the covariance matrix. The variable scores holds a matrix of each vector variable projected into all its principle components. We can check that this is so by comparing the two outputs

t(as.matrix(pca_del$loadings[,])) %*% t( delta_yield[1,] – pca_del$center )

2

Comp.1 0.1905643953

Comp.2 0.0375662026

Comp.3 0.0438591813

Comp.4 – 0.0179855611

Comp.5 0.0002473111

Comp.6 0.0002924385

Comp.7 0.0101975886

Comp.8 – 0.0093768514

Comp.9 – 0.0036798653

Comp.10 0.0004287954

Comp.11 – 0.0005602180

with

pca_del$scores[1,]