Question

ONLY NEED VALUES FOR C AND D. PLUS FINAL PLOT

Use the following code to show that the power method can be used to calculate the largest eigenvalue and corresponding eigenvector of a covariance matrix.

A. Generate the data:

x <- rnorm(100000)

dim(x) <- c(500,200)

x <- x* 1:9

x[,1] <- x[,1]*2+ x[,3] + x[,20]

x[,5] <- x[,5]*3+ x[,3] +2* x[,20]

B.Calculate the covariance matrix and its powers

vx <- var(x)

vx2 <- vx%*%vx

vx4 <- vx2%*%vx2

vx8 <- vx4%*%vx4

vx16 <- vx8%*%vx8

vx32 <- vx16%*%vx16

vx64 <- vx32%*%vx32

C. Approximate the largest eigenvalue and see how it converges.

# By the (1,1) element

vx2[1,1]^(1/2)

vx4[1,1]^(1/4)

vx8[1,1]^(1/8)

vx16[1,1]^(1/16)

vx32[1,1]^(1/32)

vx64[1,1]^(1/64)

# By the trace

sum(diag(vx2))^(1/2)

sum(diag(vx4))^(1/4)

sum(diag(vx8))^(1/8)

sum(diag(vx32))^(1/32)

D.Check it with the true value.

ei <- eigen(vx)

ei$values[1]

E.Calculate the 1st eigenvector using vx16

# Add the norm command first

norm <- function(x) sqrt(sum(x^2))

h <- vx16[,1]

h <- h/norm(h)

F. Check it with the true values

h0 <- ei$vectors[,1]

plot(h,h0)

abline(0,1)

Answer 1