2 #Gaussian Trivariate 4 x N. 3 (mu, sigma) 5 mu c(1, 2, 3) 6 sigma matrix 7 (4,

Question

2 #Gaussian Trivariate 4 x N. 3 (mu, sigma) 5 mu c(1, 2, 3) 6 sigma matrix 7 (4, 1, 1 1, 6, 4, 1, 4, 9), 10 nnow 3, byrow T) 12 Eigen-decomposition of sigma 13 ES eigen sigma, symmetric T) 14 lambda ES$values 15 U ES$vectors 16 17 square-root of Sigma 18 Sigma. half u 96 diag sqrt (lambda)) %96 t (u) 19 21 Generating MVN random variates 23 1. Generate n p standard normal random variates 24 100 25 26 Z matrix (rnorm n" p),ncol n, nr ow p) 27 2 Then transform Z to X 28 x Sigma. half z mu 30 library (rgl)

Explanation / Answer

library(MASS)

Sigma <- matrix(c(10,3,3,2),2,2)
Sigma
var(mvrnorm(n = 10, rep(0, 2), Sigma))
var(mvrnorm(n = 10, rep(0, 2), Sigma, empirical = TRUE))

mu=c(5,5,5,5,5,5,5,5,5,5)
sigma=025*diag(10)+046*11^(22/7)*diag(10) # Second term is not defined in
#standard form
# Eigen-decomposition of Sigma
ES=eigen(sigma, symmetric=T)
lambda=ES$values
U=ES$vectors
# Square root of Sigma
Sigma.half=U%*%diag(sqrt(lambda))%*%t(U)
# Generating MVN standard normal distribution
# 1. Generate n8p SNRV
n=100
p=10
Z=matrix(rnorm (n*p), ncol=n, nrow=p)
# 2. Then transform Z to X
X=Sigma.half%*%Z+mu

library(rgl)
# 1 dimentional

plot(X[1,])
# 2 dimentional

plot(X[1,], X[2,])
# dimentional

plot3d(X[1,], X[2,], X[3,])

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