DATA 605 - Week 10 Exercise

Section 11.1 Exercise 13

Write a program to compute \(u^{(n)}\) given u and P. Use this program to compute \(u^{(10)}\) for the Land of Oz example, with u = (0,1,0), and with u = (1/3, 1/3, 1/3).

# Setup
library("expm")
options(digits=4)

# Transition matrix
A <- matrix(c(0.5,0.25,0.25,0.5,0,0.5,0.25,0.25,0.5), nrow=3, ncol=3, byrow=TRUE)
A

##      [,1] [,2] [,3]
## [1,] 0.50 0.25 0.25
## [2,] 0.50 0.00 0.50
## [3,] 0.25 0.25 0.50

# Initial probability vectors
u1 <- matrix(c(0,1,0), ncol=3, nrow=1)
u2 <- matrix(c(1/3,1/3,1/3), ncol=3, nrow=1)

land_of_oz <- function(u, n) {
  P_n <- A %^% n
  return (u %*% P_n)
}

land_of_oz(u1,10)

##      [,1] [,2] [,3]
## [1,]  0.4  0.2  0.4

land_of_oz(u2,10)

##      [,1] [,2] [,3]
## [1,]  0.4  0.2  0.4

The two results converge to the same numbers despite the different initial probabilities vectors.