Data609_HW4

Question 2, Page 194

Use the linear congruence method to generate

1. 10 random numbers using a = 5, b =1, and c = 8
2. 15 random numbers using a = 1, b = 7, and c = 10
3. 20 random numbers using a = 5, b = 3, and c = 16
4. Comment about the results of each sequence. Was there cycling? If so, when did it occur? There were definitely cycling of numbers and they do not appear to have much randomness in the generated number. This is also due to the parameters that were chosen.

linear congruence formula is as follow where a is the multiplier, b is the increment and c is the remainder An example from the textbook (A First Course in Mathematical Modeling, page 193), if \(x_n = 115\) then $x_{n+1} $= remainder (122/10) = 2

$ x_{n+1} = ( a * x_n + b) mod (c) $

library(knitr)
library(animation)
library(plot3D)

#install.ImageMagick()

lcg.rand <- function(n=10, a, b,c) {
  
  rng <- vector(length = n)
  
  a <- a # mulitplier
  b <- b #increment
  c <- c #modulus 

  # Set the seed using the current system time in microseconds
  d <- as.numeric(Sys.time()) * 1000
  
  for (i in 1:n) {
    d <- (a * d + b) %% c
    rng[i] <- d / c
  }
  
  return(rng)
}

lcg.rand(10,5,1,8)

##  [1] 0.2037354 0.1436768 0.8433838 0.3419189 0.8345947 0.2979736 0.6148682
##  [8] 0.1993408 0.1217041 0.7335205

n <- c(10,15,20, 100, 500, 1000, 3000, 5000, 10000, 20000)

saveGIF({
  for (i in 1:length(n)) {
    x <- lcg.rand(n[i],5,1,8)
    y <- lcg.rand(n[i],5,1,8)
    z <- lcg.rand(n[i],5,1,8)
    
    scatter3D(x, y, z, colvar = NULL, pch=20, cex = 0.5, theta=20, main = paste('n = ', n[i]))
  }
}, movie.name = 'lcg.gif')

## Executing: 
## "convert -loop 0 -delay 100 Rplot1.png Rplot2.png Rplot3.png
##     Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png
##     Rplot9.png Rplot10.png "lcg.gif""

## Output at: lcg.gif

## [1] TRUE

Different Sample Size of Random Number

2. 15 random numbers using a = 1, b = 7, and c = 10
3. 20 random numbers using a = 5, b = 3, and c = 16

saveGIF({
  for (i in 1:length(n)) {
    x <- lcg.rand(n[i],1,7,10)
    y <- lcg.rand(n[i],1,7,10)
    z <- lcg.rand(n[i],1,7,10)
    
    scatter3D(x, y, z, colvar = NULL, pch=20, cex = 0.5, theta=20, main = paste('n = ', n[i]))
  }
}, movie.name = 'lcg2.gif')

## Executing: 
## "convert -loop 0 -delay 100 Rplot1.png Rplot2.png Rplot3.png
##     Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png
##     Rplot9.png Rplot10.png "lcg2.gif""

## Output at: lcg2.gif

## [1] TRUE

B. Radom numbers by Different Parameters

saveGIF({
  for (i in 1:length(n)) {
    x <- lcg.rand(n[i],5,3,16)
    y <- lcg.rand(n[i],5,3,16)
    z <- lcg.rand(n[i],5,3,16)
    
    scatter3D(x, y, z, colvar = NULL, pch=20, cex = 0.5, theta=20, main = paste('n = ', n[i]))
  }
}, movie.name = 'lcg3.gif')

## Executing: 
## "convert -loop 0 -delay 100 Rplot1.png Rplot2.png Rplot3.png
##     Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png
##     Rplot9.png Rplot10.png "lcg3.gif""

## Output at: lcg3.gif

## [1] TRUE

C. Radom numbers by Different Parameters

Reference https://rpubs.com/aaronsc32/linear-congruential-generator-r