In brucker3/math4800:

---

title: "math4800 project an R packing for distribution sampling" author: "Bayard Rucker" date: "r Sys.Date()" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Vignette Title} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(math4800)

Vignettes are long form documentation commonly included in packages. Because they are part of the distribution of the package, they need to be as compact as possible. The html_vignette output type provides a custom style sheet (and tweaks some options) to ensure that the resulting html is as small as possible. The html_vignette format:

General description

this R packaged allows you to perform rejection sampling from pdf and joint pdf. the user has to give an input pdf and specify the size of the sample they would like as well as bounds and the the maximum of the pdf

Rejection Sampling

Rejection sampling is a type of exact simulation algorithem that works for all real numbers and in higher deminsions

algorithem steps 1. selecte a random proposed candiate from a uniforma rv 2. put the propsed candiate into the given pdf 3. selecte your check value from a uniform between 0 and the maximum with in this case is C 4. compare if the output of the pdf and the original value is less than the check value then you accept it.

simple rejction examples

Here is a simple example of how the rejection sampler would run with a pdf from the package. this is a example if you run rejection sampler with a small n and an example output

          holder <- rejectionSampler(n = 5, pdf = mypdf, lower = 0, upper = 2, C = 1)
           hist(holder)
          # 0.01618789 0.24490906 0.05359767 0.49077322 1.70218396
          #pdf used in rejction ssampler
          mypdf <- function(x){
             if( x <= 2  && x >= 0) {
                x/2
             }else{
                 0
             }
          }

           holder1 <- rejectionSampler(n = 5, pdf = normpdf, lower = 0, upper = 3, C = 1)
          hist(holder1)
          # 2.2596883 0.9969329 0.8917946 1.2306571 1.4431262
           #pdf used in rejction ssampler
         normpdf <- function(x){
              dnorm(x)
          }

2D rejection sampler

this rejection sampler works on join distributions that are defined on a square. the underlying algorithem works the same way expect that here we are dealing with joint distributions that are defined for a specific region of the plain

Here is a simpler joint rejection smapler with a joint normal distirbtion

    holder <- jointRejectionSampler(n = 10, jointpdf = jointnorm, lower = 0, upper = 1, C = 1)
     hist(holder)
     # jointpdf used in rejction ssampler
    jointnorm <- function(x,y){
      if( x == y) {
        if(x >=0 && x <= 1 && y >= 0 && y <= 1) {
          joint <- dnorm(x) * dnorm(y)
        }else{
          0
        }
      }else{
        0
      }
    }

out put from the rejction sampler
x y 1 0.3267715 0.8545568 2 0.7263313 0.4751195 3 0.6225835 0.4948918 4 0.5576389 0.2399189 5 0.4323270 0.8654610 6 0.3047128 0.8800500 7 0.2844174 0.8304783 8 0.6518205 0.6154928 9 0.5631891 0.4660881 10 0.3482138 0.4825210

Another example with a joint uniform function

    holder <- jointRejectionSampler(n = 1000, jointpdf = jointunif, lower = 0, upper = 1, C = 1)
    hist(holder)
    # jointpdf used in rejction ssampler
    jointunif <- function(x,y){
       if(0 <= x && x <= 1 && 0 <= y && y <= 1) {
          .5
      }else {
          0
      }

    }

out put of the 2D rejection sampler using a joint uniform R.V. x y 1 0.43593755 0.70838476 2 0.09677250 0.96729673 3 0.13023959 0.33197155 4 0.05804621 0.56934939 5 0.85796676 0.91361058 6 0.73084249 0.47628922 7 0.85954057 0.24595708 8 0.15807758 0.03623352 9 0.33162128 0.24503933 10 0.15412658 0.26729462

brucker3/math4800 documentation built on May 16, 2019, 7:14 p.m.