如何建模多个（＆gt; 2）彩色多边形过程？

Question

我正在尝试构建一个Polya Urn模型。两种颜色很好，有三种颜色然而我遇到了麻烦。

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我的问题在于翻译这行代码：

ndraws<-1000; nexps<-2000; Distribution.yellow<-matrix(0,ndraws,1); for (k in       1:nexps){
red<- 1;
yellow<- 1;
blue<-1 ;
for (n in 1:ndraws){
    drawn<-sample(0:2,size=1,prob=c(red,yellow,blue)/(red+yellow +blue))
    red<-?? ;
    blue<-?? ;
    yellow<-?? ;
  }
  Distribution.yellow[k]<-yellow/(red+yellow+blue) }

添加到骨灰盒的各个额外球。 （因此问号）。

我用两种颜色做了如下：

 drawn<-sample(0:2,size=1,prob=c(red,yellow,blue)/(red+yellow +blue))

但是当这两种颜色超过两种颜色时，这显然不起作用。我应该如何处理三种或更多颜色？

Answer 1

根据Wikipedia，Pólya骨灰盒过程的规则是：

  从瓮中随机抽取一个球并观察其颜色;然后将其返回到骨灰盒中，并将另一个相同颜色的球添加到骨灰盒中，并重复选择过程。

换句话说，画一个球会使该colo（u）r的球数增加一个。

所以我们可以设置一个if语句，如果drawn==0添加一个红球，drawn==1增加一个黄球，否则添加一个蓝球......

ndraws <- 1000
nexps <- 500
set.seed(101)
yellow_final <- numeric(nexps)
for (k in 1:nexps) {
    red <- 1; yellow <- 1; blue<-1
    for (n in 1:ndraws) {
        drawn <- sample(0:2,size=1,prob=c(red,yellow,blue)/(red+yellow+blue))
        if (drawn==0) {
            red <- red+1
        } else if (drawn==1) {
            yellow <- yellow+1
        } else blue <- blue+1
    }
    yellow_final[k]<-yellow/(red+yellow+blue)
}

图片：

par(las=1,bty="l")
hist(yellow_final,col="gray",freq=FALSE,
     xlab="Prop. yellow after 1000 draws")

Answer 2

一般化解决方案：

ndraws<-1000 
nexps<-2000
colors <- c('red', 'blue', 'yellow') # add balls with other colors
initial.num.balls <- c(1,1,1) # can have different numbers of balls to start with
ball.to.observe <- 'yellow'
distribution.ball.to.observe <- replicate(nexps, {
  urn <- rep(colors, initial.num.balls) # polya's urn
  count.balls <- as.list(initial.num.balls)
  names(count.balls) <- colors
  for (i in 1:ndraws) {
    drawn <- sample(urn, 1)
    count.balls[[drawn]] <- count.balls[[drawn]] + 1
    urn <- c(urn, drawn)
  }
  count.balls[[ball.to.observe]] / sum(as.numeric(count.balls))
})
library(ggplot2)
ggplot() + stat_density(aes(distribution.ball.to.observe), bw=0.01)