# How to use Bioconductor to find empirical evidence in support of π being a normal number

March 14, 2014
By

(This article was first published on Simply Statistics » R, and kindly contributed to R-bloggers)

Happy π day everybody!

I wanted to write some simple code (included below) to the test parallelization capabilities of my  new cluster. So, in honor of  π day, I decided to check for evidence that π is a normal number. A normal number is a real number whose infinite sequence of digits has the property that picking any given random m digit pattern is 10−m. For example, using the Poisson approximation, we can predict that the pattern “123456789” should show up between 0 and 3 times in the first billion digits of π (it actually shows up twice starting, at the 523,551,502-th and  773,349,079-th decimal places).

To test our hypothesis, let Y1, …, Y100 be the number of “00”, “01”, …,”99″ in the first billion digits of  π. If  π is in fact normal then the Ys should be approximately IID binomials with N=1 billon and p=0.01.  In the qq-plot below I show Z-scores (Y – 10,000,000) /  √9,900,000) which appear to follow a normal distribution as predicted by our hypothesis. Further evidence for π being normal is provided by repeating this experiment for 3,4,5,6, and 7 digit patterns (for 5,6 and 7 I sampled 10,000 patterns). Note that we can perform a chi-square test for the uniform distribution as well. For patterns of size 1,2,3 the p-values were 0.84, 0.89, 0.92, and 0.99. Another test we can perform is to divide the 1 billion digits into 100,000 non-overlapping segments of length 10,000. The vector of counts for any given pattern should also be binomial. Below I also include these qq-plots. These observed counts should also be independent, and to explore this we can look at autocorrelation plots: To do this in about an hour and with just a few lines of code (included below), I used the Bioconductor Biostrings package to match strings and the `foreach` function to parallelize.

```library(Biostrings)
library(doParallel)
registerDoParallel(cores = 48)
x=scan("pi-billion.txt",what="c")
x=substr(x,3,nchar(x)) ##remove 3.
x=BString(x)
n<-length(x)
p <- 1/(10^d)
par(mfrow=c(2,3))
for(d in 2:4){
if(d<5){
patterns<-sprintf(paste0("%0",d,"d"),seq(0,10^d-1))
} else{
patterns<-sprintf(paste0("%0",d,"d"),sample(10^d,10^4)-1)
}
res <- foreach(pat=patterns,.combine=c) %dopar% countPattern(pat,x)
z <- (res - n*p ) / sqrt( n*p*(1-p) )
qqnorm(z,xlab="Theoretical quantiles",ylab="Observed z-scores",main=paste(d,"digits"))
abline(0,1)
##correction: original post had length(res)
if(d<5) print(1-pchisq(sum ((res - n*p)^2/(n*p)),length(res)-1))
}
###Now count in segments
d <- 1
m <-10^5

patterns <-sprintf(paste0("%0",d,"d"),seq(0,10^d-1))
res <- foreach(pat=patterns,.combine=cbind) %dopar% {
tmp<-start(matchPattern(pat,x))
tmp2<-floor( (tmp-1)/m)
return(tabulate(tmp2+1,nbins=n/m))
}
##qq-plots
par(mfrow=c(2,5))
p <- 1/(10^d)
for(i in 1:ncol(res)){
z <- (res[,i] - m*p) / sqrt( m*p*(1-p)  )
qqnorm(z,xlab="Theoretical quantiles",ylab="Observed z-scores",main=paste(i-1))
abline(0,1)
}
##ACF plots
par(mfrow=c(2,5))
for(i in 1:ncol(res)) acf(res[,i])```

NB: A normal number has the above stated property in any base. The examples above a for base 10.

To leave a comment for the author, please follow the link and comment on their blog: Simply Statistics » R.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...

If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.

# Never miss an update! Subscribe to R-bloggers to receive e-mails with the latest R posts.(You will not see this message again.)

Click here to close (This popup will not appear again)