π day special: How to use Bioconductor to find empirical evidence in support of π being a normal number

Editor’s note: Today 3/14/15 at some point between  9:26:53 and 9:26:54 it was the most π day of them all. Below is a repost from last year. 

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.

pi

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.

pi2

These observed counts should also be independent, and to explore this we can look at autocorrelation plots:

piacf

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.

`Editor’s note: Today 3/14/15 at some point between  9:26:53 and 9:26:54 it was the most π day of them all. Below is a repost from last year. 

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.

pi

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.

pi2

These observed counts should also be independent, and to explore this we can look at autocorrelation plots:

piacf

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.

`

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

 
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