Gas station without pumps

2013 June 5

Poisson Petals

Filed under: Uncategorized — gasstationwithoutpumps @ 22:37
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Robert Jernigan, on his statpics blogs, has a couple of nice posts based on an image he posted of cherry blossoms on paving blocks:

statpics: Poisson Petals gives a cursory analysis of the mean number of petals on the large blocks, divided by the mean number of the petals on the smaller blocks.

statpics: Testing Poissonness Petals provides a much more detailed analysis of the same data (the number of petals on each block), showing first that the data is well described as a Poisson process, then that the ratio of means (1.56) is not significantly different from the ratio of the areas of the blocks (1.5) with a p-value of 0.39.

I had not see the Poissonness plot [David C. Hoaglin. A Poissonness Plot. The American Statistician. 34(3):146–149, 1980] before, but it looks like a handy technique.  Because the expected frequency of k petals on a block is $N e^{-\lambda} \lambda^k k!$, if we plot $\ln(x_k) +\ln(k!)$ vs. k, we should get a straight line from a Poisson process, with the slope of the line being $\ln(\lambda)$.  Jernigan’s data makes a very nice straight line.

His significance test does an exact computation of the probability distribution for the ratio of the means, but I didn’t quite follow how he set up the computation.  The connection I missed was between

The larger stones have a length of 9.375. The shorter, square stones have a length of 6.25, for a ratio of 1.5.

and

We take these as the null parameters of two independent Poisson distributions.

I’m not sure what he is taking as “the null parameters”, as the length of the paving blocks is not the λ of the Poisson process, though they are linearly related.  Is he scaling by the overall petals per area measure?

Assuming that he does some scaling like that to get two parameters $\lambda_{\mbox{\scriptsize large}} = 1.5 \lambda_{\mbox{\scriptsize small}}$, then the rest of his computation makes sense to me.  He gets a Poisson distribution for the total number of petals on large blocks $\lambda = 21 \lambda_{\mbox{\scriptsize large}}$ and for the total numbers of petals on small blocks $\lambda= 12 \lambda_{\mbox{\scriptsize small}}$.  A simple nested loop going out to sufficiently high values would enumerate the probabilities for all pairs of petal counts, and from that the discrete probability distribution for the ratio of the means can be computed.

I wonder if a Bayesian approach with a Gamma conjugate prior would give a different posterior distribution for the ratio of the means.