Fit Poisson ditribution to histogram

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1

Is there a way how to fit a Poisson distribution (determine the lambda parameter) from a histogram of an experimental process?

In the experimental setup, in each event we are recording the number of detected photons and the experiment records the histogram of a huge number of the observation. So, I don’t have access to the recordings of each observations - I can only acces the histogram of a huge number of observations.

However, the Distributions.fit(D::Distribution, x) expects that I can provide an array of observations. Is there a way how to use the histogram instead and fit lambda parameter of the Poisson distribution to the histogram?

2

There might be more ready-to-go options, but otherwise you can use LsqFit to fit your data as you wish:

3

For a Poisson distribution lambda is just the mean of your data, which you can compute from your histogram.That said it would be nice to have interface that can take histogram-like data, since those are quite common. Otherwise better use maximum likelihood than least-square to fit a distribution.

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4

Adding to previous answers, if you only have access to samples in a stream of data, consider using OnlineStats.jl to compute the mean as the parameter of the distribution you are after.

5

As mentioned above the MLE would just be the mean, and then Poisson(lambda) would be the fitted distribution.

One perhaps more general possibility is to think about what a histogram is at the core and how you can potentially simulate data from the histogram itself. You are basically binning the data, and then each bin can be approximated by a Uniform(a,b) distribution. And if you have the proportion of the data in each bin (based on the Y-axis) you can approximate the empirical CDF. You now will also have the empirical CDF cutoffs for each bin.

Then you just simulate P from a Uniform(0,1) distribution and where P lands determines the bin to simulate from Uniform(a,b).

This leads to approximate data mimicking, and you can repeat the procedure many times to get a simulated mean (or whatever else).

But really if you have the mean value then that already gives you the maximum likelihood Poisson fit. The algorithm above is more of a nonparametric method that can be useful for other things too.