Determine function from Poisson PDF with lambda < 1

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J Toole 2023-2-17

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评论： J Toole 2023-2-17

I am trying to fit a function to a set of data with a Poisson distribution. Using poissfit, I determined the lambda (mean) is less than 1.

Looking at the Poisson PDF, it only handles integer values (i.e. counting), so how do I determine a Poisson function when lambda < 1?

For example, say I have a data set of 20 values consisting of only 0's and 1's. Perhaps 15 total 0's and 5 total 1's, so the lambda = 0.25.

This example is a realistic distribution for counting. However, lambda < 1 doesn't make sense because one would need at least 1 "event" to make sense of a Poisson distribution.

My only work-around thought would be to scale the non-zero values. Using the example dataset above, say I scaled the 20 values by a factor of 100: 15 zero's and 5 100's. Lambda = 25, so I could plot a smooth function and then scale/normalize it back. Counting only 0's and 100's is not realistic, so this shouldn't be valid. Right?

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Answer 1

the cyclist 2023-2-17

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编辑：the cyclist 2023-2-17

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It is not accurate to say "one would need at least 1 'event' to make sense of a Poisson distribution". The events can be very rare (low probability of observation during a particular interval), but still be Poisson.

Taking your case ...

lambda = 0.25;

count = 0:10;

prob = poisspdf(count,lambda);

bar(count,prob)

xlabel("Count during interval")

ylabel("Probability of this count")

If the mean probability is 0.25, then most of the time, you see zero events, sometimes 1, and quite rarely more than 1.

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J Toole 2023-2-17

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Answer 2

John D'Errico 2023-2-17

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编辑：John D'Errico 2023-2-17

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I don't see the problem. Yes, you need good data. Not having sufficient data will kill any chances for do anything, but what do you expect?

A Poisson distribution is a counting distribution. Think of it as counting the number of events that are seen in some interval of time.

https://en.wikipedia.org/wiki/Poisson_distribution

The Poisson rate lambda is equal to the mean of the distribution. Large rates are equivalent to seeing many arrivals in a unit time. Small rates tell you that you expect to see few arrivals. For example:

poissrnd(1,[1,12])
ans = 1×12
     0     2     4     0     1     0     1     0     0     0     0     2

In the above example, we see a significant number of times where there would be more than 1 Poisson arrival in a unit time. But with a smaller Poisson rate

poissrnd(0.25,[1,12])
ans = 1×12
     0     1     0     1     0     1     0     0     0     1     0     0

With that small of a poisson rate, you expect it to be a moderately rare event where you would see any arrivals at all in a unit of time. And that makes complete sense.

So I'm not at all sure what is your problem. lambda<1 is perfectly reasonable. It probably means that you might have considered using a longer time span. It certainly means more data would help.

And no, scaling your data will not help at all.