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I have foot traffic data for how many people entered a building for every hour, for several days. This SOUNDS like it should follow a poisson process.

Problem:
I need to statistically confirm that my process is poisson, so that I can estimate utilization by looking at lambda (average arrival rate in time t) divided by service rate, mu.

Data Issue/Nuance:
The data is fairly sparse so there are a lot of zeros.

What I have:
I have the average number of arrivals per hour, i.e. lambda
I also have the inter-arrival time, and average inter-arrival time.

Where I'm stuck:
I'm not sure how to go from what I have now, to confirm this process is poisson.

I read in one article to do a chi-square, goodness of fit test of my inter-arrival times, and compare that with sampling from an exponential distribution with my lambda as the parameter. Especially given that my data has a lot of zeros I'm not really sure how to adapt.

Trying to do this in either Python or R

Any help is appreciated

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    $\begingroup$ What do you mean by 'a process is Poisson'? Are you suggesting that the foot traffic is constant (e.g. during the night you have as much traffic as during the day) and that traffic is not clustered (people do not enter buildings in groups)? $\endgroup$
    – Sextus Empiricus
    Commented Nov 12, 2020 at 7:04
  • $\begingroup$ So that's what I was trying to figure out, each person is treated independently. And during business hours the traffic can be fairly constant. But because of the hours late at night there are a lot of zeros in the data. I understand it's not a perfect fit for poisson but the prototypical example is usually a call center, which would have the same issue. Do you have any suggestions on if I can test if my data behaves like a poisson distribution? $\endgroup$
    – Jamalan
    Commented Nov 12, 2020 at 7:08
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    $\begingroup$ You can use Pearson's chi-squared test to compare the observed distribution with some expected distribution. The main question seems to me 'what is your expected distribution?'. When you are comparing with a regular homogeneous Poisson process then you are obviously gonna fail, because you know there are discrepancies like inhomogeneous rates as function of time (and those create the zero's). Why do you need to know whether the process is Poisson? What is the point? $\endgroup$
    – Sextus Empiricus
    Commented Nov 12, 2020 at 7:17
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    $\begingroup$ I am still a bit puzzled about what you are trying to achieve. Mostly, why you are trying to find out whether the process is Poisson. Depending on what you are gonna do with it the answer to the question will be different. It is a bit of an XY problem and in statistics the underlying background can be very important for the strategy to solve the problem. There is not gonna be a single solution, and as George Box said 'all models are wrong but some are useful'. The problem here is to figure out what is useful for you, and that requires context. $\endgroup$
    – Sextus Empiricus
    Commented Nov 12, 2020 at 8:00
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    $\begingroup$ Would it not be easier to tackle the applicability of the queuing theory directly? The question is not whether your process is Poisson (it will not be exactly Poisson in most practical situations, but does that really matter?) but whether you can use queuing theory with reasonable accuracy. Or actually, the question is about 'utilisation rate of different places'. $\endgroup$
    – Sextus Empiricus
    Commented Nov 12, 2020 at 8:05

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