# Use of Poisson distribution to analyse distribution of individuals in space

Dytham 2010 suggests using the Poisson distribution to establish whether individuals are evenly distributed in space.

Say we end up with a map of individuals in a study site that looks like the pattern below. The black boundary is the study site boundary and the black points are the location of individuals. Dytham implies dividing the study site into sub-units, counting the number of individuals in each sub-unit and calculating the mean/variance of number of individuals across sub-units. We could then simulate a Poisson distribution with the same mean/variance and seeing how well the observed distribution of individuals matches this simulated Poisson distribution. Given the pattern of spacing of individuals seen below, we would expect a poor fit to the simulated Poisson distribution.

For a long time, I couldn't understand what this poor fit would actually tell us. However, I think the penny has dropped. I think what Dytham is saying is that if the assumptions of the Poisson distribution are met (events are independent and events are random), then the spacing of individuals will be a good fit to a Poisson distribution. But in the image below, we will see a poor fit to the Poisson distribution, therefore indicating that the spacing of individuals isn't random and not independent of each other.

Does it sound like my understanding is correct? • – kjetil b halvorsen Jun 23 '15 at 12:58
• For a literary/historical reference, consider Thomas Pynchon's Gravity's Rainbow, which covers 2-dimensional Poisson processes in some detail. – EdM Jun 23 '15 at 13:57

## 1 Answer

Your understanding is basically correct, and this kind of analysis is much older than your reference http://www.amazon.co.uk/Choosing-Using-Statistics-Biologists-Guide/dp/1405198397/ref=sr_1_1?ie=UTF8&qid=1435059697&sr=8-1&keywords=Dytham%20statistics Such a model is called a poisson point process, so I have added the tag point-process to your post. You could search this site for posts with that tag! There is a huge literature for spatial point processes, and such point processes are poisson point process under assumptions of independence and homogeneity.

But your quadrat count alluded to in your post, could fail to have an poisson distribution either becaue of lack of independence (clustering or inhibitions between points) or because of lack of homogeneity (base rates varying in space).

Today there are better analysis methods around than quadrat counts (which have a certain degree of sunjectivity because there is arbitrariness in defing quadrats). A gentle introduction is chapter 15 of Venables & Ripley: "Modern Applied Statistics with S (fourth edition)".

If you will post your data we could have a look.