# Is there a distribution which describes counts of hexagons in a spatial plane

From the context of bird surveys (how many breeding bird territories overlap a circular sampling area), is there a more suitable distribution to use, rather than poisson?

In this case, a poisson distribution is commonly used to describe the distribution of counts over various sampling points.

My issue is that poisson represents counts of spatial points (which zero area/volume). When doing bird surveys, territories are hexagon shaped (for the sake of simplicity). The area of these hexagons provide a ceiling which is not present in a poisson distribution (the right hand tail of the poisson can represent theoretically impossible values).

In my mind, a better suited distribution would be derived from a spatial map of adjacent hexagons, and while randomly placing circles on this space (representing observers), the Probability Density Function would represent the distribution of the number of hexagons which overlaps each of the randomly placed observer circles. The relative size of the hexagons vs the size of observer sampling area would describe the mean of this distribution.

Any thoughts preexisting probability density functions which may address this need? In my mind (which is somewhat uninformed about statistics), if this would be combined with a binomial distribution which represents the occupancy of the said territories, this would be a more suitable tool for species distribution modelling.

## Edit (comment)

Indeed, it seems that perhaps I should move this onto the math forum, as there seems to be closely related questions on there, as I'm more interested in the proof of the distribution rather than the practical aspects of calculating how well alternate distributions fit this situation. I'm thinking that there are no commonly used distributions for this purpose, and I'll have to write my own probability density function based on geometry proofs.

• Your question is far more general. For instance any normal distribution, as commonly used to model (say) measurement errors, assigns nonzero probability to ridiculously impossible values. The right question to ask yourself concerns whether this matters: provided those probabilities remain so small that they do not affect your analysis, why should you be concerned? The issue before you, then, is to assess whether the Poisson models used in your analysis are reasonable or not. – whuber May 16 '17 at 18:00
• Hi @Whuber, I'm not sure I get your point. I use poisson distributions now, but I have no way of knowing how it affects my analysis, as I have no information about the true population distribution being sampled. Trying to fit alternate distributions is part of model selection, as I understand it... so my idea is to try alternate distributions which theoretically better resemble the underlying process to see if they result in better fitting models. – RTbecard May 16 '17 at 19:45
• Sure you have a way of estimating how big the ceiling effect is. The fiducial quantity is (mean count)/(maximum count). So if the mean count is 100 birds per hexagon and at most 1000 birds would fit in a hexagon, you are talking about ~10% effects. If the mean count is 10 birds per hexagon and at most 10000 birds would fit in a hexagon, you are talking about ~0.1% effects. Depending on your accuracy requirements, you can estimate whether the effects are likely to be important. – David Wright May 16 '17 at 21:11
• I believe you have excellent, simple ways to assess how using the Poisson distribution affects the analysis. Take any of the Poisson parameter estimates. What probability does it assign to reasonable upper limits to the counts? I suspect that probability may be astronomically small. – whuber May 16 '17 at 23:35
• @Glen_b, thanks for the distribution suggestions. Indeed, a more practical approach is just find a distribution which can approximate the given situation, and that's something I'll likely have to do, and its just a matter of figuring out the appropriate for the parameters. But I was hoping to get some theoretical arguments for a given distribution, rather than just using a distribution that may approximate my situation, and not worrying about the underlying theoretical arguments. – RTbecard May 18 '17 at 14:08