the spatial Poisson point process

Take the idea of 'completely random events in time' and scatter them across a map instead of along a clock. Stars in a patch of sky, trees in a forest, typos on a page, cell-phone users across a city, defects on a silicon wafer — when points fall with no preference and no interaction, the spatial Poisson point process is the model. It is the two- and three-dimensional cousin of the Poisson process, the benchmark for 'complete spatial randomness'.

It is governed by an intensity: for a uniform (homogeneous) process, a constant lambda meaning expected points per unit area (or volume). The two defining rules mirror the time case. First, the number of points falling in any region A is Poisson with mean lambda times the area of A. Second, the counts in non-overlapping regions are independent. A wonderful consequence, just like in time: given that exactly n points land in a region, those n points are positioned independently and uniformly over that region — completely random placement, no pattern. The intensity can also vary across space, lambda(x), giving a non-homogeneous spatial process where the mean count in A is the integral of lambda over A.

This model is the workhorse of spatial statistics, ecology, astronomy, materials science, and wireless network design. Crucially, it is the null hypothesis against which real point patterns are tested: ecologists ask whether trees are MORE clustered (seedlings near parents) or MORE regular (competition spacing them out) than spatial-Poisson randomness predicts. The honest caveat to remember: under complete spatial randomness points often LOOK clustered to the eye — random scatter naturally produces clumps and voids — so apparent clusters are not by themselves evidence of any attracting mechanism.