Complete spatial randomness (CSR) describes a point process whereby point events occur within a given study area in a completely random fashion.
It is synonymous with a homogeneous spatial Poisson process.
The term complete spatial randomness is commonly used in Applied Statistics in the context of examining certain point patterns, whereas in most other statistical contexts it is referred to the concept of a spatial Poisson process.
[1] Data in the form of a set of points, irregularly distributed within a region of space, arise in many different contexts; examples include locations of trees in a forest, of nests of birds, of nuclei in tissue, of ill people in a population at risk.
We call any such data-set a spatial point pattern and refer to the locations as events, to distinguish these from arbitrary points of the region in question.
The hypothesis of complete spatial randomness for a spatial point pattern asserts that the number of events in any region follows a Poisson distribution with given mean count per uniform subdivision.
"Uniform" is used in the sense of following a uniform probability distribution across the study region, not in the sense of “evenly” dispersed across the study region.
is therefore: The first moment of which, the average number of points in the area, is simply
can be derived via the use of the gamma function using statistical moments.
The first moment is the mean distance between randomly distributed particles in
The study of CSR is essential for the comparison of measured point data from experimental sources.
[3] CSR is often the standard against which data sets are tested.
Roughly described one approach to test the CSR hypothesis is the following:[4] In cases where computing test statistics analytically is difficult, numerical methods, such as the Monte Carlo method simulation are employed, by simulating a stochastic process a large number of times.