Complete spatial randomness

Complete spatial randomness (CSR) describes a point process whereby point events occur within a given study area in a completely random fashion. Such a process is often modeled using only one parameter, i.e. the density of points, $\rho$ within the defined area. This is also called a spatial Poisson process.

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.

Model

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. The intensity of events does not vary over the plane. This implies that there are no interactions amongst the events. For example, the independence assumption would be violated if the existence of one event either encouraged or inhibited the occurrence of other events in the neighborhood. The study CSR is essential for the comparison of measured point data from experimental sources. As a statistical testing method, the test for CSR has many applications in the social sciences and in astronomical examinations.

Distribution

The probability of finding exactly $k$ points within the area $V$ with event density $\rho$ is therefore:

$P(k,\rho,V) = \frac{(V\rho)^k e^{-(V\rho) }}{k!} . \,\!$

The first moment of which, the average number of points in the area, is simply $\rho V$. This value is intuitive as it is the Poisson rate parameter.

The probability of locating the $N^{\mathrm{th}}$ neighbor of any given point, at some radial distance $r$ is:

$P_N(r) = \frac{D}{(N-1)!} {\lambda}^N r^{DN-1} e^{- \lambda r^D} ,$

where $D$ is the number of dimensions, and $\Gamma$ is the gamma function, which when its argument is integral, is simply the factorial function. $\lambda$ is a density dependent parameter given by:

$\lambda = \frac{\rho \pi ^{\frac{D}{2}}}{\Gamma (\frac{D}{2} +1)} .$

The expected value of $P_N(r)$ 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 $D$ dimensions.

Bibliography

• Improvement of Inter-event Distance Tests of Randomness in Spatial Point Processes
• Diggle, P. J. (2003) "Statistical Analysis of Spatial Point Patterns", 2nd ed., Academic Press, New York.