Complete spatial randomness

Distribution

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

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

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

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

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

where ${\displaystyle D}$ is the number of dimensions, ${\displaystyle \lambda }$ is a density-dependent parameter given by ${\displaystyle \lambda =\frac{\rho {\pi }^{\frac{D}{2}}}{\mathrm{\Gamma }(\frac{D}{2}+1)}}$ and ${\displaystyle \mathrm{\Gamma }}$ is the gamma function, which when its argument is integer, is simply the factorial function - i.e. ${\displaystyle \mathrm{\Gamma }(n+1)=n!}$ for integral ${\displaystyle n}$.

The expected value of ${\displaystyle P_N(r)}$ 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 ${\displaystyle D}$ dimensions.

Applications

The study of 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. CSR is often the standard against which data sets are tested. Roughly described one approach to test the CSR hypothesis is the following:

1. Use statistics that are a function of the distance from every event to the next nearest event.
2. Firstly focus on a specific event and formulate a method for testing whether the event and the next nearest event are significantly close (or distant).
3. Next consider all events and formulate a method for testing whether the average distance from every event to the next nearest event is significantly short (or long).

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.