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. It is synonymous with a homogeneous spatial Poisson process. Such a process is modeled using only one parameter \rho , i.e. the density of points within the defined area. 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.

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. Such a process is modeled using only one parameter \rho , i.e. the density of points within the defined area. 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. 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, \lambda is a density-dependent parameter given by \lambda = \frac{\rho \pi ^{\frac{D}{2}}}{\Gamma (\frac{D}{2} +1)} and \Gamma is the gamma function, which when its argument is integer, is simply the factorial function - i.e. \Gamma(n+1) = n! for integral n . 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.

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