Point pattern analysis
Point pattern analysis (PPA) is the study of the spatial arrangements of points in (usually 2-dimensional) space. The simplest formulation is a set X = {x in D} where D, which can be called the 'study region,' is a subset of Rn, an coordinate system.
Description
The easiest way to visualize a 2-D point pattern is a map of the locations, which is simply a scatterplot but with the provision that the axes are equally scaled. If D is not the boundary of the map then it should also be indicated.
Modeling
The null model for point patterns is complete spatial randomness (CSR), modeled as a Poisson process in Rn, which implies that the number of points in any arbitrary region A in D will be proportional to the area of A. Exploring models is generally iterative: if CSR is accepted not much more can be said, but if rejected, there two avenues. First, one must decide which models are worth exploring, such as investigations of clustering, density, trends, etc. And for each of these models there are appropriate scale ranges, from the finest, which essentially mirrors the point pattern, to the coarsest, which aggregates D. It is generally interesting to explore a range of scales within these limits.
Estimation
A fundamental problem of PPA is inferring whether a given arrangement is merely random or the result of some process. The picture illustrates patterns of 256 points using four point processes. The clustered process results in all points having the same location. Popular models are those based on simple circles and ellipses, inter-point (and especially nearest neighbor) distances, quadrats, and intensity functions. Each model yields estimates (that can increase insights into the underlying real-world processes) as well as associated goodness-of-fit diagnostics.
References
Cressie, N. A. C. and C. K. Wikle (2011) Statistics for spatio-temporal data. Hoboken, N.J., Wiley. ISBN 9780471692744