Point pattern analysis
Encyclopedia
Point pattern analysis is the study of the spatial arrangements of points in (usually 2-dimensional) space. The simplest formulation is a set X = {x ∈ D} where D, which can be called the 'study region,' is a subset of Rn, a n-dimensional Euclidean space
.
(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 or volume 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.
A particularly robust model of clustered point patterns is diffusion
, which can also be thought of as the trajectory of a point doing a random walk
.
s and ellipse
s, 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.
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
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. An empirical definition of D would be the convex hull of the points, or at least their bounding box, a matrix of the ranges of the coordinates. Another straightforward way to visualize the points is a 2D histogram (sometimes called a quadrats) that bins the points into rectangular regions. A benefit of quadrat analysis is that it forces the analysis to take into account possible scales within which statistically significant inhomogeneities may be occurring.Modeling
The null model for point patterns is complete spatial randomnessComplete spatial randomness
Complete spatial randomness 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...
(CSR), modeled as a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
in Rn, which implies that the number of points in any arbitrary region A in D will be proportional to the area or volume 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.
A particularly robust model of clustered point patterns is diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
, which can also be thought of as the trajectory of a point doing a random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
.
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 circleCircle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s and ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
s, 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
Goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g...
diagnostics.
Applications
PPA has applications in a wide range of areas, including astronomy, geography, ecology, and epidemiology. A few topics in the last area are discussed here.- A case control study compares the point patterns of organisms both with and without some condition to determine if there were significant differences in their arrangements.
- Environmental exposure examines the locations of cases and possible sources (e.g. of pollution or carcinogens).
- ContagionContagious diseaseA contagious disease is a subset category of infectious diseases , which are easily transmitted by physical contact with the person suffering the disease, or by their secretions or objects touched by them....
explores the temporal unfolding of the pattern, asking about such phenomena as the location of the 'index case.' - Examination of infectionInfectious diseaseInfectious diseases, also known as communicable diseases, contagious diseases or transmissible diseases comprise clinically evident illness resulting from the infection, presence and growth of pathogenic biological agents in an individual host organism...
compares the arrangements of parasites and hosts (predators and prey, agents and organisms).