30810_Spatial_point

A point process is a stochastic process which generates a countable set of events within a bounded region (Bivand et al. 2008, Diggle 2013). The locations of the events in a study region generated by a point process are called spatial point pattern. The points may have extra information called marks attached to them, which represent a categorical or continuous attribute of the point. The spatial locations and the values attached to them are treated as response variables. In addition point pattern may also include covariates that are treated as explanatory variables, for example a spatial function $Z(s)$ defined at all spatial locations $s$, such as altitude, soil pH, etc.

A spatial process model is written as

$$\{\mathbf Z(\mathbf s): \mathbf s \in \mathbf D\}$$

where $D$ is a spatial point process. Although, in general, $D$ is contained in $d$-dimensional Euclidean space $\mathbb R^d$, most applications consider $d = 2$ or $d = 3$ (Cressie 1993).

The Poisson process¶

Poisson point process or a Poisson process or a Poisson point field is a type of random object that consists of randomly positioned points located on some underlying mathematical space (Keeler 2016).

The Poisson point process is related to the Poisson distribution, which implies that the probability of a Poisson random variable $N$ equal to $n$ is:

$$P\{N=n\} = \frac{\lambda^n}{n!}e^{-\lambda}$$

where $n!$ denotes $n$ factorial and $\lambda$ is the single Poisson parameter that is used to define the Poisson distribution.

The Poisson point process is sometimes called a purely or completely random process. This process has the property that the number of events $N(A)$ in a bounded region $A \in \mathbb R^d$ are independently and uniformly distributed over $A$. This means that the location of one point does not affect the probabilities of other points appearing nearby and that there are no regions where events are more likely to appear.

Homogeneous and inhomogeneous Poisson point processes¶

If a Poisson point process has a constant parameter, say, $\lambda$, then it is called a homogeneous (or stationary) Poisson process (HPP). The parameter $\lambda$, called intensity, is related to the expected (or average) number of Poisson points existing in some bounded region. The parameter $\lambda$ can be interpreted as the average number of points per some unit of length, area or volume, depending on the underlying mathematical space, hence it is sometimes called the mean density (Keeler 2016).

The HPP is stationary and spatial point patterns and processes isotropic. It is stationary because the intensity is constant and in addition, it is isotropic because the intensity is invariant to rotation of $\mathbb R^d$ (Baddeley 2007).

A generalization of HPP which allows for non-constant intensity $\lambda$ is called an inhomogeneous Poisson process (IPP). Both HPP and IPP assume that the events occur independently and are distributed according to a given intensity, $\lambda$. The main difference is that the HPP assumes that the intensity function is constant $(\lambda = \text{const.})$, while the intensity of an IPP varies spatially $(\lambda = Z(u))$.

The spatial Poisson process¶

In the plane $(\mathbb R^2)$, the Poisson point process, is referred to as spatial Poisson process. In a bounded region $A$ on a plane $(\mathbb R^2)$, with $N(A)$ being the (random) number of $N$ points existing in the region $A \subset \mathbb R^2$, a homogeneous Poisson process with parameter $\lambda > 0$ describes the probability of $n$ points existing in $A$ by:

$$P\{N(A)=n\}= \frac{\lambda(\vert A\vert)^n}{n!}e^{-\lambda \vert A \vert}$$

where $\vert A \vert$ denotes the area of $A$ (Keeler 2016).

Other point process models¶

Please note that the Poisson point process model does not adequately describe phenomena such as spatial clustering of points and spatial regularity. Hence, there exist a variety of of other point process models to capture these different data generation processes, such as Cox process, the Neyman-Scott process, Markov point process (Strauss process and Gibbs process), among others (see e.g. Baddeley 2007 , Cressie 1993, Daley and Vere-Jones 2003, Daley and Vere-Jones 2008, Gelfand at al. 2010)