# Spatial Poisson process

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In statistics and probability theory the spatial Poisson process (SPP) is a multidimensional generalization of the Poisson process, which can be described as a counting process where the number of points (events) in disjoint intervals are independent and have a Poisson distribution. Similarly, one can think of "points" being scattered over a $d$ -dimensional space in some random manner and of the spatial Poisson process as counting the number of points in a given set. It is also common to speak of a Poisson point process instead of a SPP.

## Definition

More generally it is possible to consider a state space $S\not \subset \mathbb {R} ^{d}$ in which the points of a Poisson process sit. Though, it is naturally assumed that $S$ is a measurable space and that its measurable sets form a $\sigma$ -field. It is also possible to define the SPP with a general measure $\mu$ instead of using the Lebesgue measure. In that case $|\Pi \cap A|$ is replaced by $\mu (\Pi \cap A)$ . 

It is common to distinguish between the homogeneous and inhomogeneous case:

### Inhomogeneous spatial Poisson process

Roughly speaking, the inhomogeneous case differs from the homogeneous case by the intensity $\lambda$ , which is not constant anymore. As indicated above, it is useful to have a definition of a Poisson process with other measures than Lebesgue measure. In order to get another measure $\Lambda (A)$ than $|A|$ the Euclidean element $\lambda d{\textbf {x}}$ is replaced by the element $\lambda ({\textbf {x}})d{\textbf {x}}$ . As a consequence the definition follows $\Lambda (A):=\int _{A}\lambda ({\textbf {x}})\,d{\textbf {x}},\quad A\in {\mathcal {B}}^{d}.$ ## Examples

Besides the application of the Poisson process in one dimension, there are many examples in two and higher dimensions. Modeling with a spatial Poisson process can be done in the following situations:

• The distribution of stars in a galaxy or of galaxies in the universe,
• Positions of animals in their habitat,
• The locations of active sites in a chemical reaction or of the weeds in your lawn,
• Defects on a surface or in a volume in reliability engineering.
• Positionally resolved photo-electron events on a photo-cathode focal plane array.Template:Cn

Even when a Poisson process is not a perfect description of such a system, it can provide a relatively simple yardstick against which to measure the improvements which may be offered by more sophisticated but often less tractable models.

## Mathematical properties

Many properties known from the Poisson Process hold also true in the multidimensional process. The Poisson point process is also characterized by the single parameter $\lambda$ . It is a simple, stationary point process with mean measure $\lambda$ . 

### Equivalent formulation

It can be shown, that because of the two essential conditions the distribution of the spatial Poisson process is given by

$\mathbb {P} (N(A_{i})=k_{i},1\leq i\leq n)={\dfrac {(\lambda A_{1})^{k_{1}}}{k_{1}!}}\cdot e^{-\lambda |A_{1}|}\cdots {\dfrac {(\lambda A_{n})^{k_{n}}}{k_{n}!}}\cdot e^{-\lambda |A_{n}|},$ ### Derivation from physically postulates

Using the law of rare events the Poisson process can be concluded by certain physically plausible postulates. Let $N(A)$ be a random point process fulfilling these postulates, then $N(A)$ is a homogeneous Poisson Point Process with intensity $\lambda$ derived from the postulates and the distribution is given as above in the Equivalent Formulation. Namely the four postulates are:

While postulate 1 excludes extreme or trivial cases, the second one asserts that the probability distribution of $N(A)$ does depend only on the size of $A$ , not on the shape or location. Thirdly it is postulated, that disjoint regions are independent regarding the outcome of the process. Finally, postulate 4 requires that there cannot be tow points occupying the same location.

### Distribution of n points in a given set

We are interested in the distribution of a point from which is supposed to be contained in a region $A$ with positive size $|A|>0$ . In other words: $N(A)=1$ . The question where the point can be found in $A$ is answered by a uniform distribution:Template:Disambiguation needed

$\mathbb {P} (N(B)=1\mid N(A)=1)={\dfrac {|B|}{|A|}}$ for any set $B\subset A$ $\mathbb {P} (N(A_{1})=k_{1},\ldots ,N(A_{m})=k_{m}\mid N(A)=n)={\dfrac {n!}{k_{1}!\cdots k_{m}!}}\left({\dfrac {|A_{1}|}{|A|}}\right)^{k_{1}}\cdots \left({\dfrac {|A_{m}|}{|A|}}\right)^{k_{m}}.$ Thus, the conditional distribution follows a multinomial distribution.

## Generalization

The Spatial Poisson Process is a very common example of a Point process.