Definition and Basic Properties

 Before studying more advanced point process models, we begin with the mathematical definition and some simple properties of homogeneous Poisson processes which can be introduced in a similar way as for the Poisson process on the real line via the counting variables X(B).

Let $\nu$ denote the (2-dimensional) Lebesgue measure in $\RL^2$.Then, X is called a homogeneous Poisson process if there is a constant $\lambda \gt$ such that 1) the number of points X(B) is Poisson distributed with parameter $\lambda \nu(B)$, for each bounded Borel set B, and 2) the random variables $X(B_1),\ldots,X(B_n)$ are independent for each finite sequence $B_1,\ldots,B_n$ of disjoint bounded Borel sets. Note that property (2) is the complete spatial randomness mentioned above. Furthermore, the parameter $\lambda$ occurring in property (1) is the expected number of points per unit area, that is $\Exp X(B) = \lambda \nu(B)$for all bounded $B \in \calB$. Thus, $\lambda$ is called the intensity of X. Another (so-called local) characterization of $\lambda$ is connected with the fact that $\Prob (X(B)\gt) = \lambda \nu(B) + {\rm o}(\nu(B))$as $\nu(B) \to 0$. That is, for small sets B, the probability that there is at least one point in B is nearly proportional to $\lambda$.

The defining properties (1) and (2) of a homogeneous Poisson process immediately imply the following conditional uniformity property: under the condition that in the test set B with $0 < \nu(B) < \infty$ there is a fixed number n of points, the locations of these points are independent and uniformly distributed in B, i.e., for each sequence $x_1,\ldots,x_n \in B$ of pairwise distinct points  

$$\Prob (X(dx_1) \gt 0, \ldots, X(dx_n)\gt \, \vert \, X(B)=n) \, = \, \frac{dx_1 \cdots dx_n}{(\nu(B))^n}.$$ (1)

Assume as before that X is a homogeneous Poisson process with intensity $\lambda$. Then X is stationary, i.e., the translated point processes $X_x = \{ X_n - x\}$ (or equivalently the random vectors $(X(B_1 + x) , \ldots, X(B_n+x))$) have the same distribution for all $x \in \RL^2$ and for all finite sequences $B_1,\ldots,B_n$ of bounded Borel sets. Furthermore, X is isotropic, i.e., the rotated point processes $\varphi (X) = \{ \varphi (X_n)\}$have the same distribution for all rotations $\varphi$ about the origin. Another interesting property of homogeneous Poisson processes is connected with compression/stretching of point patterns: for each r>0, the scaled point process $rX=\{rX_n\}$ is again a homogeneous Poisson process where the intensity of rX equals $\lambda r^{-2}$. In particular, if B is a compact Borel set containing the origin o and is star-shaped with respect to the origin (i.e., $rB \subseteq B$ for all $r \in [0,1]$), then the contact distribution function $H_B(r) = \Prob (X(rB) \gt 0)$ with respect to the 'structuring set' B is given by $H_B(r) = 1 - \exp(-\lambda r^2 \nu(B))$.In the special case where B is the unit circle $B = \{ x \, : \, \vert x\vert \leq 1\}$, $H_B(r) \equiv H(r)$ is called the spherical contact distribution function. Then H_B(r) is the distribution function of the distance from an arbitrary fixed sampling point, say the origin, to the nearest point of X.

Another closely related characteristic of a stationary isotropic point process $X=\{X_n\}$ is the nearest neighbor distance distribution function D(r) which in many cases of practical interest can be given by  

$$D(r) = 1 - \lim_{\varepsilon \downarrow 0} \Prob \left(X(b(... ... b(o,\varepsilon)) =0\, \vert \, X(b(o,\varepsilon)) =1 \right)$$ (2)

where $b(o,r) = \{x : \vert x\vert \leq r\}$ denotes the circle with radius r and center at the origin. Note that (2.2) suggests to interpret D(r) as the conditional contact distribution function under the condition that the sampling point is a point of the point process. Another possibility is the interpretation of D(r) as the distribution function of the distance from the `typical point' of the point process to its nearest neighbor. Roughly speaking, the typical point of the point process is a point which is randomly chosen among all points. The mathematical definition of this notion is always connected with a given property of the typical point. In the case of D(r), this is the property that the distance to its nearest neighbor is not larger than r. We return to this question in Section 3. If $X=\{X_n\}$ is a homogeneous Poisson process, then

 

H(r) = D(r)

for all $r \geq 0$. This coincidence property plays an important role in the statistical analysis of Poisson processes, see Section 2.2.

For a rectangular area $A = [0,s] \times [0,t]$ and a fixed scanning set C, the scan statistic L is defined as $L= \max_{x \in \RL^2} X(C(x) \cap A)$, where C(x) is the translate of C by $x \in \RL^2$, i.e., the scan statistic L is the largest count of points of A which can be seen as the window C is moved around. In Alm (1997), the distribution of L is accurately approximated for rectangular scanning sets.

Note that the class of homogeneous Poisson processes is closed with respect to various other operations applied independently to the individual points. For example, if the points of a homogeneous Poisson process with intensity $\lambda$ are shifted by independent and indentically distributed random vectors, the translated point process remains Poisson with the same intensity $\lambda$. Similarly, if each point, independently of the other points, is deleted with a fixed probability p, the thinned point process is Poisson with intensity $\lambda (1-p)$. Finally, the superposition of two independent homogeneous Poisson processes with intensities $\lambda_1$ and $\lambda_2$respectively, is a homogeneous Poisson process with intensity $\lambda_1 + \lambda_2$.

The operations of translating, thinning and superposition as described above can also be modeled by so-called independent marking. Besides the homogeneous Poisson process $X=\{X_1,X_2,\ldots\}$ with intensity $\lambda$, consider a sequence $M= \{M_1,M_2,\ldots\}$of i.i.d. random variables, independent of X. Then, the sequence $(X,M) = \{(X_1,M_1), (X_2,M_2),\ldots\}$is called an independently marked Poisson process. If the M_n are random vectors with values in $\RL^2$, then the translated point process $\{X_1+M_1,X_2+M_2,\ldots\}$ is Poisson with the same intensity $\lambda$. If $\Prob (M_n=0) = 1 - \Prob (M_n=1) = p$, then the subsequence of those points X_n with M_n = 1 is the (thinned) Poisson process of surviving (nondeleted) points which has intensity $\lambda (1-p)$.If $\lambda = \lambda_1 + \lambda_2$, $\Prob (M_n = 1) = \frac{\lambda_1}{\lambda_1 + \lambda_2}$ and $\Prob (M_n = 2) = \frac{\lambda_2}{\lambda_1 + \lambda_2}$, then the subsequence X⁽ⁱ⁾ of those points X_n with M_n=i is a Poisson process with intensity $\lambda_i$; i=1,2. Furthermore, X can be seen as the superposition of X⁽¹⁾ and X⁽²⁾.