Spatial Poisson process
Section of Poisson processes, part of Stochastic processes
Spatial Poisson process
| Type | Section |
| Taught in | Poisson processes |
| Part | Stochastic processes |
A homogeneous spatial Poisson process of intensity \lambda assigns to each bounded region A a \mathrm{Poisson}(\lambda|A|) count, and assigns independent counts to disjoint regions. Here \lambda is in points per unit area.
Let R be the distance from a fixed origin to the nearest point, and let B_r be the disk of radius r. The event \{R>r\} is the event that B_r is empty:
\mathbb{P}(R>r)=\mathbb{P}\{N(B_r)=0\}=e^{-\lambda\pi r^2}, \qquad r\ge 0,
so the distribution function is F_R(r)=1-e^{-\lambda\pi r^2}. For a varying spatial intensity \lambda(x), replace \lambda|A| by \int_A\lambda(x)\,dx. The void probability is still the exponential of minus that integral.
The same marking idea used in time works in space. If points of intensity \lambda are kept independently with probability r(x), the kept intensity is \lambda r(x), and the mean count in A is \int_A\lambda r(x)\,dx. That is a model for a detection probability that varies across a survey region. Clustering in the original pattern is a different departure from the Poisson process, and thinning does not remove it.
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See also
Homogeneous Poisson process · Forward equations of the Poisson process · Nonhomogeneous Poisson process · Superposition and thinning · Intensity function · Interarrival time