Analysing spatial point patterns in R - CSIRO

204 Multitype Poisson models
33 Multitype Poisson models
This section covers multitype Poisson process models: basic properties, simulation, and fitting
models to data.
33.1 Theory
33.1.1 Complete spatial randomness and independence
A uniform Poisson marked point process in R 2 with marks in M can be defined in the following
equivalent ways.
randomly marked Poisson process (Poisson [X], iid [M|X]): a Poisson point process of
locations X with intensity β is first generated. Then each point x i is labelled with a
random mark m i , independently of other points, with distribution P {M i = m} = p m for
m ∈ M.
superposition of independent Poisson processes (iid [M], Poisson [X|M]): for each possible
mark m ∈ M, a Poisson process X m is generated, with intensity β m . The points of X m
are tagged with the mark m. Then the processes X m with different marks m ∈ M are
superimposed, to yield a marked point process.
Poisson marked point process (jointly Poisson [X,M]): a Poisson process on R 2 × M is
generated, with intensity function λ(u,m) = β m at location u and mark m.
These constructions are equivalent when β m = p m β. See the lovely book by Kingman [45].
Since the established term CSR (‘complete spatial randomness’) is used to refer to the
uniform Poisson point process, I propose that the uniform marked Poisson point process should
be called ‘complete spatial randomness and independence’ (CSRI).
33.1.2 Inhomogeneous Poisson marked point processes
A inhomogeneous Poisson marked point process Y with ‘joint’ intensity λ(u,m) for locations u
and mark values m is simply defined as an inhomogeneous Poisson point process on R 2 × M
with intensity function λ(u,m).
Let’s restrict attention to the case of categorical marks, where M is finite. Then the process
Y has the following properties:
The locations X, obtained by removing the marks, constitute an inhomogeneous Poisson
process in R 2 with intensity function
β(u) = ∑ m
λ(u,m).
Conditional on the locations X, the marks attached to the points are independent. For a
point x i the conditional distribution of the mark m i is P{M i = m} = λ(x i ,m)/β(x i ).
The sub-process X m of points with mark m, is an inhomogeneous Poisson point process
with intensity β m (u) = λ(u,m).
The sub-processes X m of points with different marks m are independent processes.
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