The Hierarchical Strauss Hard Core Point Process Model
Creates an instance of the hierarchical Strauss-hard core point process model which can then be fitted to point pattern data.
HierStraussHard(iradii, hradii=NULL, types=NULL, archy=NULL)
iradii |
Matrix of interaction radii |
hradii |
Optional matrix of hard core distances |
types |
Optional; vector of all possible types (i.e. the possible levels
of the |
archy |
Optional: the hierarchical order. See Details. |
This is a hierarchical point process model for a multitype point pattern (Hogmander and Sarkka, 1999; Grabarnik and Sarkka, 2009). It is appropriate for analysing multitype point pattern data in which the types are ordered so that the points of type j depend on the points of type 1,2,...,j-1.
The hierarchical version of the (stationary) Strauss hard core process with m types, with interaction radii r[i,j], hard core distances h[i,j] and parameters beta[j] and gamma[i,j] is a point process in which each point of type j contributes a factor beta[j] to the probability density of the point pattern, and a pair of points of types i and j closer than r[i,j] units apart contributes a factor gamma[i,j] to the density provided i <= j. If any pair of points of types i and j lies closer than h[i,j] units apart, the configuration of points is impossible (probability density zero).
The nonstationary hierarchical Strauss hard core process is similar except that the contribution of each individual point x[i] is a function beta(x[i]) of location and type, rather than a constant beta.
The function ppm(),
which fits point process models to
point pattern data, requires an argument
of class "interact" describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the hierarchical
Strauss hard core process pairwise interaction is
yielded by the function HierStraussHard(). See the examples below.
The argument types need not be specified in normal use.
It will be determined automatically from the point pattern data set
to which the HierStraussHard interaction is applied,
when the user calls ppm.
However, the user should be confident that
the ordering of types in the dataset corresponds to the ordering of
rows and columns in the matrix radii.
The argument archy can be used to specify a hierarchical
ordering of the types. It can be either a vector of integers
or a character vector matching the possible types.
The default is the sequence
1,2, ..., m meaning that type j
depends on types 1,2, ..., j-1.
The matrices iradii and hradii must be square, with entries
which are either positive numbers or zero or NA.
A value of zero or NA indicates that no interaction term
should be included for this combination of types.
Note that only the interaction radii and hard core distances are
specified in HierStraussHard. The canonical
parameters log(beta[j]) and
log(gamma[i,j]) are estimated by
ppm(), not fixed in HierStraussHard().
An object of class "interact"
describing the interpoint interaction
structure of the hierarchical Strauss-hard core process with
interaction radii iradii[i,j] and hard core distances
hradii[i,j].
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
, Rolf Turner r.turner@auckland.ac.nz
and Ege Rubak rubak@math.aau.dk.
Grabarnik, P. and Sarkka, A. (2009) Modelling the spatial structure of forest stands by multivariate point processes with hierarchical interactions. Ecological Modelling 220, 1232–1240.
Hogmander, H. and Sarkka, A. (1999) Multitype spatial point patterns with hierarchical interactions. Biometrics 55, 1051–1058.
MultiStraussHard for the corresponding
symmetrical interaction.
r <- matrix(c(30, NA, 40, 30), nrow=2,ncol=2) h <- matrix(c(4, NA, 10, 15), 2, 2) HierStraussHard(r, h) # prints a sensible description of itself ppm(ants ~1, HierStraussHard(r, h)) # fit the stationary hierarchical Strauss-hard core process to ants data
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