spatstat (version 1.42-2)

HierHard: The Hierarchical Hard Core Point Process Model

Description

Creates an instance of the hierarchical hard core point process model which can then be fitted to point pattern data.

Usage

HierHard(hradii=NULL, types=NULL, archy=NULL)

Arguments

hradii

Optional matrix of hard core distances

types

Optional; vector of all possible types (i.e. the possible levels of the marks variable in the data)

archy

Optional: the hierarchical order. See Details.

Value

• An object of class "interact" describing the interpoint interaction structure of the hierarchical hard core process with hard core distances $hradii[i,j]$.

Details

This is a hierarchical point process model for a multitype point pattern (latex{H{"o}gmander}{Hogmander} and latex{S{"a}rkk{"a}}{Sarkka}, 1999; Grabarnik and latex{S"{a}rkk"{a}}{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 $1,2,\ldots,j-1$. The hierarchical version of the (stationary) hard core process with $m$ types, with hard core distances $h_{ij}$ and parameters $\beta_j$, is a point process in which each point of type $j$ contributes a factor $\beta_j$ to the probability density of the point pattern. If any pair of points of types $i$ and $j$ lies closer than $h_{ij}$ units apart, the configuration of points is impossible (probability density zero). The nonstationary hierarchical 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 hard core process pairwise interaction is yielded by the function HierHard(). 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 HierHard 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 $1,2, \ldots, m$ meaning that type $j$ depends on types $1,2, \ldots, j-1$. The matrix iradii must be square, with entries which are either positive numbers, or zero or NA. A value of zero or NA indicates that no hard core interaction term should be included for this combination of types. Note that only the hard core distances are specified in HierHard. The canonical parameters $\log(\beta_j)$ are estimated by ppm(), not fixed in HierHard().

References

Grabarnik, P. and latex{S"{a}rkk"{a}}{Sarkka}, A. (2009) Modelling the spatial structure of forest stands by multivariate point processes with hierarchical interactions. Ecological Modelling 220, 1232--1240.

latex{H{"o}gmander}{Hogmander}, H. and latex{S{"a}rkk{"a}}{Sarkka}, A. (1999) Multitype spatial point patterns with hierarchical interactions. Biometrics 55, 1051--1058.

See Also

MultiHard for the corresponding symmetrical interaction.

HierStrauss, HierStraussHard.

Examples

h <- matrix(c(4, NA, 10, 15), 2, 2)
   HierHard(h)
   # prints a sensible description of itself
   ppm(ants ~1, HierHard(h))
   # fit the stationary hierarchical hard core process to ants data