| Iest {spatstat.explore} | R Documentation |
Estimate the I-function
Description
Estimates the summary function I(r) for a multitype point pattern.
Usage
Iest(X, ..., eps=NULL, r=NULL, breaks=NULL, correction=NULL)
Arguments
X |
The observed point pattern,
from which an estimate of |
... |
Ignored. |
eps |
the resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default. |
r |
Optional. Numeric vector of values for the argument |
breaks |
This argument is for internal use only. |
correction |
Optional. Vector of character strings specifying the edge correction(s)
to be used by |
Details
The I function
summarises the dependence between types in a multitype point process
(Van Lieshout and Baddeley, 1999)
It is based on the concept of the J function for an
unmarked point process (Van Lieshout and Baddeley, 1996).
See Jest for information about the J function.
The I function is defined as
%
I(r) = \sum_{i=1}^m p_i J_{ii}(r) %
- J_{\bullet\bullet}(r)
where J_{\bullet\bullet} is the J function for
the entire point process ignoring the marks, while
J_{ii} is the J function for the
process consisting of points of type i only,
and p_i is the proportion of points which are of type i.
The I function is designed to measure dependence between
points of different types, even if the points are
not Poisson. Let X be a stationary multitype point process,
and write X_i for the process of points of type i.
If the processes X_i are independent of each other,
then the I-function is identically equal to 0.
Deviations I(r) < 1 or I(r) > 1
typically indicate negative and positive association, respectively,
between types.
See Van Lieshout and Baddeley (1999)
for further information.
An estimate of I derived from a multitype spatial point pattern dataset
can be used in exploratory data analysis and formal inference
about the pattern. The estimate of I(r) is compared against the
constant function 0.
Deviations I(r) < 1 or I(r) > 1
may suggest negative and positive association, respectively.
This algorithm estimates the I-function
from the multitype point pattern X.
It assumes that X can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial marked point process in the plane, observed through
a bounded window.
The argument X is interpreted as a point pattern object
(of class "ppp", see ppp.object) and can
be supplied in any of the formats recognised by
as.ppp(). It must be a multitype point pattern
(it must have a marks vector which is a factor).
The function Jest is called to
compute estimates of the J functions in the formula above.
In fact three different estimates are computed
using different edge corrections. See Jest for
information.
Value
An object of class "fv", see fv.object,
which can be plotted directly using plot.fv.
Essentially a data frame containing
r |
the vector of values of the argument |
rs |
the “reduced sample” or “border correction”
estimator of |
km |
the spatial Kaplan-Meier estimator of |
han |
the Hanisch-style estimator of |
un |
the uncorrected estimate of |
theo |
the theoretical value of |
Note
Sizeable amounts of memory may be needed during the calculation.
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner rolfturner@posteo.net
References
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344–361.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Examples
Ic <- Iest(amacrine)
plot(Ic, main="Amacrine Cells data")
# values are below I= 0, suggesting negative association
# between 'on' and 'off' cells.