Iest {spatstat.explore}R Documentation

Estimate the I-function

Description

Estimates the summary function I(r)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 I(r)I(r) will be computed. An object of class "ppp", or data in any format acceptable to as.ppp().

...

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 rr at which I(r)I(r) should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

correction

Optional. Vector of character strings specifying the edge correction(s) to be used by Jest.

Details

The II function summarises the dependence between types in a multitype point process (Van Lieshout and Baddeley, 1999) It is based on the concept of the JJ function for an unmarked point process (Van Lieshout and Baddeley, 1996). See Jest for information about the JJ function.

The II function is defined as

I(r)=i=1mpiJii(r)J(r) % I(r) = \sum_{i=1}^m p_i J_{ii}(r) % - J_{\bullet\bullet}(r)

where JJ_{\bullet\bullet} is the JJ function for the entire point process ignoring the marks, while JiiJ_{ii} is the JJ function for the process consisting of points of type ii only, and pip_i is the proportion of points which are of type ii.

The II function is designed to measure dependence between points of different types, even if the points are not Poisson. Let XX be a stationary multitype point process, and write XiX_i for the process of points of type ii. If the processes XiX_i are independent of each other, then the II-function is identically equal to 00. Deviations I(r)<1I(r) < 1 or I(r)>1I(r) > 1 typically indicate negative and positive association, respectively, between types. See Van Lieshout and Baddeley (1999) for further information.

An estimate of II 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)I(r) is compared against the constant function 00. Deviations I(r)<1I(r) < 1 or I(r)>1I(r) > 1 may suggest negative and positive association, respectively.

This algorithm estimates the II-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 JJ 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 rr at which the function II has been estimated

rs

the “reduced sample” or “border correction” estimator of I(r)I(r) computed from the border-corrected estimates of JJ functions

km

the spatial Kaplan-Meier estimator of I(r)I(r) computed from the Kaplan-Meier estimates of JJ functions

han

the Hanisch-style estimator of I(r)I(r) computed from the Hanisch-style estimates of JJ functions

un

the uncorrected estimate of I(r)I(r) computed from the uncorrected estimates of JJ

theo

the theoretical value of I(r)I(r) for a stationary Poisson process: identically equal to 00

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

Jest

Examples

   Ic <- Iest(amacrine)
   plot(Ic, main="Amacrine Cells data")
   # values are below I= 0, suggesting negative association
   # between 'on' and 'off' cells.

[Package spatstat.explore version 3.3-1 Index]