Estimates the summary function \(I(r)\) for a multitype point pattern.

`Iest(X, ..., eps=NULL, r=NULL, breaks=NULL, correction=NULL)`

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 \(r\) at which the function \(I\) has been estimated

- rs
the ``reduced sample'' or ``border correction'' estimator of \(I(r)\) computed from the border-corrected estimates of \(J\) functions

- km
the spatial Kaplan-Meier estimator of \(I(r)\) computed from the Kaplan-Meier estimates of \(J\) functions

- han
the Hanisch-style estimator of \(I(r)\) computed from the Hanisch-style estimates of \(J\) functions

- un
the uncorrected estimate of \(I(r)\) computed from the uncorrected estimates of \(J\)

- theo
the theoretical value of \(I(r)\) for a stationary Poisson process: identically equal to \(0\)

- X
The observed point pattern, from which an estimate of \(I(r)\) 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 \(r\) at which \(I(r)\) 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`

.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz

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.

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.

`Jest`

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

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