# Iest

##### Estimate the I-function

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

- Keywords
- spatial, nonparametric

##### Usage

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

##### Arguments

- 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()`

. - 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
- An alternative to the argument
`r`

. Not normally invoked by the user. See Details section.

##### 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 $r$ 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 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 $0$

##### Note

Sizeable amounts of memory may be needed during the calculation.

##### 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

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

*Documentation reproduced from package spatstat, version 1.11-7, License: GPL version 2 or newer*