# Jcross

##### Multitype J Function (i-to-j)

For a multitype point pattern, estimate the multitype \(J\) function summarising the interpoint dependence between points of type \(i\) and of type \(j\).

- Keywords
- spatial, nonparametric

##### Usage

`Jcross(X, i, j, eps=NULL, r=NULL, breaks=NULL, …, correction=NULL)`

##### Arguments

- X
The observed point pattern, from which an estimate of the multitype \(J\) function \(J_{ij}(r)\) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.

- i
The type (mark value) of the points in

`X`

from which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of`marks(X)`

.- j
The type (mark value) of the points in

`X`

to which distances are measured. A character string (or something that will be converted to a character string). Defaults to the second level of`marks(X)`

.- eps
A positive number. The resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.

- r
Optional. Numeric vector. The values of the argument \(r\) at which the function \(J_{ij}(r)\) 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.

- …
Ignored.

- correction
Optional. Character string specifying the edge correction(s) to be used. Options are

`"none"`

,`"rs"`

,`"km"`

,`"Hanisch"`

and`"best"`

. Alternatively`correction="all"`

selects all options.

##### Details

This function `Jcross`

and its companions
`Jdot`

and `Jmulti`

are generalisations of the function `Jest`

to multitype point patterns.

A multitype point pattern is a spatial pattern of points classified into a finite number of possible ``colours'' or ``types''. In the spatstat package, a multitype pattern is represented as a single point pattern object in which the points carry marks, and the mark value attached to each point determines the type of that point.

The argument `X`

must be a point pattern (object of class
`"ppp"`

) or any data that are acceptable to `as.ppp`

.
It must be a marked point pattern, and the mark vector
`X$marks`

must be a factor.
The argument `i`

will be interpreted as a
level of the factor `X$marks`

. (Warning: this means that
an integer value `i=3`

will be interpreted as the number 3,
**not** the 3rd smallest level).

The ``type \(i\) to type \(j\)'' multitype \(J\) function of a stationary multitype point process \(X\) was introduced by Van lieshout and Baddeley (1999). It is defined by $$J_{ij}(r) = \frac{1 - G_{ij}(r)}{1 - F_{j}(r)}$$ where \(G_{ij}(r)\) is the distribution function of the distance from a type \(i\) point to the nearest point of type \(j\), and \(F_{j}(r)\) is the distribution function of the distance from a fixed point in space to the nearest point of type \(j\) in the pattern.

An estimate of \(J_{ij}(r)\) is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the subprocess of type \(i\) points is independent of the subprocess of points of type \(j\), then \(J_{ij}(r) \equiv 1\). Hence deviations of the empirical estimate of \(J_{ij}\) from the value 1 may suggest dependence between types.

This algorithm estimates \(J_{ij}(r)\)
from the point pattern `X`

. It assumes that `X`

can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in `X`

as `Window(X)`

)
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in `Jest`

,
using the Kaplan-Meier and border corrections.
The main work is done by `Gmulti`

and `Fest`

.

The argument `r`

is the vector of values for the
distance \(r\) at which \(J_{ij}(r)\) should be evaluated.
The values of \(r\) must be increasing nonnegative numbers
and the maximum \(r\) value must not exceed the radius of the
largest disc contained in the window.

##### Value

An object of class `"fv"`

(see `fv.object`

).

Essentially a data frame containing six numeric columns

the recommended estimator of \(J_{ij}(r)\), currently the Kaplan-Meier estimator.

the values of the argument \(r\) at which the function \(J_{ij}(r)\) has been estimated

the Kaplan-Meier estimator of \(J_{ij}(r)\)

the ``reduced sample'' or ``border correction'' estimator of \(J_{ij}(r)\)

the Hanisch-style estimator of \(J_{ij}(r)\)

the ``uncorrected''
estimator of \(J_{ij}(r)\)
formed by taking the ratio of uncorrected empirical estimators
of \(1 - G_{ij}(r)\)
and \(1 - F_{j}(r)\), see
`Gdot`

and `Fest`

.

the theoretical value of \(J_{ij}(r)\) for a marked Poisson process, namely 1.

##### Warnings

The arguments `i`

and `j`

are always interpreted as
levels of the factor `X$marks`

. They are converted to character
strings if they are not already character strings.
The value `i=1`

does **not**
refer to the first level of the factor.

##### 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

```
# NOT RUN {
# Lansing woods data: 6 types of trees
woods <- lansing
# }
# NOT RUN {
Jhm <- Jcross(woods, "hickory", "maple")
# diagnostic plot for independence between hickories and maples
plot(Jhm)
# synthetic example with two types "a" and "b"
pp <- runifpoint(30) %mark% factor(sample(c("a","b"), 30, replace=TRUE))
J <- Jcross(pp)
# }
```

*Documentation reproduced from package spatstat, version 1.63-3, License: GPL (>= 2)*