R: Multitype J Function (i-to-j)

Jcross {spatstat} | R Documentation

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

### Description

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

### 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 Jij(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 Jij(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

Jij(r) = (1 - Gij(r))/(1-Fj(r))

where Gij(r) is the distribution function of the distance from a type i point to the nearest point of type j, and Fj(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 Jij(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 Jij(r) = 1. Hence deviations of the empirical estimate of Jij from the value 1 may suggest dependence between types.

This algorithm estimates Jij(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 Jij(r) should be evaluated. The values of r must be increasing nonnegative numbers and the maximum r value must 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

`J` | the recommended estimator of Jij(r), currently the Kaplan-Meier estimator.
`r` | the values of the argument r at which the function Jij(r) has been estimated
`km` | the Kaplan-Meier estimator of Jij(r)
`rs` | the “reduced sample” or “border correction” estimator of Jij(r)
`han` | the Hanisch-style estimator of Jij(r)
`un` | the “uncorrected” estimator of Jij(r) formed by taking the ratio of uncorrected empirical estimators of 1 - Gij(r) and 1 - Fj(r), see `Gdot` and `Fest`.
`theo` | the theoretical value of Jij(r) for a marked Poisson process, namely 1.

The result also has two attributes `"G"` and `"F"` which are respectively the outputs of `Gcross` and `Fest` for the point pattern.

### 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.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### 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

`Jdot`, `Jest`, `Jmulti`

### Examples

         # Lansing woods data: 6 types of trees
        woods <- lansing

        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)

[Package spatstat version 1.54-0 Index]