R: Marked J Function

Jmulti {spatstat} | R Documentation

## Marked J Function

### Description

For a marked point pattern, estimate the multitype J function summarising dependence between the points in subset I and those in subset J.

### Usage

      Jmulti(X, I, J, eps=NULL, r=NULL, breaks=NULL, ..., disjoint=NULL,
             correction=NULL)

### Arguments

`X` | The observed point pattern, from which an estimate of the multitype distance distribution function J[IJ](r) will be computed. It must be a marked point pattern. See under Details.
`I` | Subset of points of `X` from which distances are measured. See Details.
`J` | Subset of points in `X` to which distances are measured. See Details.
`eps` | A positive number. The pixel resolution of the discrete approximation to Euclidean distance (see `Jest`). There is a sensible default.
`r` | numeric vector. The values of the argument r at which the distribution 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.
`disjoint` | Optional flag indicating whether the subsets `I` and `J` are disjoint. If missing, this value will be computed by inspecting the vectors `I` and `J`.
`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

The function `Jmulti` generalises `Jest` (for unmarked point patterns) and `Jdot` and `Jcross` (for multitype point patterns) to arbitrary marked point patterns.

Suppose X[I], X[J] are subsets, possibly overlapping, of a marked point process. Define

J[IJ](r) = (1 - G[IJ](r))/(1 - F[J](r))

where F[J](r) is the cumulative distribution function of the distance from a fixed location to the nearest point of X[J], and GJ(r) is the distribution function of the distance from a typical point of X[I] to the nearest distinct point of X[J].

The argument `X` must be a point pattern (object of class `"ppp"`) or any data that are acceptable to `as.ppp`.

The arguments `I` and `J` specify two subsets of the point pattern. They may be any type of subset indices, for example, logical vectors of length equal to `npoints(X)`, or integer vectors with entries in the range 1 to `npoints(X)`, or negative integer vectors.

Alternatively, `I` and `J` may be functions that will be applied to the point pattern `X` to obtain index vectors. If `I` is a function, then evaluating `I(X)` should yield a valid subset index. This option is useful when generating simulation envelopes using `envelope`.

It is assumed 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`.

The argument `r` is the vector of values for the distance r at which J[IJ](r) should be evaluated. It is also used to determine the breakpoints (in the sense of `hist`) for the computation of histograms of distances. The reduced-sample and Kaplan-Meier estimators are computed from histogram counts. In the case of the Kaplan-Meier estimator this introduces a discretisation error which is controlled by the fineness of the breakpoints.

First-time users would be strongly advised not to specify `r`. However, if it is specified, `r` must satisfy `r[1] = 0`, and `max(r)` must be larger than the radius of the largest disc contained in the window. Furthermore, the successive entries of `r` must be finely spaced.

### Value

An object of class `"fv"` (see `fv.object`).

Essentially a data frame containing six numeric columns

`r` | the values of the argument r at which the function J[IJ](r) has been estimated
`rs` | the “reduced sample” or “border correction” estimator of J[IJ](r)
`km` | the spatial Kaplan-Meier estimator of J[IJ](r)
`han` | the Hanisch-style estimator of J[IJ](r)
`un` | the uncorrected estimate 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`.
`theo` | the theoretical value of J[IJ](r) for a marked Poisson process with the same estimated intensity, namely 1.

### 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. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.

### See Also

`Jcross`, `Jdot`, `Jest`

### Examples

        trees <- longleaf
         # Longleaf Pine data: marks represent diameter

        Jm <- Jmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
        plot(Jm)

[Package spatstat version 1.54-0 Index]