markcrosscorr {spatstat.explore}

Mark Cross-Correlation Function

Description

Given a spatial point pattern with several columns of marks, this function computes the mark correlation function between each pair of columns of marks.

Usage

  markcrosscorr(X, r = NULL,
                correction = c("isotropic", "Ripley", "translate"),
                method = "density", ..., normalise = TRUE, Xname = NULL)

Arguments

X

The observed point pattern. An object of class "ppp" or something acceptable to as.ppp.

r

Optional. Numeric vector. The values of the argument r at which the mark correlation function k_f(r) should be evaluated. There is a sensible default.

correction

A character vector containing any selection of the options "isotropic", "Ripley", "translate", "translation", "none" or "best". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.

method

A character vector indicating the user's choice of density estimation technique to be used. Options are "density", "loess", "sm" and "smrep".

...

Arguments passed to the density estimation routine (density, loess or sm.density) selected by method.

normalise

If normalise=FALSE, compute only the numerator of the expression for the mark correlation.

Xname

Optional character string name for the dataset X.

Details

First, all columns of marks are converted to numerical values. A factor with m possible levels is converted to m columns of dummy (indicator) values.

Next, each pair of columns is considered, and the mark cross-correlation is defined as

k_{mm}(r) = \frac{E_{0u}[M_i(0) M_j(u)]}{E[M_i,M_j]}

where E_{0u} denotes the conditional expectation given that there are points of the process at the locations 0 and u separated by a distance r. On the numerator, M_i(0) and M_j(u) are the marks attached to locations 0 and u respectively in the ith and jth columns of marks respectively. On the denominator, M_i and M_j are independent random values drawn from the ith and jth columns of marks, respectively, and E is the usual expectation.

Note that k_{mm}(r) is not a “correlation” in the usual statistical sense. It can take any nonnegative real value. The value 1 suggests “lack of correlation”: if the marks attached to the points of X are independent and identically distributed, then k_{mm}(r) \equiv 1.

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.

The cross-correlations are estimated in the same manner as for markcorr.

Value

A function array (object of class "fasp") containing the mark cross-correlation functions for each possible pair of columns of marks.

Author(s)

Adrian Baddeley [e-mail], Rolf Turner [e-mail] and Ege Rubak [e-mail].

See Also

markcorr

Examples

  # The dataset 'betacells' has two columns of marks:
  #       'type' (factor)
  #       'area' (numeric)
  if(interactive()) plot(betacells)
  plot(markcrosscorr(betacells))