markcrosscorr: 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)

Value

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

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.

Author

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net and Ege Rubak rubak@math.aau.dk.

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 $i$th and $j$th columns of marks respectively. On the denominator, $M_i$ and $M_j$ are independent random values drawn from the $i$th and $j$th 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.

Examples

Run this code

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

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