alltypes: Calculate Summary Statistic for All Types in a Multitype Point Pattern

Description

Given a marked point pattern, this computes the estimates of a selected summary function (\(F\),\(G\), \(J\), \(K\) etc) of the pattern, for all possible combinations of marks, and returns these functions in an array.

Usage

alltypes(X, fun="K", ...,
           dataname=NULL,verb=FALSE,envelope=FALSE,reuse=TRUE)

Arguments

Value

A function array (an object of class "fasp", see fasp.object). This can be plotted using plot.fasp.

If the pattern is not marked, the resulting ``array'' has dimensions

\(1 \times 1\). Otherwise the following is true:

If fun="F", the function array has dimensions \(m \times 1\)

where \(m\) is the number of different marks in the point pattern. The entry at position [i,1] in this array is the result of applying Fest to the points of type i only.

If fun is "Gdot", "Jdot", "Kdot"

or "Ldot", the function array again has dimensions \(m \times 1\). The entry at position [i,1] in this array is the result of Gdot(X, i), Jdot(X, i)

Kdot(X, i) or Ldot(X, i) respectively.

If fun is "Gcross", "Jcross", "Kcross"

or "Lcross"

(or their abbreviations "G", "J", "K" or "L"), the function array has dimensions \(m \times m\). The [i,j] entry of the function array (for \(i \neq j\)) is the result of applying the function Gcross,

Jcross, Kcross orLcross to the pair of types (i,j). The diagonal

[i,i] entry of the function array is the result of applying the univariate function Gest,

Jest, Kest or Lest to the points of type i only.

If envelope=FALSE, then each function entry fns[[i]] retains the format of the output of the relevant estimating routine

Fest, Gest, Jest,

Kest, Lest, Gcross,

Jcross ,Kcross, Lcross,

Gdot, Jdot, Kdot or

Ldot

The default formulae for plotting these functions are

cbind(km,theo) ~ r for F, G, and J functions, and

cbind(trans,theo) ~ r for K and L functions.

If envelope=TRUE, then each function entry fns[[i]]

has the same format as the output of the envelope command.

Details

This routine is a convenient way to analyse the dependence between types in a multitype point pattern. It computes the estimates of a selected summary function of the pattern, for all possible combinations of marks. It returns these functions in an array (an object of class "fasp") amenable to plotting by plot.fasp().

The argument fun specifies the summary function that will be evaluated for each type of point, or for each pair of types. It may be either an R function or a character string.

Suppose that the points have possible types \(1,2,\ldots,m\) and let \(X_i\) denote the pattern of points of type \(i\) only.

If fun="F" then this routine calculates, for each possible type \(i\), an estimate of the Empty Space Function \(F_i(r)\) of \(X_i\). See Fest for explanation of the empty space function. The estimate is computed by applying Fest to \(X_i\) with the optional arguments ....

If fun is "Gcross", "Jcross", "Kcross" or "Lcross", the routine calculates, for each pair of types \((i,j)\), an estimate of the ``i-toj'' cross-type function \(G_{ij}(r)\), \(J_{ij}(r)\), \(K_{ij}(r)\) or \(L_{ij}(r)\) respectively describing the dependence between \(X_i\) and \(X_j\). See Gcross, Jcross, Kcross or Lcross respectively for explanation of these functions. The estimate is computed by applying the relevant function (Gcross etc) to X using each possible value of the arguments i,j, together with the optional arguments ....

If fun is "pcf" the routine calculates the cross-type pair correlation function pcfcross between each pair of types.

If fun is "Gdot", "Jdot", "Kdot" or "Ldot", the routine calculates, for each type \(i\), an estimate of the ``i-to-any'' dot-type function \(G_{i\bullet}(r)\), \(J_{i\bullet}(r)\) or \(K_{i\bullet}(r)\) or \(L_{i\bullet}(r)\) respectively describing the dependence between \(X_i\) and \(X\). See Gdot, Jdot, Kdot or Ldot respectively for explanation of these functions. The estimate is computed by applying the relevant function (Gdot etc) to X using each possible value of the argument i, together with the optional arguments ....

The letters "G", "J", "K" and "L" are interpreted as abbreviations for Gcross, Jcross, Kcross and Lcross respectively, assuming the point pattern is marked. If the point pattern is unmarked, the appropriate function Fest, Jest, Kest or Lest is invoked instead.

If envelope=TRUE, then as well as computing the value of the summary function for each combination of types, the algorithm also computes simulation envelopes of the summary function for each combination of types. The arguments ... are passed to the function envelope to control the number of simulations, the random process generating the simulations, the construction of envelopes, and so on.

When envelope=TRUE it is possible that errors could occur because the simulated point patterns do not satisfy the requirements of the summary function (for example, because the simulated pattern is empty and fun requires at least one point). If the number of such errors exceeds the maximum permitted number maxnerr, then the envelope algorithm will give up, and will return the empirical summary function for the data point pattern, fun(X), in place of the envelope.

Examples

Run this code

   # bramblecanes (3 marks).
   bram <- bramblecanes
   # \testonly{
      bram <- bram[c(seq(1, 744, by=20), seq(745, 823, by=4))]
   # }
   bF <- alltypes(bram,"F",verb=TRUE)
   plot(bF)
   if(interactive()) {
     plot(alltypes(bram,"G"))
     plot(alltypes(bram,"Gdot"))
   }
   
   # Swedishpines (unmarked).
  swed <- swedishpines
   # \testonly{
     swed <- swed[1:25]
   # }
   plot(alltypes(swed,"K"))

   plot(alltypes(amacrine, "pcf"), ylim=c(0,1.3))

   # envelopes
   bKE <- alltypes(bram,"K",envelope=TRUE,nsim=19)
   # global version:
   # \donttest{
     bFE <- alltypes(bram,"F",envelope=TRUE,nsim=19,global=TRUE)
   # }

   # extract one entry
   as.fv(bKE[1,1])