R: Local Multitype K Function (Cross-Type)

localKcross {spatstat} | R Documentation

## Local Multitype K Function (Cross-Type)

### Description

for a multitype point pattern, computes the cross-type version of the local K function.

### Usage

      localKcross(X, from, to, ..., rmax = NULL,
                  correction = "Ripley", verbose = TRUE, rvalue=NULL)
      localLcross(X, from, to, ..., rmax = NULL, correction = "Ripley")

### Arguments

`X` | A multitype point pattern (object of class `"ppp"` with marks which are a factor).
`...` | Further arguments passed from `localLcross` to `localKcross`.
`rmax` | Optional. Maximum desired value of the argument r.
`from` | Type of points from which distances should be measured. A single value; one of the possible levels of `marks(X)`, or an integer indicating which level.
`to` | Type of points to which distances should be measured. A single value; one of the possible levels of `marks(X)`, or an integer indicating which level.
`correction` | String specifying the edge correction to be applied. Options are `"none"`, `"translate"`, `"translation"`, `"Ripley"`, `"isotropic"` or `"best"`. Only one correction may be specified.
`verbose` | Logical flag indicating whether to print progress reports during the calculation.
`rvalue` | Optional. A single value of the distance argument r at which the function L or K should be computed.

### Details

Given a multitype spatial point pattern `X`, the local cross-type K function `localKcross` is the local version of the multitype K function `Kcross`. Recall that `Kcross(X, from, to)` is a sum of contributions from all pairs of points in `X` where the first point belongs to `from` and the second point belongs to type `to`. The local cross-type K function is defined for each point `X[i]` that belongs to type `from`, and it consists of all the contributions to the cross-type K function that originate from point `X[i]`:

K[i,from,to](r) = sqrt( (a/((n-1)* pi)) * sum[j] e[i,j])

where the sum is over all points j != i belonging to type `to`, that lie within a distance r of the ith point, a is the area of the observation window, n is the number of points in `X`, and e[i,j] is an edge correction term (as described in `Kest`). The value of K[i,from,to](r) can also be interpreted as one of the summands that contributes to the global estimate of the `Kcross` function.

By default, the function K[i,from,to](r) is computed for a range of r values for each point i belonging to type `from`. The results are stored as a function value table (object of class `"fv"`) with a column of the table containing the function estimates for each point of the pattern `X` belonging to type `from`.

Alternatively, if the argument `rvalue` is given, and it is a single number, then the function will only be computed for this value of r, and the results will be returned as a numeric vector, with one entry of the vector for each point of the pattern `X` belonging to type `from`.

The local cross-type L function `localLcross` is computed by applying the transformation L(r) = sqrt(K(r)/(2*pi)).

### Value

If `rvalue` is given, the result is a numeric vector of length equal to the number of points in the point pattern that belong to type `from`.

If `rvalue` is absent, the result is an object of class `"fv"`, see `fv.object`, which can be plotted directly using `plot.fv`. Essentially a data frame containing columns

`r` | the vector of values of the argument r at which the function K has been estimated
`theo` | the theoretical value K(r) = pi * r^2 or L(r)=r for a stationary Poisson process

together with columns containing the values of the neighbourhood density function for each point in the pattern. Column `i` corresponds to the `i`th point of type `from`. The last two columns contain the `r` and `theo` values.

### Author(s)

Ege Rubak rubak@math.aau.dk and Adrian Baddeley Adrian.Baddeley@curtin.edu.au.

### See Also

`Kcross`, `Lcross`, `localK`, `localL`.

Inhomogeneous counterparts of `localK` and `localL` are computed by `localKcross.inhom` and `localLinhom`.

### Examples

      X <- amacrine

      # compute all the local Lcross functions
      L <- localLcross(X)

      # plot all the local Lcross functions against r
      plot(L, main="local Lcross functions for amacrine", legend=FALSE)

      # plot only the local L function for point number 7
      plot(L, iso007 ~ r)

      # compute the values of L(r) for r = 0.1 metres
      L12 <- localLcross(X, rvalue=0.1)

[Package spatstat version 1.63-3 Index]