R: Inhomogeneous Linear K Function

linearKinhom {spatstat.linnet} | R Documentation

## Inhomogeneous Linear K Function

### Description

Computes an estimate of the inhomogeneous linear K function for a point pattern on a linear network.

### Usage

    linearKinhom(X, lambda=NULL, r=NULL, ..., correction="Ang",
                 normalise=TRUE, normpower=1,
    	     update=TRUE, leaveoneout=TRUE, ratio=FALSE)

### Arguments

`X` | Point pattern on linear network (object of class `"lpp"`).
`lambda` | Intensity values for the point pattern. Either a numeric vector, a `function`, a pixel image (object of class `"im"` or `"linim"`) or a fitted point process model (object of class `"ppm"` or `"lppm"`).
`r` | Optional. Numeric vector of values of the function argument r. There is a sensible default.
`...` | Ignored.
`correction` | Geometry correction. Either `"none"` or `"Ang"`. See Details.
`normalise` | Logical. If `TRUE` (the default), the denominator of the estimator is data-dependent (equal to the sum of the reciprocal intensities at the data points, raised to `normpower`), which reduces the sampling variability. If `FALSE`, the denominator is the length of the network.
`normpower` | Integer (usually either 1 or 2). Normalisation power. See Details.
`update` | Logical value indicating what to do when `lambda` is a fitted model (class `"lppm"` or `"ppm"`). If `update=TRUE` (the default), the model will first be refitted to the data `X` (using `update.lppm` or `update.ppm`) before the fitted intensity is computed. If `update=FALSE`, the fitted intensity of the model will be computed without re-fitting it to `X`.
`leaveoneout` | Logical value (passed to `fitted.lppm` or `fitted.ppm`) specifying whether to use a leave-one-out rule when calculating the intensity, when `lambda` is a fitted model. Supported only when `update=TRUE`.
`ratio` | Logical. If `TRUE`, the numerator and denominator of the estimate will also be saved, for use in analysing replicated point patterns.

### Details

This command computes the inhomogeneous version of the linear K function from point pattern data on a linear network.

If `lambda = NULL` the result is equivalent to the homogeneous K function `linearK`. If `lambda` is given, then it is expected to provide estimated values of the intensity of the point process at each point of `X`. The argument `lambda` may be a numeric vector (of length equal to the number of points in `X`), or a `function(x,y)` that will be evaluated at the points of `X` to yield numeric values, or a pixel image (object of class `"im"`) or a fitted point process model (object of class `"ppm"` or `"lppm"`).

If `lambda` is a fitted point process model, the default behaviour is to update the model by re-fitting it to the data, before computing the fitted intensity. This can be disabled by setting `update=FALSE`.

If `correction="none"`, the calculations do not include any correction for the geometry of the linear network. If `correction="Ang"`, the pair counts are weighted using Ang's correction (Ang, 2010).

Each estimate is initially computed as

K^inhom(r)= (1/length(L)) sum[i] sum[j] 1(d[i,j] <= r) * e(x[i],x[j])/(lambda(x[i]) * lambda(x[j]))

where `L` is the linear network, d[i,j] is the distance between points x[i] and x[j], and e(x[i],x[j]) is a weight. If `correction="none"` then this weight is equal to 1, while if `correction="Ang"` the weight is e(x[i],x[j],r) = 1/m(x[i],d[i,j]) where m(u,t) is the number of locations on the network that lie exactly t units distant from location u by the shortest path.

If `normalise=TRUE` (the default), then the estimates described above are multiplied by c^normpower where c = length(L)/sum[i] (1/lambda(x[i])). This rescaling reduces the variability and bias of the estimate in small samples and in cases of very strong inhomogeneity. The default value of `normpower` is 1 (for consistency with previous versions of spatstat) but the most sensible value is 2, which would correspond to rescaling the `lambda` values so that sum[i] (1/lambda(x[i])) = area(W).

### Value

Function value table (object of class `"fv"`).

### Author(s)

Ang Qi Wei aqw07398@hotmail.com and Adrian Baddeley Adrian.Baddeley@curtin.edu.au

### References

Ang, Q.W. (2010) Statistical methodology for spatial point patterns on a linear network. MSc thesis, University of Western Australia.

Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of events on a linear network, with applications to ecology and criminology. Scandinavian Journal of Statistics 39, 591–617.

### See Also

`lpp`

### Examples

      data(simplenet)
      X <- rpoislpp(5, simplenet)
      fit <- lppm(X ~x)
      K <- linearKinhom(X, lambda=fit)
      plot(K)

[Package spatstat.linnet version 2.3-1 Index]