linearKinhom

Inhomogeneous Linear K Function

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)

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 $$ \widehat K_{\rm inhom}(r) = \frac{1}{\mbox{length}(L)} \sum_i \sum_j \frac{1\{d_{ij} \le r\} e(x_i,x_j)}{\lambda(x_i)\lambda(x_j)} $$ where L is the linear network, \(d_{ij}\) 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 \(e(x_i,x_j,r) = 1/m(x_i, d_{ij})\) 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 \(c^{\mbox{normpower}}\) where \( c = \mbox{length}(L)/\sum (1/\lambda(x_i)). \) 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 \( \sum (1/\lambda(x_i)) = \mbox{area}(W). \)

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

Aliases

• linearKinhom

Examples

# NOT RUN {
  data(simplenet)
  X <- rpoislpp(5, simplenet)
  fit <- lppm(X ~x)
  K <- linearKinhom(X, lambda=fit)
  plot(K)
# }