Anisotropic and Inhomogeneous K function, cylinder version — Kest_anin

Estimate a cylinder-K function for second order reweighted ("inhomogeneous") pattern.

Usage

Kest_anin_cylinder(
  x,
  u,
  epsilon,
  r,
  lambda = NULL,
  lambda_h,
  renormalise = TRUE,
  border = 1,
  aspect = 1/3,
  ...
)

Arguments

x: pp, list with $x~coordinates $bbox~bounding box
u: unit vector(s) of direction, as row vectors. Default: x and y axes, viz. c(1,0) and c(0,1).
epsilon: The cylinder half-width. Will be extended to the length of r, so can be given per r.
r: radius vector at which to evaluate K
lambda: optional vector of intensity estimates at points
lambda_h: if lambda missing, use this bandwidth in a kernel estimate of lambda(x)
renormalise: See details.
border: Use border correction? Default=1, yes.
aspect: Instead of using a fixed halfwidth (epsilon) take the halfwidth to be 'aspect * r/2' (so an increasing vector). Default : 1/3
...: passed on to e.g. intensity_at_points

Value

Returns a dataframe.

Details

Computes a second order reweighted version of the cylinder-K. In short, we count how many pairs of points in the pattern has both a) their difference vector inside a cylinder with major-axial direction 'u' and radius epsilon, and b) difference vector length less than range r. Usually r is a vector and the output is then a vector as well.

Note the default behaviour is to use a fixed aspect ratio cylinder with aspect = 2*epsilon/range = 1/3.

An estimate of the intensity Lambda(x) at points can be given ('lambda'). If it is a single value, the pattern is assumed to be homogeneous. If it is a vector the same length as there are points, the pattern is taken to be second-order stationary. In this case the the sum over the pairs (i,j) is weighted with 1/(lambda[i]*lambda[j]). If 'lambda' is missing, 'lambda_h', a single positive number, should be given, which is then used for estimating the non-constant Lambda(x) via Epanechnikov kernel smoothing (see intensity_at_points). If 'renormalise=TRUE', we normalise the intensity estimate so that sum(1/lambda(x))=|W|. This corresponds in spatstat's Kinhom to setting 'normpower=2'.

About border correction: If x$bbox is a a simple bounding box, the algorithm uses the translation corrected weighting 1/area(Wx intersect Wy) with Wx=W+x. If x$bbox is a bbquad-object, for example rotated polygon, the algorithm uses simple minus border correction.