Kinhom {spatstat}R Documentation

Inhomogeneous K-function

Description

Estimates the inhomogeneous K function of a non-stationary point pattern.

Usage

  Kinhom(X, lambda)
  Kinhom(X, lambda, r)
  Kinhom(X, lambda, breaks)

Arguments

X The observed data point pattern, from which an estimate of the inhomogeneous K function will be computed. An object of class "ppp" or in a format recognised by as.ppp()
lambda Vector of values of the estimated intensity function, evaluated at the points of the pattern X
r vector of values for the argument r at which the inhomogeneous K function should be evaluated. There is a sensible default.
breaks An alternative to the argument r. Not normally invoked by the user. See Details.

Details

This computes a generalisation of the K function for inhomogeneous point patterns, proposed by Baddeley, Moller and Waagepetersen (2000).

The ``ordinary'' K function (variously known as the reduced second order moment function and Ripley's K function), is described under Kest. It is defined only for stationary point processes.

The inhomogeneous K function Kinhom(r) is a direct generalisation to nonstationary point processes. Suppose x is a point process with non-constant intensity lambda(u) at each location u. Define Kinhom(r) to be the expected value, given that u is a point of x, of the sum of all terms 1/lambda(u)lambda(x[j]) over all points x[j] in the process separated from u by a distance less than r. This reduces to the ordinary K function if lambda() is constant. If x is an inhomogeneous Poisson process with intensity function lambda(u), then Kinhom(r) = pi * r^2.

This allows us to inspect a point pattern for evidence of interpoint interactions after allowing for spatial inhomogeneity of the pattern. Values Kinhom(r) > pi * r^2 are suggestive of clustering.

The argument lambda must be a vector of length equal to the number of points in the pattern X. It will be interpreted as giving the (estimated) values of lambda(x[i]) for each point x[i] of the pattern x.

The pair correlation function can also be applied to the result of Kinhom; see pcf.

Value

A data frame containing

r the vector of values of the argument r at which the pair correlation function g(r) has been estimated
K vector of values of Kinhom(r)
theo vector of values of pi * r^2, the theoretical value of Kinhom(r) for an inhomogeneous Poisson process

Author(s)

Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf

References

Baddeley, A., Moller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.

See Also

Kest, pcf

Examples

  
  library(spatstat)
  

  data(lansing)
  # inhomogeneous pattern of maples
  X <- unmark(lansing[lansing$marks == "maple",])
  
  # fit spatial trend
  fit <- mpl(X, ~ polynom(x,y,2), Poisson())
  # predict intensity values at points themselves
  lambda <- predict(fit, newdata=data.frame(x=X$x, y=X$y), type="trend")
  # inhomogeneous K function
  Ki <- Kinhom(X, lambda)
  
    conspire(Ki, cbind(K, theo) ~ r, subset="r <= 0.4")
  

  # SIMULATED DATA
  # known intensity function
  lamfun <- function(x,y) { 100 * x }
  # inhomogeneous Poisson process
  Y <- rpoispp(lamfun, 100, owin())
  # evaluate intensity at points of pattern
  lambda <- lamfun(Y$x, Y$y)
  # inhomogeneous K function
  Ki <- Kinhom(Y, lambda)
  
    conspire(Ki, cbind(K, theo) ~ r, subset="r <= 0.2")
  

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