Kinhom

Inhomogeneous K-function

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

Usage

Kinhom(X, lambda, r = NULL, breaks = NULL, slow = FALSE,
    correction=c("border", "bord.modif", "isotropic", "translate"),
    ..., lambda2
)

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 $K_{\rm inhom}(r)$ 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 $K_{\rm inhom}(r)$ 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 $K_{\rm inhom}(r) = \pi r^2$.

This allows us to inspect a point pattern for evidence of interpoint interactions after allowing for spatial inhomogeneity of the pattern. Values $K_{\rm inhom}(r) > \pi r^2$ are suggestive of clustering.

The argument lambda should 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 $x$.

Edge corrections are used to correct bias in the estimation of $K_{\rm inhom}$. Each edge-corrected estimate of $K_{\rm inhom}(r)$ is of the form $$\widehat K_{\rm inhom}(r) = \sum_i \sum_j \frac{1{d_{ij} \le r} e(x_i,x_j,r)}{\lambda(x_i)\lambda(x_j)}$$ where $d_{ij}$ is the distance between points $x_i$ and $x_j$, and $e(x_i,x_j,r)$ is an edge correction factor. For the `border' correction, $$e(x_i,x_j,r) = \frac{1(b_i > r)}{\sum_j 1(b_j > r)/\lambda(x_j)}$$ where $b_i$ is the distance from $x_i$ to the boundary of the window. For the `modified border' correction, $$e(x_i,x_j,r) = \frac{1(b_i > r)}{\mbox{area}(W \ominus r)}$$ where $W \ominus r$ is the eroded window obtained by trimming a margin of width $r$ from the border of the original window. For the `translation' correction, $$e(x_i,x_j,r) = \frac 1 {\mbox{area}(W \cap (W + (x_j - x_i)))}$$ and for the `isotropic' correction, $$e(x_i,x_j,r) = \frac 1 {\mbox{area}(W) g(x_i,x_j)}$$ where $g(x_i,x_j)$ is the fraction of the circumference of the circle with centre $x_i$ and radius $||x_i - x_j||$ which lies inside the window. The pair correlation function can also be applied to the result of Kinhom; see pcf.

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

Aliases

• Kinhom

Examples

data(lansing)
  # inhomogeneous pattern of maples
  X <- unmark(lansing[lansing$marks == "maple",])
  <testonly>sub <- sample(c(TRUE,FALSE), X$n, replace=TRUE, prob=c(0.1,0.9))
     X <- X[sub , ]</testonly>
  # fit spatial trend
  fit <- ppm(X, ~ polynom(x,y,2), Poisson())
  # predict intensity values at points themselves
  lambda <- predict(fit, locations=X, type="trend")
  # inhomogeneous K function
  Ki <- Kinhom(X, lambda)
  plot(Ki)

  # 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)
  plot(Ki)