Kinhom

Inhomogeneous K-function

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

Usage

Kinhom(X, lambda=NULL, …, r = NULL, breaks = NULL,
    correction=c("border", "bord.modif", "isotropic", "translate"),
    renormalise=TRUE,
    normpower=1,
    update=TRUE,
    leaveoneout=TRUE,
    nlarge = 1000,
    lambda2=NULL, reciplambda=NULL, reciplambda2=NULL,
    diagonal=TRUE,
    sigma=NULL, varcov=NULL,
    ratio=FALSE)

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 \(K_{\mbox{\scriptsize\rm inhom}}(r)\) 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 \(K_{\mbox{\scriptsize\rm inhom}}(r)\) to be the expected value, given that \(u\) is a point of \(x\), of the sum of all terms \(1/\lambda(x_j)\) 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 \(K_{\mbox{\scriptsize\rm inhom}}(r) = \pi r^2\).

Given a point pattern dataset, the inhomogeneous \(K\) function can be estimated essentially by summing the values \(1/(\lambda(x_i)\lambda(x_j))\) for all pairs of points \(x_i, x_j\) separated by a distance less than \(r\).

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

The argument lambda should supply the (estimated) values of the intensity function \(\lambda\). It may be either

a numeric vector

containing the values of the intensity function at the points of the pattern X.

a pixel image

(object of class "im") assumed to contain the values of the intensity function at all locations in the window.

a fitted point process model

(object of class "ppm", "kppm" or "dppm") whose fitted trend can be used as the fitted intensity. (If update=TRUE the model will first be refitted to the data X before the trend is computed.)

a function

which can be evaluated to give values of the intensity at any locations.

omitted:

if lambda is omitted, then it will be estimated using a `leave-one-out' kernel smoother.

If lambda is a numeric vector, then its length should be equal to the number of points in the pattern X. The value lambda[i] is assumed to be the the (estimated) value of the intensity \(\lambda(x_i)\) for the point \(x_i\) of the pattern \(X\). Each value must be a positive number; NA's are not allowed.

If lambda is a pixel image, the domain of the image should cover the entire window of the point pattern. If it does not (which may occur near the boundary because of discretisation error), then the missing pixel values will be obtained by applying a Gaussian blur to lambda using blur, then looking up the values of this blurred image for the missing locations. (A warning will be issued in this case.)

If lambda is a function, then it will be evaluated in the form lambda(x,y) where x and y are vectors of coordinates of the points of X. It should return a numeric vector with length equal to the number of points in X.

If lambda is omitted, then it will be estimated using a `leave-one-out' kernel smoother, as described in Baddeley, Moller and Waagepetersen (2000). The estimate lambda[i] for the point X[i] is computed by removing X[i] from the point pattern, applying kernel smoothing to the remaining points using density.ppp, and evaluating the smoothed intensity at the point X[i]. The smoothing kernel bandwidth is controlled by the arguments sigma and varcov, which are passed to density.ppp along with any extra arguments.

Edge corrections are used to correct bias in the estimation of \(K_{\mbox{\scriptsize\rm inhom}}\). Each edge-corrected estimate of \(K_{\mbox{\scriptsize\rm inhom}}(r)\) is of the form $$ \widehat K_{\mbox{\scriptsize\rm inhom}}(r) = (1/A) \sum_i \sum_j \frac{1\{d_{ij} \le r\} e(x_i,x_j,r)}{\lambda(x_i)\lambda(x_j)} $$ where A is a constant denominator, \(d_{ij}\) 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.

If renormalise=TRUE (the default), then the estimates described above are multiplied by \(c^{\mbox{normpower}}\) where \( c = \mbox{area}(W)/\sum (1/\lambda(x_i)). \) 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 \( \sum (1/\lambda(x_i)) = \mbox{area}(W). \)

If the point pattern X contains more than about 1000 points, the isotropic and translation edge corrections can be computationally prohibitive. The computations for the border method are much faster, and are statistically efficient when there are large numbers of points. Accordingly, if the number of points in X exceeds the threshold nlarge, then only the border correction will be computed. Setting nlarge=Inf or correction="best" will prevent this from happening. Setting nlarge=0 is equivalent to selecting only the border correction with correction="border".

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

# NOT RUN {
  # inhomogeneous pattern of maples
  X <- unmark(split(lansing)$maple)
  
# }
# NOT RUN {
  # (1) intensity function estimated by model-fitting
  # Fit spatial trend: polynomial in x and y coordinates
  fit <- ppm(X, ~ polynom(x,y,2), Poisson())
  # (a) predict intensity values at points themselves,
  #     obtaining a vector of lambda values
  lambda <- predict(fit, locations=X, type="trend")
  # inhomogeneous K function
  Ki <- Kinhom(X, lambda)
  plot(Ki)
  # (b) predict intensity at all locations,
  #     obtaining a pixel image
  lambda <- predict(fit, type="trend")
  Ki <- Kinhom(X, lambda)
  plot(Ki)

  # (2) intensity function estimated by heavy smoothing
  Ki <- Kinhom(X, sigma=0.1)
  plot(Ki)

  # (3) simulated data: known intensity function
  lamfun <- function(x,y) { 50 + 100 * x }
  # inhomogeneous Poisson process
  Y <- rpoispp(lamfun, 150, owin())
  # inhomogeneous K function
  Ki <- Kinhom(Y, lamfun)
  plot(Ki)

  # How to make simulation envelopes:
  #      Example shows method (2)
  
# }
# NOT RUN {
  smo <- density.ppp(X, sigma=0.1)
  Ken <- envelope(X, Kinhom, nsim=99,
                  simulate=expression(rpoispp(smo)),
                  sigma=0.1, correction="trans")
  plot(Ken)
  
# }