Locally Scaled K-function

Estimates the locally-rescaled \(K\)-function of a point process.

spatial, nonparametric
Kscaled(X, lambda=NULL, …, r = NULL, breaks = NULL,
    rmax = 2.5, 
    correction=c("border", "isotropic", "translate"),
    renormalise=FALSE, normpower=1,
    sigma=NULL, varcov=NULL)



The observed data point pattern, from which an estimate of the locally scaled \(K\) function will be computed. An object of class "ppp" or in a format recognised by as.ppp().


Optional. Values of the estimated intensity function. Either a vector giving the intensity values at the points of the pattern X, a pixel image (object of class "im") giving the intensity values at all locations, a function(x,y) which can be evaluated to give the intensity value at any location, or a fitted point process model (object of class "ppm").

Arguments passed from Lscaled to Kscaled and from Kscaled to density.ppp if lambda is omitted.


vector of values for the argument \(r\) at which the locally scaled \(K\) function should be evaluated. (These are rescaled distances.) Not normally given by the user; there is a sensible default.


This argument is for internal use only.


maximum value of the argument \(r\) that should be used. (This is the rescaled distance).


A character vector containing any selection of the options "border", "isotropic", "Ripley", "translate", "translation", "none" or "best". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.


Logical. Whether to renormalise the estimate. See Details.


Integer (usually either 1 or 2). Normalisation power. See Details.


Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.


Kscaled computes an estimate of the \(K\) function for a locally scaled point process. Lscaled computes the corresponding \(L\) function \(L(r) = \sqrt{K(r)/\pi}\).

Locally scaled point processes are a class of models for inhomogeneous point patterns, introduced by Hahn et al (2003). They include inhomogeneous Poisson processes, and many other models.

The template \(K\) function of a locally-scaled process is a counterpart of the ``ordinary'' Ripley \(K\) function, in which the distances between points of the process are measured on a spatially-varying scale (such that the locally rescaled process has unit intensity).

The template \(K\) function is an indicator of interaction between the points. For an inhomogeneous Poisson process, the theoretical template \(K\) function is approximately equal to \(K(r) = \pi r^2\). Values \(K_{\rm scaled}(r) > \pi r^2\) are suggestive of clustering.

Kscaled computes an estimate of the template \(K\) function and Lscaled computes the corresponding \(L\) function \(L(r) = \sqrt{K(r)/\pi}\).

The locally scaled interpoint distances are computed using an approximation proposed by Hahn (2007). The Euclidean distance between two points is multiplied by the average of the square roots of the intensity values at the two points.

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 function

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


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.

If renormalise=TRUE, the estimated intensity lambda is multiplied by \(c^(normpower/2)\) before performing other calculations, where \(c = area(W)/sum[i] (1/lambda(x[i]))\). This renormalisation has about the same effect as in Kinhom, reducing the variability and bias of the estimate in small samples and in cases of very strong inhomogeneity.

Edge corrections are used to correct bias in the estimation of \(K_{\rm scaled}\). First the interpoint distances are rescaled, and then edge corrections are applied as in Kest. See Kest for details of the edge corrections and the options for the argument correction.

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


An object of class "fv" (see fv.object).

Essentially a data frame containing at least the following columns,


the vector of values of the argument \(r\) at which the pair correlation function \(g(r)\) has been estimated


vector of values of \(\pi r^2\), the theoretical value of \(K_{\rm scaled}(r)\) for an inhomogeneous Poisson process

and containing additional columns according to the choice specified in the correction argument. The additional columns are named border, trans and iso and give the estimated values of K_{\rm scaled}(r)Kscaled(r) using the border correction, translation correction, and Ripley isotropic correction, respectively.


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

Hahn, U. (2007) Global and Local Scaling in the Statistics of Spatial Point Processes. Habilitationsschrift, Universitaet Augsburg.

Hahn, U., Jensen, E.B.V., van Lieshout, M.N.M. and Nielsen, L.S. (2003) Inhomogeneous spatial point processes by location-dependent scaling. Advances in Applied Probability 35, 319--336.

Prokesova, M., Hahn, U. and Vedel Jensen, E.B. (2006) Statistics for locally scaled point patterns. In A. Baddeley, P. Gregori, J. Mateu, R. Stoica and D. Stoyan (eds.) Case Studies in Spatial Point Pattern Modelling. Lecture Notes in Statistics 185. New York: Springer Verlag. Pages 99--123.

See Also

Kest, pcf

  • Kscaled
  • Lscaled
  X <- unmark(bronzefilter)
  K <- Kscaled(X)
  fit <- ppm(X, ~x)
  lam <- predict(fit)
  K <- Kscaled(X, lam)
# }
Documentation reproduced from package spatstat, version 1.63-0, License: GPL (>= 2)

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