Kscaled

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Percentile

Locally Scaled K-function

Estimates the template $K$ function of a locally-scaled point process.

Keywords
spatial, nonparametric
Usage
Kscaled(X, lambda=NULL, ..., r = NULL, breaks = NULL,
    correction=c("border", "isotropic", "translate"),
    sigma=NULL, varcov=NULL)
  Lscaled(...)
Arguments
X
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().
lambda
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 locatio
...
Arguments passed from Lscaled to Kscaled and from Kscaled to density.ppp if lambda is omitted.
r
vector of values for the argument $r$ at which the locally scaled $K$ function should be evaluated. Not normally given by the user; there is a sensible default.
breaks
An alternative to the argument r. Not normally invoked by the user. See Details.
correction
A character vector containing any selection of the options "border", "isotropic", "Ripley", "translate", "none" or "best". It specifies the edge correct
sigma,varcov
Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.
Details

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 [object Object],[object Object],[object Object],[object Object] 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 $K_{\rm scaled}$. 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.

Value

  • An object of class "fv" (see fv.object). Essentially a data frame containing at least the following columns,
  • rthe vector of values of the argument $r$ at which the pair correlation function $g(r)$ has been estimated
  • theovector 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 $K_{\rm scaled}(r)$ using the border correction, translation correction, and Ripley isotropic correction, respectively.

References

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

Aliases
  • Kscaled
  • Lscaled
Examples
data(bronzefilter)
  X <- unmark(bronzefilter)
  K <- Kscaled(X)
  fit <- ppm(X, ~x)
  lam <- predict(fit)
  K <- Kscaled(X, lam)
Documentation reproduced from package spatstat, version 1.21-0, License: GPL (>= 2)

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