Inhomogeneous J-function

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Description

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

Usage

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  Jinhom(X, lambda = NULL, lmin = NULL, ...,
        sigma = NULL, varcov = NULL,
        r = NULL, breaks = NULL, update = TRUE)

Arguments

X

The observed data point pattern, from which an estimate of the inhomogeneous J 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 locations, a fitted point process model (object of class "ppm" or "kppm") or a function(x,y) which can be evaluated to give the intensity value at any location.

lmin

Optional. The minimum possible value of the intensity over the spatial domain. A positive numerical value.

sigma,varcov

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

...

Extra arguments passed to as.mask to control the pixel resolution, or passed to density.ppp to control the smoothing bandwidth.

r

vector of values for the argument r at which the inhomogeneous K function should be evaluated. Not normally given by the user; there is a sensible default.

breaks

This argument is for internal use only.

update

Logical. If lambda is a fitted model (class "ppm" or "kppm") and update=TRUE (the default), the model will first be refitted to the data X (using update.ppm or update.kppm) before the fitted intensity is computed. If update=FALSE, the fitted intensity of the model will be computed without fitting it to X.

Details

This command computes estimates of the inhomogeneous J-function (Van Lieshout, 2010) of a point pattern. It is the counterpart, for inhomogeneous spatial point patterns, of the J function for homogeneous point patterns computed by Jest.

The argument X should be a point pattern (object of class "ppp").

The inhomogeneous J function is computed as Jinhom(r) = (1 - Ginhom(r))/(1-Finhom(r)) where Ginhom, Finhom are the inhomogeneous G and F functions computed using the border correction (equations (7) and (6) respectively in Van Lieshout, 2010).

The argument lambda should supply the (estimated) values of the intensity function lambda of the point process. 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" or "kppm") 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.

Value

An object of class "fv", see fv.object, which can be plotted directly using plot.fv.

Author(s)

Original code by Marie-Colette van Lieshout. C implementation and R adaptation by \adrian

and \ege.

References

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

van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344–361.

van Lieshout, M.N.M. (2010) A J-function for inhomogeneous point processes. Statistica Neerlandica 65, 183–201.

See Also

Ginhom, Finhom, Jest

Examples

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  ## Not run: 
    plot(Jinhom(swedishpines, sigma=bw.diggle, adjust=2))
  
## End(Not run)
  plot(Jinhom(swedishpines, sigma=10))

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