Kcross.inhom {spatstat.explore}

Inhomogeneous Cross K Function

Description

For a multitype point pattern, estimate the inhomogeneous version of the cross K function, which counts the expected number of points of type j within a given distance of a point of type i, adjusted for spatially varying intensity.

Usage

Kcross.inhom(X, i, j, lambdaI=NULL, lambdaJ=NULL, ...,  r=NULL, breaks=NULL,
         correction = c("border", "isotropic", "Ripley", "translate"),
         sigma=NULL, varcov=NULL,
         lambdaIJ=NULL,
         lambdaX=NULL, update=TRUE, leaveoneout=TRUE)

Arguments

X

The observed point pattern, from which an estimate of the inhomogeneous cross type K function K_{ij}(r) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

lambdaI

Optional. Values of the estimated intensity of the sub-process of points of type i. Either a pixel image (object of class "im"), a numeric vector containing the intensity values at each of the type i points in X, a fitted point process model (object of class "ppm" or "kppm" or "dppm"), or a function(x,y) which can be evaluated to give the intensity value at any location.

lambdaJ

Optional. Values of the the estimated intensity of the sub-process of points of type j. Either a pixel image (object of class "im"), a numeric vector containing the intensity values at each of the type j points in X, a fitted point process model (object of class "ppm" or "kppm" or "dppm"), or a function(x,y) which can be evaluated to give the intensity value at any location.

r

Optional. Numeric vector giving the values of the argument r at which the cross K function K_{ij}(r) should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions on r.

breaks

This argument is for advanced use only.

correction

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

...

Ignored.

sigma

Standard deviation of isotropic Gaussian smoothing kernel, used in computing leave-one-out kernel estimates of lambdaI, lambdaJ if they are omitted.

varcov

Variance-covariance matrix of anisotropic Gaussian kernel, used in computing leave-one-out kernel estimates of lambdaI, lambdaJ if they are omitted. Incompatible with sigma.

lambdaIJ

Optional. A matrix containing estimates of the product of the intensities lambdaI and lambdaJ for each pair of points of types i and j respectively.

lambdaX

Optional. Values of the intensity for all points of X. Either a pixel image (object of class "im"), a numeric vector containing the intensity values at each of the points in X, a fitted point process model (object of class "ppm" or "kppm" or "dppm"), or a function(x,y) which can be evaluated to give the intensity value at any location. If present, this argument overrides both lambdaI and lambdaJ.

update

Logical value indicating what to do when lambdaI, lambdaJ or lambdaX is a fitted point process model (class "ppm", "kppm" or "dppm"). If 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 re-fitting it to X.

leaveoneout

Logical value (passed to density.ppp or fitted.ppm) specifying whether to use a leave-one-out rule when calculating the intensity.

Details

This is a generalisation of the function Kcross to include an adjustment for spatially inhomogeneous intensity, in a manner similar to the function Kinhom.

The inhomogeneous cross-type K function is described by Moller and Waagepetersen (2003, pages 48-49 and 51-53).

Briefly, given a multitype point process, suppose the sub-process of points of type j has intensity function \lambda_j(u) at spatial locations u. Suppose we place a mass of 1/\lambda_j(\zeta) at each point \zeta of type j. Then the expected total mass per unit area is 1. The inhomogeneous “cross-type” K function K_{ij}^{\mbox{inhom}}(r) equals the expected total mass within a radius r of a point of the process of type i.

If the process of type i points were independent of the process of type j points, then K_{ij}^{\mbox{inhom}}(r) would equal \pi r^2. Deviations between the empirical K_{ij} curve and the theoretical curve \pi r^2 suggest dependence between the points of types i and j.

The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor.

The arguments i and j will be interpreted as levels of the factor X$marks. (Warning: this means that an integer value i=3 will be interpreted as the number 3, not the 3rd smallest level). If i and j are missing, they default to the first and second level of the marks factor, respectively.

The argument lambdaI supplies the values of the intensity of the sub-process of points of type i. It may be either

a pixel image

(object of class "im") which gives the values of the type i intensity at all locations in the window containing X;

a numeric vector

containing the values of the type i intensity evaluated only at the data points of type i. The length of this vector must equal the number of type i points in X.

a function

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

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.)

omitted:

if lambdaI is omitted then it will be estimated using a leave-one-out kernel smoother.

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

Similarly lambdaJ should contain estimated values of the intensity of the sub-process of points of type j. It may be either a pixel image, a function, a numeric vector, or omitted.

Alternatively if the argument lambdaX is given, then it specifies the intensity values for all points of X, and the arguments lambdaI, lambdaJ will be ignored.

The optional argument lambdaIJ is for advanced use only. It is a matrix containing estimated values of the products of these two intensities for each pair of data points of types i and j respectively.

The argument r is the vector of values for the distance r at which K_{ij}(r) should be evaluated. The values of r must be increasing nonnegative numbers and the maximum r value must not exceed the radius of the largest disc contained in the window.

The argument correction chooses the edge correction as explained e.g. in Kest.

The pair correlation function can also be applied to the result of Kcross.inhom; see pcf.

Value

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

Essentially a data frame containing numeric columns

r

the values of the argument r at which the function K_{ij}(r) has been estimated

theo

the theoretical value of K_{ij}(r) for a marked Poisson process, namely \pi r^2

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function K_{ij}(r) obtained by the edge corrections named.

Warnings

The arguments i and j are always interpreted as levels of the factor X$marks. They are converted to character strings if they are not already character strings. The value i=1 does not refer to the first level of the factor.

Author(s)

Adrian Baddeley [e-mail], Rolf Turner [e-mail] and Ege Rubak [e-mail].

References

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

Moller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes Chapman and Hall/CRC Boca Raton, 2003.

See Also

Kcross, Kinhom, Kdot.inhom, Kmulti.inhom, pcf

Examples

    # Lansing Woods data
    woods <- lansing
    
    ma <- split(woods)$maple
    wh <- split(woods)$whiteoak

    # method (1): estimate intensities by nonparametric smoothing
    lambdaM <- density.ppp(ma, sigma=0.15, at="points")
    lambdaW <- density.ppp(wh, sigma=0.15, at="points")
    K <- Kcross.inhom(woods, "whiteoak", "maple", lambdaW, lambdaM)

    # method (2): leave-one-out
    K <- Kcross.inhom(woods, "whiteoak", "maple", sigma=0.15)

    # method (3): fit parametric intensity model
    if(require("spatstat.model")) {
    fit <- ppm(woods ~marks * polynom(x,y,2))
    # alternative (a): use fitted model as 'lambda' argument
    online <- interactive()
    K <- Kcross.inhom(woods, "whiteoak", "maple",
                      lambdaI=fit, lambdaJ=fit,
                      update=online, leaveoneout=online)
    K <- Kcross.inhom(woods, "whiteoak", "maple",
                      lambdaX=fit,
                      update=online, leaveoneout=online)
    # alternative (b): evaluate fitted intensities at data points
    # (these are the intensities of the sub-processes of each type)
    inten <- fitted(fit, dataonly=TRUE, leaveoneout=FALSE)
    # split according to types of points
    lambda <- split(inten, marks(woods))
    K <- Kcross.inhom(woods, "whiteoak", "maple",
              lambda$whiteoak, lambda$maple)
    }
    
    # synthetic example: type A points have intensity 50,
    #                    type B points have intensity 100 * x
    lamB <- as.im(function(x,y){50 + 100 * x}, owin())
    X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
    K <- Kcross.inhom(X, "A", "B",
        lambdaI=as.im(50, Window(X)), lambdaJ=lamB)