R: Inhomogeneous Multitype K Dot Function

Kdot.inhom {spatstat.core} | R Documentation

## Inhomogeneous Multitype K Dot Function

### Description

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

### Usage

    Kdot.inhom(X, i, lambdaI=NULL, lambdadot=NULL, ..., r=NULL, breaks=NULL,
             correction = c("border", "isotropic", "Ripley", "translate"),
             sigma=NULL, varcov=NULL, lambdaIdot=NULL,
             lambdaX=NULL, update=TRUE, leaveoneout=TRUE)

### Arguments

`X` | The observed point pattern, from which an estimate of the inhomogeneous cross type K function Ki.(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)`.
`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.
`lambdadot` | Optional. Values of the estimated intensity of the entire point process, 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.
`...` | Ignored.
`r` | Optional. Numeric vector giving the values of the argument r at which the cross K function Kij(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 internal 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.
`sigma` | Standard deviation of isotropic Gaussian smoothing kernel, used in computing leave-one-out kernel estimates of `lambdaI`, `lambdadot` if they are omitted.
`varcov` | Variance-covariance matrix of anisotropic Gaussian kernel, used in computing leave-one-out kernel estimates of `lambdaI`, `lambdadot` if they are omitted. Incompatible with `sigma`.
`lambdaIdot` | Optional. A matrix containing estimates of the product of the intensities `lambdaI` and `lambdadot` for each pair of points, the first point of type `i` and the second of any type.
`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 `lambdadot`.
`update` | Logical value indicating what to do when `lambdaI`, `lambdadot` 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 `Kdot` to include an adjustment for spatially inhomogeneous intensity, in a manner similar to the function `Kinhom`.

Briefly, given a multitype point process, consider the points without their types, and suppose this unmarked point process has intensity function lambda(u) at spatial locations u. Suppose we place a mass of 1/lambda(z) at each point z of the process. Then the expected total mass per unit area is 1. The inhomogeneous “dot-type” K function K[i.]inhom(r) equals the expected total mass within a radius r of a point of the process of type i, discounting this point itself.

If the process of type i points were independent of the points of other types, then K[i.]inhom(r) would equal pi * r^2. Deviations between the empirical Ki. curve and the theoretical curve pi * r^2 suggest dependence between the points of types i and j for j != i.

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 argument `i` will be interpreted as a level of the factor `X$marks`. (Warning: this means that an integer value `i=3` will be interpreted as the number 3, not the 3rd smallest level). If `i` is missing, it defaults to the first level of the marks factor, `i = levels(X$marks)[1]`.

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

  of the form `function(x,y)` 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 the argument `lambdadot` should contain estimated values of the intensity of the entire point process. It may be either a pixel image, a numeric vector of length equal to the number of points in `X`, a function, or omitted.

Alternatively if the argument `lambdaX` is given, then it specifies the intensity values for all points of `X`, and the arguments `lambdaI`, `lambdadot` will be ignored. (The two arguments `lambdaI`, `lambdadot` allow the user to specify two different methods for calculating the intensities of the two kinds of points, while `lambdaX` ensures that the same method is used for both kinds of points.)

For advanced use only, the optional argument `lambdaIdot` is a matrix containing estimated values of the products of these two intensities for each pair of points, the first point of type `i` and the second of any type.

The argument `r` is the vector of values for the distance r at which Ki.(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 Ki.(r) has been estimated
`theo` | the theoretical value of Ki.(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 Ki.(r) obtained by the edge corrections named.

### Warnings

The argument `i` is interpreted as a level of the factor `X$marks`. It is converted to a character string if it is not already a character string. The value `i=1` does not refer to the first level of the factor.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk

### References

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

### See Also

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

### Examples

        # Lansing Woods data
        woods <- lansing
        woods <- woods[seq(1,npoints(woods), by=10)]
        ma <- split(woods)$maple
        lg <- unmark(woods)

        # Estimate intensities by nonparametric smoothing
        lambdaM <- density.ppp(ma, sigma=0.15, at="points")
        lambdadot <- density.ppp(lg, sigma=0.15, at="points")
        K <- Kdot.inhom(woods, "maple", lambdaI=lambdaM,
                                          lambdadot=lambdadot)

        # Equivalent
        K <- Kdot.inhom(woods, "maple", sigma=0.15)

        # Fit model
        fit <- ppm(woods ~ marks * polynom(x,y,2))
        K <- Kdot.inhom(woods, "maple", lambdaX=fit, update=FALSE)

        # synthetic example: type A points have intensity 50,
        #                    type B points have intensity 50 + 100 * x
        lamB <- as.im(function(x,y){50 + 100 * x}, owin())
        lamdot <- as.im(function(x,y) { 100 + 100 * x}, owin())
        X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
        K <- Kdot.inhom(X, "B",  lambdaI=lamB,     lambdadot=lamdot)

[Package spatstat.core version 2.4-4 Index]