R: Multitype K Function (i-to-any)

Kdot {spatstat} | R Documentation

## Multitype K Function (i-to-any)

### Description

For a multitype point pattern, estimate the multitype K function which counts the expected number of other points of the process within a given distance of a point of type i.

### Usage

    Kdot(X, i, r=NULL, breaks=NULL, correction, ..., ratio=FALSE, from)

### Arguments

`X` | The observed point pattern, from which an estimate of the multitype 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)`.
`r` | numeric vector. The values of the argument r at which the distribution function Ki.(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.
`...` | Ignored.
`ratio` | Logical. If `TRUE`, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.
`from` | An alternative way to specify `i`.

### Details

This function `Kdot` and its companions `Kcross` and `Kmulti` are generalisations of the function `Kest` to multitype point patterns.

A multitype point pattern is a spatial pattern of points classified into a finite number of possible “colours” or “types”. In the spatstat package, a multitype pattern is represented as a single point pattern object in which the points carry marks, and the mark value attached to each point determines the type of that point.

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$marks`. If `i` is missing, it defaults to the first level of the marks factor, `i = levels(X$marks)[1]`.

The “type i to any type” multitype K function of a stationary multitype point process X is defined so that lambda Ki.(r) equals the expected number of additional random points within a distance r of a typical point of type i in the process X. Here lambda is the intensity of the process, i.e. the expected number of points of X per unit area. The function Ki. is determined by the second order moment properties of X.

An estimate of Ki.(r) is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the subprocess of type i points were independent of the subprocess of points of all types not equal to i, then Ki.(r) would equal pi * r^2. Deviations between the empirical Ki. curve and the theoretical curve pi * r^2 may suggest dependence between types.

This algorithm estimates the distribution function Ki.(r) from the point pattern `X`. It assumes that `X` can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in `X` as `Window(X)`) may have arbitrary shape. Biases due to edge effects are treated in the same manner as in `Kest`, using the border correction.

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 exceed the radius of the largest disc contained in the window.

The pair correlation function can also be applied to the result of `Kdot`; 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.

If `ratio=TRUE` then the return value also has two attributes called `"numerator"` and `"denominator"` which are `"fv"` objects containing the numerators and denominators of each estimate of K(r).

### 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.

The reduced sample estimator of Ki. is pointwise approximately unbiased, but need not be a valid distribution function; it may not be a nondecreasing function of r. Its range is always within [0,1].

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

### References

Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.

Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants' nests. Applied Statistics 32, 293–303

Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.

Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.

### See Also

`Kdot`, `Kest`, `Kmulti`, `pcf`

### Examples

         # Lansing woods data: 6 types of trees
        woods <- lansing

        Kh. <- Kdot(woods, "hickory")
        # diagnostic plot for independence between hickories and other trees
        plot(Kh.)

        ## Not run:
        # synthetic example with two marks "a" and "b"
        pp <- runifpoispp(50)
        pp <- pp %mark% factor(sample(c("a","b"), npoints(pp), replace=TRUE))
        K <- Kdot(pp, "a")

    ## End(Not run)

[Package spatstat version 1.54-0 Index]