R: K-function

Kest {spatstat.core} | R Documentation

## K-function

### Description

Estimates Ripley's reduced second moment function K(r) from a point pattern in a window of arbitrary shape.

### Usage

      Kest(X, ..., r=NULL, rmax=NULL, breaks=NULL,
         correction=c("border", "isotropic", "Ripley", "translate"),
        nlarge=3000, domain=NULL, var.approx=FALSE, ratio=FALSE)

### Arguments

`X` | The observed point pattern, from which an estimate of K(r) will be computed. An object of class `"ppp"`, or data in any format acceptable to `as.ppp()`.
`...` | Ignored.
`r` | Optional. Vector of values for the argument r at which K(r) should be evaluated. Users are advised not to specify this argument; there is a sensible default. If necessary, specify `rmax`.
`rmax` | Optional. Maximum desired value of the argument r.
`breaks` | This argument is for internal use only.
`correction` | Optional. A character vector containing any selection of the options `"none"`, `"border"`, `"bord.modif"`, `"isotropic"`, `"Ripley"`, `"translate"`, `"translation"`, `"rigid"`, `"none"`, `"good"` or `"best"`. It specifies the edge correction(s) to be applied. Alternatively `correction="all"` selects all options.
`nlarge` | Optional. Efficiency threshold. If the number of points exceeds `nlarge`, then only the border correction will be computed (by default), using a fast algorithm.
`domain` | Optional. Calculations will be restricted to this subset of the window. See Details.
`var.approx` | Logical. If `TRUE`, the approximate variance of Kest(r) under CSR will also be computed.
`ratio` | Logical. If `TRUE`, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

### Details

The K function (variously called “Ripley's K-function” and the “reduced second moment function”) of a stationary point process X is defined so that lambda K(r) equals the expected number of additional random points within a distance r of a typical random point of X. Here lambda is the intensity of the process, i.e. the expected number of points of X per unit area. The K function is determined by the second order moment properties of X.

An estimate of K derived from a spatial point pattern dataset can be used in exploratory data analysis and formal inference about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1977, 1988). In exploratory analyses, the estimate of K is a useful statistic summarising aspects of inter-point “dependence” and “clustering”. For inferential purposes, the estimate of K is usually compared to the true value of K for a completely random (Poisson) point process, which is K(r) = pi * r^2. Deviations between the empirical and theoretical K curves may suggest spatial clustering or spatial regularity.

This routine `Kest` estimates the K function of a stationary point process, given observation of the process inside a known, bounded window. The argument `X` is interpreted as a point pattern object (of class `"ppp"`, see `ppp.object`) and can be supplied in any of the formats recognised by `as.ppp()`.

The estimation of K is hampered by edge effects arising from the unobservability of points of the random pattern outside the window. An edge correction is needed to reduce bias (Baddeley, 1998; Ripley, 1988). The corrections implemented here are

border

  the border method or “reduced sample” estimator (see Ripley, 1988). This is the least efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary shape.

isotropic/Ripley

  Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented for rectangular and polygonal windows (not for binary masks).

translate/translation

  Translation correction (Ohser, 1983). Implemented for all window geometries, but slow for complex windows.

rigid

  Rigid motion correction (Ohser and Stoyan, 1981). Implemented for all window geometries, but slow for complex windows.

none

  Uncorrected estimate. An estimate of the K function without edge correction. (i.e. setting e[i,j] = 1 in the equation below. This estimate is biased and should not be used for data analysis, unless you have an extremely large point pattern (more than 100,000 points).

best

  Selects the best edge correction that is available for the geometry of the window. Currently this is Ripley's isotropic correction for a rectangular or polygonal window, and the translation correction for masks.

good

  Selects the best edge correction that can be computed in a reasonable time. This is the same as `"best"` for datasets with fewer than 3000 points; otherwise the selected edge correction is `"border"`, unless there are more than 100,000 points, when it is `"none"`.

The estimates of K(r) are of the form

Kest(r) = (a/(n * (n-1))) * sum[i,j] I(d[i,j] <= r) e[i,j])

where a is the area of the window, n is the number of data points, and the sum is taken over all ordered pairs of points i and j in `X`. Here d[i,j] is the distance between the two points, and I(d[i,j] <= r) is the indicator that equals 1 if the distance is less than or equal to r. The term e[i,j] is the edge correction weight (which depends on the choice of edge correction listed above).

Note that this estimator assumes the process is stationary (spatially homogeneous). For inhomogeneous point patterns, see `Kinhom`.

If the point pattern `X` contains more than about 3000 points, the isotropic and translation edge corrections can be computationally prohibitive. The computations for the border method are much faster, and are statistically efficient when there are large numbers of points. Accordingly, if the number of points in `X` exceeds the threshold `nlarge`, then only the border correction will be computed. Setting `nlarge=Inf` or `correction="best"` will prevent this from happening. Setting `nlarge=0` is equivalent to selecting only the border correction with `correction="border"`.

If `X` contains more than about 100,000 points, even the border correction is time-consuming. You may want to consider setting `correction="none"` in this case. There is an even faster algorithm for the uncorrected estimate.

Approximations to the variance of Kest(r) are available, for the case of the isotropic edge correction estimator, assuming complete spatial randomness (Ripley, 1988; Lotwick and Silverman, 1982; Diggle, 2003, pp 51-53). If `var.approx=TRUE`, then the result of `Kest` also has a column named `rip` giving values of Ripley's (1988) approximation to var(Kest(r)), and (if the window is a rectangle) a column named `ls` giving values of Lotwick and Silverman's (1982) approximation.

If the argument `domain` is given, the calculations will be restricted to a subset of the data. In the formula for K(r) above, the first point i will be restricted to lie inside `domain`. The result is an approximately unbiased estimate of K(r) based on pairs of points in which the first point lies inside `domain` and the second point is unrestricted. This is useful in bootstrap techniques. The argument `domain` should be a window (object of class `"owin"`) or something acceptable to `as.owin`. It must be a subset of the window of the point pattern `X`.

The estimator `Kest` ignores marks. Its counterparts for multitype point patterns are `Kcross`, `Kdot`, and for general marked point patterns see `Kmulti`.

Some writers, particularly Stoyan (1994, 1995) advocate the use of the “pair correlation function”

g(r) = K'(r)/ ( 2 * pi * r)

where K'(r) is the derivative of K(r). See `pcf` on how to estimate this function.

### Value

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

Essentially a data frame containing columns

`r` | the vector of values of the argument r at which the function K has been estimated
`theo` | the theoretical value K(r) = pi * r^2 for a stationary Poisson process

together with columns named `"border"`, `"bord.modif"`, `"iso"` and/or `"trans"`, according to the selected edge corrections. These columns contain estimates of the function K(r) obtained by the edge corrections named.

If `var.approx=TRUE` then the return value also has columns `rip` and `ls` containing approximations to the variance of Kest(r) under CSR.

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).

### Envelopes, significance bands and confidence intervals

To compute simulation envelopes for the K-function under CSR, use `envelope`.

To compute a confidence interval for the true K-function, use `varblock` or `lohboot`.

### Warnings

The estimator of K(r) is approximately unbiased for each fixed r, for point processes which do not have very strong interaction. (For point processes with a strong clustering interaction, the estimator is negatively biased; for point processes with a strong inhibitive interaction, the estimator is positively biased.)

Bias increases with r and depends on the window geometry. For a rectangular window it is prudent to restrict the r values to a maximum of 1/4 of the smaller side length of the rectangle (Ripley, 1977, 1988; Diggle, 1983). Bias may become appreciable for point patterns consisting of fewer than 15 points.

While K(r) is always a non-decreasing function, the estimator of K is not guaranteed to be non-decreasing. This is rarely a problem in practice, except for the border correction estimators when the number of points is small.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz

### References

Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and Hall, 1998. Chapter 2, pages 37–78.

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.

Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 – 71.

Ohser, J. and Stoyan, D. (1981) On the second-order and orientation analysis of planar stationary point processes. Biometrical Journal 23, 523–533.

Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society, Series B, 39, 172 – 212.

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

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

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

### See Also

`localK` to extract individual summands in the K function.

`pcf` for the pair correlation.

`Fest`, `Gest`, `Jest` for alternative summary functions.

`Kcross`, `Kdot`, `Kinhom`, `Kmulti` for counterparts of the K function for multitype point patterns.

`reduced.sample` for the calculation of reduced sample estimators.

### Examples

     X <- runifpoint(50)
     K <- Kest(X)
     K <- Kest(cells, correction="isotropic")
     plot(K)
     plot(K, main="K function for cells")
     # plot the L function
     plot(K, sqrt(iso/pi) ~ r)
     plot(K, sqrt(./pi) ~ r, ylab="L(r)", main="L function for cells")

[Package spatstat.core version 2.0-0 Index]