Estimates Ripley's reduced second moment function \(K(r)\) from a point pattern in a window of arbitrary shape.
Kest(X, ..., r=NULL, rmax=NULL, breaks=NULL,
correction=c("border", "isotropic", "Ripley", "translate"),
nlarge=3000, domain=NULL, var.approx=FALSE, ratio=FALSE)An object of class "fv", see fv.object,
which can be plotted directly using plot.fv.
Essentially a data frame containing columns
the vector of values of the argument \(r\) at which the function \(K\) has been estimated
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 \(\hat K(r)\) 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 \(K(r)\).
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.
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.
Optional. Maximum desired value of the argument \(r\).
This argument is for internal use only.
Optional. A character vector containing any selection of the
options "none", "border", "bord.modif",
"isotropic", "Ripley", "translate",
"translation", "rigid",
"none", "periodic", "good" or "best".
It specifies the edge correction(s) to be applied.
Alternatively correction="all" selects all options.
Optional. Efficiency threshold.
If the number of points exceeds nlarge, then only the
border correction will be computed (by default), using a fast algorithm.
Optional. Calculations will be restricted to this subset of the window. See Details.
Logical. If TRUE, the approximate
variance of \(\hat K(r)\) under CSR
will also be computed.
Logical.
If TRUE, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns.
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.
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.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz
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
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.
Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented for rectangular and polygonal windows (not for binary masks).
Translation correction (Ohser, 1983). Implemented for all window geometries, but slow for complex windows.
Rigid motion correction (Ohser and Stoyan, 1981). Implemented for all window geometries, but slow for complex windows.
Uncorrected estimate. An estimate of the K function without edge correction. (i.e. setting \(e_{ij} = 1\) 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).
Periodic (toroidal) edge correction. Defined only for rectangular windows.
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.
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
$$
\hat K(r) = \frac a {n (n-1) } \sum_i \sum_j I(d_{ij}\le r) e_{ij}
$$
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 \(d_{ij}\) is the distance between the two points,
and \(I(d_{ij} \le r)\) is the indicator
that equals 1 if the distance is less than or equal to \(r\).
The term \(e_{ij}\) 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 \(\hat K(r)\)
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
\(\mbox{var}(\hat K(r))\),
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''
$$
g(r) = \frac{K'(r)}{2\pi r}
$$
where \(K'(r)\) is the derivative of \(K(r)\).
See pcf on how to estimate this function.
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.
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.
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")
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