Kest(X, ..., r=NULL, rmax=NULL, breaks=NULL, correction=c("border", "isotropic", "Ripley", "translate"), nlarge=3000, domain=NULL, var.approx=FALSE, ratio=FALSE)"ppp", or data
in any format acceptable to as.ppp().
rmax.
"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, then only the
border correction will be computed (by default), using a fast algorithm.
TRUE, the approximate
variance of $Kest(r)$ under CSR
will also be computed.
TRUE, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns.
"fv", see fv.object,
which can be plotted directly using plot.fv.Essentially a data frame containing columnstogether 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 $K(r)$.
envelope. To compute a confidence interval for the true $K$-function,
use varblock or lohboot.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 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 $d[i,j]$ is the distance between the two points,
and $I(d[i,j] <= 1="" r)$="" is="" the="" indicator="" that="" equals="" if="" distance="" less="" than="" or="" equal="" to="" $r$.="" term="" $e[i,j]$="" edge="" correction="" weight="" (which="" depends="" on="" choice="" of="" listed="" above).<="" p="">
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''
$$
g(r) = \frac{K'(r)}{2\pi r}
$$
where $K'(r)$ is the derivative of $K(r)$.
See pcf on how to estimate this function.
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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