rknn:
Theoretical Distribution of Nearest Neighbour Distance
Description
Density, distribution function, quantile function and random
generation for the random distance to the $k$th nearest neighbour
in a Poisson point process in $d$ dimensions.
Usage
dknn(x, k = 1, d = 2, lambda = 1)
pknn(q, k = 1, d = 2, lambda = 1)
qknn(p, k = 1, d = 2, lambda = 1)
rknn(n, k = 1, d = 2, lambda = 1)
Arguments
x,q
vector of quantiles.
p
vector of probabilities.
n
number of observations to be generated.
k
order of neighbour.
d
dimension of space.
lambda
intensity of Poisson point process.
Value
A numeric vector:
dknn returns the probability density,
pknn returns cumulative probabilities (distribution function),
qknn returns quantiles,
and rknn generates random deviates.
Details
In a Poisson point process in $d$-dimensional space, let
the random variable $R$ be
the distance from a fixed point to the $k$-th nearest random point,
or the distance from a random point to the
$k$-th nearest other random point.
Then $R^d$ has a Gamma distribution with shape parameter $k$
and rate $lambda * alpha$ where
$alpha$ is a constant (equal to the volume of the
unit ball in $d$-dimensional space).
See e.g. Cressie (1991, page 61).
These functions support calculation and simulation for the
distribution of $R$.
References
Cressie, N.A.C. (1991)
Statistics for spatial data.
John Wiley and Sons, 1991.