spatstat (version 1.20-1)

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.

Examples

x <- seq(0, 5, length=20)
  densities <- dknn(x, k=3, d=2)
  cdfvalues <- pknn(x, k=3, d=2)
  randomvalues <- rknn(100, k=3, d=2)
  deciles <- qknn((1:9)/10, k=3, d=2)