spatstat (version 1.59-0)

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

# NOT RUN {
  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)
# }

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