# rknn

##### Theoretical Distribution of Nearest Neighbour Distance

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.

- Keywords
- distribution, spatial

##### 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.

##### 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 \(R\).

##### Value

A numeric vector:
`dknn`

returns the probability density,
`pknn`

returns cumulative probabilities (distribution function),
`qknn`

returns quantiles,
and `rknn`

generates random deviates.

##### 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)
# }
```

*Documentation reproduced from package spatstat, version 1.57-1, License: GPL (>= 2)*