# 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

```
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 1.42-2, License: GPL (>= 2)*