R: Hopkins-Skellam Test

hopskel {spatstat} | R Documentation

## Hopkins-Skellam Test

### Description

Perform the Hopkins-Skellam test of Complete Spatial Randomness, or simply calculate the test statistic.

### Usage

    hopskel(X)

    hopskel.test(X, ...,
                 alternative=c("two.sided", "less", "greater",
                               "clustered", "regular"),
                 method=c("asymptotic", "MonteCarlo"),
                 nsim=999)

### Arguments

`X` | Point pattern (object of class `"ppp"`).
`alternative` | String indicating the type of alternative for the hypothesis test. Partially matched.
`method` | Method of performing the test. Partially matched.
`nsim` | Number of Monte Carlo simulations to perform, if a Monte Carlo p-value is required.
`...` | Ignored.

### Details

Hopkins and Skellam (1954) proposed a test of Complete Spatial Randomness based on comparing nearest-neighbour distances with point-event distances.

If the point pattern `X` contains `n` points, we first compute the nearest-neighbour distances P[1], ..., P[n] so that P[i] is the distance from the ith data point to the nearest other data point. Then we generate another completely random pattern `U` with the same number `n` of points, and compute for each point of `U` the distance to the nearest point of `X`, giving distances I[1], ..., I[n]. The test statistic is

A = (sum[i] P[i]^2) / (sum[i] I[i]^2)

The null distribution of A is roughly an F distribution with shape parameters (2n,2n). (This is equivalent to using the test statistic H=A/(1+A) and referring H to the Beta distribution with parameters (n,n)).

The function `hopskel` calculates the Hopkins-Skellam test statistic A, and returns its numeric value. This can be used as a simple summary of spatial pattern: the value H=1 is consistent with Complete Spatial Randomness, while values H < 1 are consistent with spatial clustering, and values H > 1 are consistent with spatial regularity.

The function `hopskel.test` performs the test. If `method="asymptotic"` (the default), the test statistic H is referred to the F distribution. If `method="MonteCarlo"`, a Monte Carlo test is performed using `nsim` simulated point patterns.

### Value

The value of `hopskel` is a single number.

The value of `hopskel.test` is an object of class `"htest"` representing the outcome of the test. It can be printed.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### References

Hopkins, B. and Skellam, J.G. (1954) A new method of determining the type of distribution of plant individuals. Annals of Botany 18, 213–227.

### See Also

`clarkevans`, `clarkevans.test`, `nndist`, `nncross`

### Examples

      hopskel(redwood)
      hopskel.test(redwood, alternative="clustered")

[Package spatstat version 1.63-3 Index]