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The R Book by Michael J. Crawley

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Tests for Spatial Randomness

Clark and Evans (1954) give a very simple test of spatial randomness. Making the strong assumption that you know the population density of the individuals, ρ (generally you do not know this, and would need to estimate it independently), then the expected mean distance to the nearest neighbour is

images

In our example we have 100 individuals in a unit square, so ρ = 0.01 and E(r) = 0.05. The actual mean nearest neighbour distance was

mean(r)
[1] 0.05404338

which is very close to expectation: this clearly is a random distribution of individuals (as we constructed it to be). An index of randomness is given by the ratio images. This takes the value 1 for random patterns, more than 1 for regular (spaced-out) patterns, and less than 1 for aggregated patterns.

One problem with such first-order estimates of spatial pattern (including measures such as the variance–mean ratio) is that they can give no feel for the way that spatial distribution changes within an area.

Ripley's K

The second-order properties of a spatial point process describe the way that spatial interactions change through space. These are computationally intensive measures that take a range of distances within the area, calculate a pattern measure, then plot a graph of the function against distance, to show ...

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