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

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 . 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.

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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