## 1st order Spatial Statistics

### Hopkins' Index of aggregation

Hopkins' index of agregation is the ratio of the distance from a tree to it's nearest neighbor and the distance from random points within the same space to their nearest neighbor tree.

$$Hop_F = \frac{\sum_{i=1}^m d^2_{(p-t_i)i}}{\sum_{i=1}^m d^2_{(t-t_i)i}}$$

where d(p-ti)i is the distance from a random point to it's nearest neighbor tree. d(t-ti)i is the distance from a tree to it's nearest neighbor tree. This test has a F distribution with F(2m,2m) (Hopkins, 1954).

Figure 1. A example of how the Hopkin data is collected. The red dots are trees. The gold dots are random points. We collect the 1st nearest neighbor trees to the point distances and the tree to tree nearest neighbors.

Byth and Ripley (1980) presented standardized index based on this test as:

$$Hop_N = \frac{1}{m} \sum_{i=1}^m \left[ \frac{d^2_{(p-t_i)i}}{(d^2_{(p-t_i)i} + d^2_{(t-t_i)i)}} \right]$$

This index has a Normal null distribution N(1/2,1/12m).

• As HopN approaches 0 it indicates a more "uniform" pattern.
• As HopN approaches 1 it indicates a more "clustered" pattern.
• A value of 0.5 is consider random.

Also See:

Hopkins, B. 1954. A new method for determining the type of distribution of plant individuals. Annals of Botany 18:213-227.
Byth, K. and B. D. Ripley. 1980. Sampling spatial patterns by distance methods. Biometrics 36:279-284.