## 1st order Spatial Statistics

### Holgate Index

The Holgate index is based on the ratio of the distance from a random point to the first and second nearest neighbors. The Holgate sre designed to have the same distributions as the Hopkins' indese.

$$Hol_F = \frac{\sum_{i=1}^m d^2_{(p-t)i}}{\sum_{i=1}^m d^2_{(p-t2)i} - \sum_{i=1}^m d^2_{(p-t)i}}$$

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) (Holgate, 1964).

Figure 1. A example of how the Holgate data is collected. The red dots are trees. The gold dots are random points. We collect the 1st and 2nd nearest neighbor trees to the point distances.
$$Hol_N = \frac{1}{m} \sum_{i=1}^m \left[ \frac{d^2_{(p-t)i}}{(d^2_{(p-t2)i} } \right]$$

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

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

Also See:

Hopkins, B. 1954. A new method for determining the type of distribution of plant individuals. Annals of Botany 18:213-227.
Holgate, P. 1964. The efficiency of nearest neighbor estimators. Biometrics 20:647-649.
Holgate, P. 1965. Some new tests of randomness. Journal of Ecology 53:261-266.
Byth, K. and B. D. Ripley. 1980. Sampling spatial patterns by distance methods. Biometrics 36:279-284.