# A protocol for Hopkins statistic

#### 2022-01-06

Hopkins statistic is used to test a Null Hypothesis of spatial randomness. Under the null distribution of spatial randomness, the value of the statistic should be 0.5.

# A protocol for using Hopkins statistic

1. Does the data need to be scaled?
2. Is the number of events $$n > 100$$ and the number of randomly-sampled events at most 10% of $$n$$? This is recommended by Cross and Jain (1982).
3. Is spatial randomness of the events even possible? If the events are known or suspected to be correlated, this violates the null hypothesis of spatial uniformity, and may also mean that the sampling frame is not shaped like a hypercube.
4. Could nearest-neighbor events have occurred outside the boundary of the sampling frame? If yes, it may be appropriate to calculate nearest-neighbor distances using a torus geometry.
5. Is the sampling frame non-rectangular? If yes, then be extremely careful with the use of Hopkins statistic in how points are samples from $$U$$.
6. Is the dimension of the data much greater than 2? Edge effects are more common in higher dimensions.

The important point of this protocol is to raise awareness of potential problems. We leave it to the practitioner to decide what do with the answers to these questions.

### Example 1

We can simulate 1000 points uniformly in a unit square and then calculate Hopkins statistic, which is 0.52.

library(hopkins)
set.seed(42)
dat1 <- matrix(runif(2000), ncol=2)
plot(dat1)

hopkins(dat1) # .52
## [1] 0.5217802

### Example 2

Simulate 1000 points from a bivariate normal distribution (with 0 covariance). The sampling frame for generating new points $$u$$ is from the minimum value to maximum value of the events in each axis. Roughly -3 to 3 for Normal data. The points form a circular “cluster” within this bounding box and the value of Hopkins statistic is 0.89.

set.seed(42)
dat2 <- matrix(rnorm(1000), ncol=2)
plot(dat2)

hopkins(dat2) # .89
## [1] 0.8927865
Cross, GR, and AK Jain. 1982. “Measurement of Clustering Tendency.” In Theory and Application of Digital Control, 315–20. Elsevier. https://doi.org/10.1016/B978-0-08-027618-2.50054-1.