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The Hopkins statistic (introduced by Brian Hopkins and John Gordon Skellam) is a way of measuring the cluster tendency of a data set. It belongs to the family of sparse sampling tests. It acts as a statistical hypothesis test where the null hypothesis is that the data is generated by a Poisson point process and are thus uniformly randomly distributed. A value close to 1 tends to indicate the data is highly clustered, random data will tend to result in values around 0.5, and uniformly distributed data will tend to result in values close to 0.
A typical formulation of the Hopkins statistic follows.
- Let be the set of data points.
- Consider a random sample (without replacement) of data points with members .
- Generate a set of uniformly randomly distributed data points.
- Define two distance measures,
- the distance of from its nearest neighbour in , and
- the distance of from its nearest neighbour in .
With the above notation, if the data is dimensional, then the Hopkins statistic is defined as:
Notes and references
- Hopkins, Brian; Skellam, John Gordon (1954). "A new method for determining the type of distribution of plant individuals". Annals of Botany. Annals Botany Co. 18 (2): 213–227.
- Banerjee, A. (2004). "Validating clusters using the Hopkins statistic". IEEE International Conference on Fuzzy Systems: 149–153. doi:10.1109/FUZZY.2004.1375706.