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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 number of randomly chosen 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.
- Aggarwal, Charu C. (2015). Data Mining. Cham: Springer International Publishing. p. 158. doi:10.1007/978-3-319-14142-8. ISBN 978-3-319-14141-1.