Hopkins statistic

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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.[1] 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.[2] 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[citation needed].

Preliminaries[edit]

A typical formulation of the Hopkins statistic follows.[2]

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 .

Definition[edit]

With the above notation, if the data is dimensional, then the Hopkins statistic is defined as:


Notes and references[edit]

  1. ^ 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.
  2. ^ a b Banerjee, A. (2004). "Validating clusters using the Hopkins statistic". IEEE International Conference on Fuzzy Systems: 149–153. doi:10.1109/FUZZY.2004.1375706.

External links[edit]