TheHopkins statistic (introduced by Brian Hopkins andJohn Gordon Skellam) is a way of measuring thecluster tendency of a data set.^[1] It belongs to the family of sparse sampling tests. It acts as astatistical hypothesis test where thenull hypothesis is that the data is generated by aPoisson point process and are thus uniformly randomly distributed.^[2] If individuals are aggregated, then its value approaches 0, and if they are randomly distributed along the value tends to 0.5.^[3]

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

Let${\displaystyle X}$ be the set of${\displaystyle n}$ data points.

Generate a random sample${\displaystyle \stackrel{\sim }{X}}$ of${\displaystyle m\ll n}$ data points sampled without replacement from${\displaystyle X}$.

Generate a set${\displaystyle Y}$ of${\displaystyle m}$ uniformly randomly distributed data points.

Define two distance measures,

${\displaystyle u_i,}$ the minimum distance (given some suitable metric) of${\displaystyle y_i\in Y}$ to its nearest neighbour in${\displaystyle X}$, and

${\displaystyle w_i,}$ the minimum distance of${\displaystyle {\stackrel{\sim }{x}}_i\in \stackrel{\sim }{X}\subseteq X}$ to its nearest neighbour${\displaystyle x_j\in X,\stackrel{\sim }{x_i}\ne x_j.}$

With the above notation, if the data is${\displaystyle d}$ dimensional, then the Hopkins statistic is defined as:^[4]

${\displaystyle H=\frac{\sum _{i=1}^m{u}_i^d}{\sum _{i=1}^m{u}_i^d+\sum _{i=1}^m{w}_i^d}}$

Under the null hypotheses, this statistic has a Beta(m,m) distribution.

• http://www.sthda.com/english/wiki/assessing-clustering-tendency-a-vital-issue-unsupervised-machine-learning

• Clustering criteria

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