Hopkins statistic

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. If individuals are aggregated, then its value approaches 0, and if they are randomly distributed along the value tends to 0.5.

A typical formulation of the Hopkins statistic follows.

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

Generate a random sample ${\displaystyle {\overset {\sim }{X}}}$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 ${\displaystyle {\overset {\sim }{x}}_{i}\in {\overset {\sim }{X}}\subseteq X}$to its nearest neighbour ${\displaystyle x_{j}\in X,\,{\overset {\sim }{x_{i}}}\neq x_{j}.}$

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

${\displaystyle H={\frac {\sum _{i=1}^{m}{u_{i}^{d}}}{\sum _{i=1}^{m}{u_{i}^{d}}+\sum _{i=1}^{m}{w_{i}^{d}}}}\,}$

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

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