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.^[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] If individuals are aggregated, then its value approaches 1, and if they are randomly distributed along the value tends to 0.5.^[3]

Preliminaries

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

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

Generate a random sample ${\displaystyle {\overset {\sim }{X}}}$ 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 ${\displaystyle x_{j}\in X,\,{\overset {\sim }{x_{i}}}\neq x_{j}.}$

Definition

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.