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, the value tends to 0.5.

Preliminaries

A typical formulation of the Hopkins statistic follows.
:Let $X$ be the set of $n$ data points.
:Generate a random sample $\overset$ of $m \ll n$ data points sampled without replacement from $X$.
:Generate a set $Y$ of $m$ uniformly randomly distributed data points.
:Define two distance measures,
::$u_i,$ the minimum distance (given some suitable metric) of $y_i \in Y$ to its nearest neighbour in $X$, and
::$w_i,$ the minimum distance of $\overset_i \in \overset\subseteq X$ to its nearest neighbour $x_j \in X,\, \overset\ne x_j.$

Definition

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

$H=\frac \,$

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

Notes and references

External links

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

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