title: Hopkins statistic
text: # 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 ${\overset {\sim }{X}}$ 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 ${\overset {\sim }{x}}_{i}\in {\overset {\sim }{X}}\subseteq X$ to its nearest neighbour $x_{j}\in X,\,{\overset {\sim }{x_{i}}}\neq x_{j}.$
  
  ## Definition
  
  With the above notation, if the data is $d$ dimensional, then the Hopkins statistic is defined as:
  
  $H={\frac {\sum _{i=1}^{m}{u_{i}^{d}}}{\sum _{i=1}^{m}{u_{i}^{d}}+\sum _{i=1}^{m}{w_{i}^{d}}}}\,$
  
  Under the null hypotheses, this statistic has a Beta(m,m) distribution.
  
  