Rand index

In statistics, the Rand index or Rand measure is a measure of the similarity between two data clusters.

Definition

Given a set of $n$ elements $S\; =\; \{O\_1,\; ldots,\; O\_n\}$ and two partitions of $S$ to compare, $X\; =\; \{x\_1,\; ldots,\; x\_r\}$ and $Y\; =\; \{y\_1,\; ldots,\; y\_s\}$, we define the following:
* $a$, the number of pairs of elements in $S$ that are in the same set in $X$ and in the same set in $Y$
* $b$, the number of pairs of elements in $S$ that are in different sets in $X$ and in different sets in $Y$
* $c$, the number of pairs of elements in $S$ that are in the same set in $X$ and in different sets in $Y$
* $d$, the number of pairs of elements in $S$ that are in different sets in $X$ and in the same set in $Y$The Rand index, $R$, is::$R\; =\; frac\{a+b\}\{a+b+c+d\}\; =\; frac\{a+b\}$n choose 2 Intuitively, one can think of $a\; +\; b$ as the number of agreements between $X$ and $Y$ and $c\; +\; d$ as the number of disagreements between $X$ and $Y$.

The Rand index has a value between 0 and 1, with 0 indicating that the two data clusters do not agree on any pair of points and 1 indicating that the data clusters are exactly the same.

References

* Cite journal
author = W. M. Rand
title = Objective criteria for the evaluation of clustering methods
journal = Journal of the American Statistical Association
volume = 66
pages = 846–850
year = 1971
doi = 10.2307/2284239
* Cite journal
author = K. Y. Yeung & W. L. Ruzzo
title = Principal component analysis for clustering gene expression data
journal = Bioinformatics
volume = 17
issue = 9
year = 2001
pages = 763–774
url = http://bioinformatics.oxfordjournals.org/cgi/content/abstract/17/9/763
doi = 10.1093/bioinformatics/17.9.763