Rand index

From formulasearchengine
Jump to navigation Jump to search

The Rand index[1] or Rand measure (named after William M. Rand) in statistics, and in particular in data clustering, is a measure of the similarity between two data clusterings. A form of the Rand index may be defined that is adjusted for the chance grouping of elements, this is the adjusted Rand index. From a mathematical standpoint, Rand index is related to the accuracy, but is applicable even when class labels are not used.

Rand index


Given a set of elements and two partitions of to compare, , a partition of S into r subsets, and , a partition of S into s subsets, define the following:

The Rand index, , is:[1][2]

Intuitively, can be considered as the number of agreements between and and as the number of disagreements between and .


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.

In mathematical terms, a, b, c, d are defined as follows:

for some

Adjusted Rand index

The adjusted Rand index is the corrected-for-chance version of the Rand index.[1][2][3] Though the Rand Index may only yield a value between 0 and +1, the Adjusted Rand Index can yield negative values if the index is less than the expected index.[4]

The contingency table

Given a set of elements, and two groupings (e.g. clusterings) of these points, namely and , the overlap between and can be summarized in a contingency table where each entry denotes the number of objects in common between and  : .

X\Y Sums


The adjusted form of the Rand Index, the Adjusted Rand Index, is , more specifically

where are values from the contingency table.


  1. 1.0 1.1 1.2 {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  2. 2.0 2.1 {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  3. {{#invoke:citation/CS1|citation |CitationClass=conference }}PDF.
  4. http://i11www.iti.uni-karlsruhe.de/extra/publications/ww-cco-06.pdf

External links