# Rand index

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

### Definition

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 number of pairs of elements in that are in the same set in and in the same set in
• , the number of pairs of elements in that are in different sets in and in different sets in
• , the number of pairs of elements in that are in the same set in and in different sets in
• , the number of pairs of elements in that are in different sets in and in the same set in

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 .

### Properties

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:

• , where
• , where
• , where
• , where

for some

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
Sums

### Definition

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

where are values from the contingency table.

## References

1. W. M. Rand (1971). "Objective criteria for the evaluation of clustering methods". Journal of the American Statistical Association. American Statistical Association. 66 (336): 846–850. doi:10.2307/2284239. JSTOR 2284239.
2. Lawrence Hubert and Phipps Arabie (1985). "Comparing partitions". Journal of Classification. 2 (1): 193–218. doi:10.1007/BF01908075.
3. Nguyen Xuan Vinh, Julien Epps and James Bailey (2009). PDF. "Information Theoretic Measures for Clustering Comparison: Is a Correction for Chance Necessary?" Check |URL= value (help) (PDF). ICML '09: Proceedings of the 26th Annual International Conference on Machine Learning. ACM. pp. 1073–1080.PDF.
4. http://i11www.iti.uni-karlsruhe.de/extra/publications/ww-cco-06.pdf