Rand index

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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[edit]

Definition[edit]

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 subset in and in the same subset in
  • , the number of pairs of elements in that are in different subsets in and in different subsets in
  • , the number of pairs of elements in that are in the same subset in and in different subsets in
  • , the number of pairs of elements in that are in different subsets in and in the same subset 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 .

Since the denominator is the total number of pairs, the Rand index represents the frequency of occurrence of agreements over the total pairs, or the probability that and will agree on a randomly chosen pair.


Properties[edit]

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

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

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

for some

Adjusted Rand index[edit]

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[edit]

Given a set of elements, and two groupings or partitions (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[edit]

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

where are values from the contingency table.

References[edit]

  1. ^ a b c 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. ^ a b 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

External links[edit]