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
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
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
:
.
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Sums |
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Sums | ![]() |
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Definition
The adjusted form of the Rand Index, the Adjusted Rand Index, is , more specifically
where are values from the contingency table.
References
- 1 2 3 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.
- 1 2 Lawrence Hubert and Phipps Arabie (1985). "Comparing partitions". Journal of Classification 2 (1): 193–218. doi:10.1007/BF01908075.
- ↑ Nguyen Xuan Vinh, Julien Epps and James Bailey (2009). PDF. "Information Theoretic Measures for Clustering Comparison: Is a Correction for Chance Necessary?" Check
value (help) (PDF). ICML '09: Proceedings of the 26th Annual International Conference on Machine Learning. ACM. pp. 1073–1080.PDF.|URL=
- ↑ http://i11www.iti.uni-karlsruhe.de/extra/publications/ww-cco-06.pdf