# 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 $n$ elements $S = \{o_1, \ldots, o_n\}$ and two partitions of $S$ to compare, $X = \{X_1, \ldots, X_r\}$, a partition of S into r subsets, and $Y = \{Y_1, \ldots, Y_s\}$, a partition of S into s subsets, 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:[1][2]

$R = \frac{a+b}{a+b+c+d} = \frac{a+b}{{n \choose 2 }}$

Intuitively, $a + b$ can be considered as the number of agreements between $X$ and $Y$ and $c + d$ as the number of disagreements between $X$ and $Y$.

### 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:

• $a = |S^{*}|$, where $S^{*} = \{ (o_{i}, o_{j}) | o_{i}, o_{j} \in X_{k}, o_{i}, o_{j} \in Y_{l}\}$
• $b = |S^{*}|$, where $S^{*} = \{ (o_{i}, o_{j}) | o_{i} \in X_{k_{1}}, o_{j} \in X_{k_{2}}, o_{i} \in Y_{l_{1}}, o_{j} \in Y_{l_{2}}\}$
• $c = |S^{*}|$, where $S^{*} = \{ (o_{i}, o_{j}) | o_{i}, o_{j} \in X_{k}, o_{i} \in Y_{l_{1}}, o_{j} \in Y_{l_{2}}\}$
• $d = |S^{*}|$, where $S^{*} = \{ (o_{i}, o_{j}) | o_{i} \in X_{k_{1}}, o_{j} \in X_{k_{2}}, o_{i}, o_{j} \in Y_{l}\}$

for some $1 \leq i,j \leq n, i \neq j, 1 \leq k, k_{1}, k_{2} \leq r, k_{1} \neq k_{2}, 1 \leq l, l_{1},l_{2} \leq s, l_{1} \neq l_{2}$

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 $S$ of $n$ elements, and two groupings (e.g. clusterings) of these points, namely $X = \{ X_1, X_2, \ldots , X_r \}$ and $Y = \{ Y_1, Y_2, \ldots , Y_s \}$, the overlap between $X$ and $Y$ can be summarized in a contingency table $\left[n_{ij}\right]$ where each entry $n_{ij}$ denotes the number of objects in common between $X_i$ and $Y_j$ : $n_{ij}=|X_i \cap Y_j|$.

X\Y $Y_1$ $Y_2$ $\ldots$ $Y_s$ Sums
$X_1$ $n_{11}$ $n_{12}$ $\ldots$ $n_{1s}$ $a_1$
$X_2$ $n_{21}$ $n_{22}$ $\ldots$ $n_{2s}$ $a_2$
$\vdots$ $\vdots$ $\vdots$ $\ddots$ $\vdots$ $\vdots$
$X_r$ $n_{r1}$ $n_{r2}$ $\ldots$ $n_{rs}$ $a_r$
Sums $b_1$ $b_2$ $\ldots$ $b_s$

### Definition

The adjusted form of the Rand Index, the Adjusted Rand Index, is $AdjustedIndex = \frac{Index - ExpectedIndex}{MaxIndex - ExpectedIndex}$, more specifically
$ARI = \frac{ \sum_{ij} \binom{n_{ij}}{2} - [\sum_i \binom{a_i}{2} \sum_j \binom{b_j}{2}] / \binom{n}{2} }{ \frac{1}{2} [\sum_i \binom{a_i}{2} + \sum_j \binom{b_j}{2}] - [\sum_i \binom{a_i}{2} \sum_j \binom{b_j}{2}] / \binom{n}{2} }$
where $n_{ij}, a_i, b_j$ are values from the contingency table.

## References

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?". 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