Adjusted Rand index

The adjusted Rand index (86) measures the extent of agreement between two different cluster structures obtained for the same set of data points. This is a useful measure when comparing two methods producing a different number of clusters. The Rand index is simply the proportion of agreement between the two methods and is defined as... 

N is the total number of genes in the data set. The index ranges from 0 (when the two cluster results are completely different) to 1 (when the two methods agree completely). The expectation of the Rand index for two random partitions is not a constant and it is preferable to use the adjusted Rand index, which corrects the Rand index for the case of random partitions of the data. The adjusted Rand index is given by... 

To obtain values for the expected and maximum values, the hypergeometric distribution is assumed. If Uij is the number of genes that are common to clusters i and j from each method, the adjusted Rand index can be calculated from the following formula ... 

The upper bound of the adjusted Rand index is unity and it takes on the value of zero when the similarity between the two clustering methods matches the expectation of the hypergeometric distribution. A higher value of the adjusted Rand index indicates a greater similarity of membership between the clusters of the two methods being compared. 

Notwithstanding its intuitive appeal, the Rand index suffers from several pitfalls. In particular, it highly depends on the number of groups in the two partitions and the number of stimuli at hand. So much so that, in some situations, it can take high values even for two random partitions (Courcoux et al., 2014 Santos and Embrechts, 2009 Youness and Saporta, 2(X)4). In order to cope with this problem, an adjusted Rand index (ARI) was proposed as a form of the Rand index that is corrected for the grouping of the stimuli by chance (Albatineh and Mihalko, 2006 Hubert and Arabie, 1985). 

Santos, J. M. and Embrechts, M. (2009). On the use of the adjusted Rand index as a metric for evaluating supervised classification. Lecture Notes in Computer Science, 5769, 175-184. 

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