Adjusted Mutual Information - Adjustment For Chance

Adjustment For Chance

Like the Rand index, the baseline value of mutual information between two random clusterings does not take on a constant value, and tends to be larger when the two partitions have a larger number of clusters (with a fixed number of set elements N). By adopting a hypergeometric model of randomness, it can be shown that the expected mutual information between two random clusterings is:

$\begin{align} E\{MI(U,V)\} = & \sum_{i=1}^R \sum_{j=1}^C \sum_{n_{ij}=(a_i+b_j-N)^+}^{\min(a_i, b_j)} \frac{n_{ij}}{N} \log \left( \frac{ N\cdot n_{ij}}{a_i b_j}\right) \times \\ & \frac{a_i!b_j!(N-a_i)!(N-b_j)!} {N!n_{ij}!(a_i-n_{ij})!(b_j-n_{ij})!(N-a_i-b_j+n_{ij})!} \\ \end{align}$

where denotes . The variables and are partial sums of the contingency table; that is,

and

The adjusted measure for the mutual information may then be defined to be:

$AMI(U,V)= \frac{MI(U,V)-E\{MI(U,V)\}} {\max{\{H(U),H(V)\}}-E\{MI(U,V)\}}$ .

The AMI takes a value of 1 when the two partitions are identical and 0 when the MI between two partitions equals to that expected by chance.

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