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In [[probability theory]] and [[information theory]], '''adjusted mutual information''', a variation of [[mutual information]] may be used for comparing clusterings.<ref name="vinh-icml09">{{cite doi|10.1145/1553374.1553511}}</ref> It corrects the effect of agreement solely due to chance between clusterings, similar to the way the [[adjusted rand index]] corrects the [[Rand index]]. It is closely related to [[variation of information]]:<ref>{{cite doi|10.1016/j.jmva.2006.11.013}}</ref> when a similar adjustment is made to the VI index, it becomes equivalent to the AMI.<ref name="vinh-icml09" /> The adjusted measure however is no longer metrical.<ref name="vinh-jmlr10">{{Citation
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| title = Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance
| journal = The Journal of Machine Learning Research
| pages = 2837–54 | volume = 11 | issue = oct | year = 2010
| last1 = Vinh | first1 = Nguyen Xuan | last2 = Epps | first2 = Julien | last3 = Bailey | first3 = James
| url=http://jmlr.csail.mit.edu/papers/volume11/vinh10a/vinh10a.pdf}}</ref>
 
==Mutual Information of two Partitions==
Given a set ''S'' of ''N'' elements <math>S=\{s_1, s_2,\ldots s_N\}</math>, consider two [[partition of a set|partitions]] of ''S'', namely <math>U=\{U_1, U_2,\ldots, U_R\}</math> with ''R'' clusters, and <math>V=\{V_1, V_2,\ldots, V_C\}</math> with ''C'' clusters.  It is presumed here that the partitions are so-called ''hard clusters;'' the partitions are pairwise disjoint:
:<math>U_i\cap U_j = V_i\cap V_j = \varnothing</math>
for all <math>i\ne j</math>, and complete:
:<math>\cup_{i=1}^RU_i=\cup_{j=1}^C V_j=S</math>
The [[mutual information]] of cluster overlap between ''U'' and ''V'' can be summarized in the form of an ''R''x''C'' [[contingency table]] <math>M=[n_{ij}]^{i=1 \ldots R}_{j=1 \ldots C}</math>, where <math>n_{ij}</math> denotes the number of objects that are common to clusters <math>U_i</math> and <math>V_j</math>. That is,
 
:<math>n_{ij}=\left|U_i\cap V_j\right|</math>
 
Suppose an object is picked at random from ''S''; the probability that the object falls into cluster <math>U_i</math> is:
:<math>P(i)=\frac{|U_i|}{N}</math>
The [[entropy]] associated with the partitioning ''U'' is:
:<math>H(U)=-\sum_{i=1}^R P(i)\log P(i)</math>
''H(U)'' is non-negative and takes the value 0 only when there is no uncertainty determining an object's cluster membership, ''i.e.'', when there is only one cluster. Similarly, the entropy of the clustering ''V'' can be calculated as:
:<math>H(V)=-\sum_{j=1}^C P'(j)\log P'(j) </math>
where <math>P'(j)=|V_j|/n</math>. The [[mutual information]] (MI) between two partitions{{citation needed|date=September 2011}}:
:<math>MI(U,V)=\sum_{i=1}^R \sum_{j=1}^C P(i,j)\log \frac{P(i,j)}{P(i)P'(j)}</math>
where ''P(i,j)'' denotes the probability that a point belongs to both the cluster <math>U_i</math> in ''U'' and cluster <math>V_j</math> in ''V'':
:<math>P(i,j)=\frac{|U_i \cap V_j|}{N}</math>
MI is a non-negative quantity upper bounded by the entropies ''H''(''U'') and ''H''(''V''). It quantifies the information shared by the two clusterings and thus can be employed as a clustering similarity measure.
 
==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 distribution|hypergeometric]] model of randomness, it can be shown that the expected mutual information between two random clusterings is:
:<math>\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}</math>
where <math>(a_i+b_j-N)^+</math>
denotes <math>\max(0,a_i+b_j-N)</math>.  The variables <math>a_i</math> and <math>b_j</math> are partial sums of the contingency table; that is,
:<math>a_i=\sum_{j=1}^Cn_{ij}</math>
and
:<math>b_j=\sum_{i=1}^Rn_{ij}</math>
 
The adjusted measure<ref name="vinh-icml09"/> for the mutual information may then be defined to be:
:<math> AMI(U,V)= \frac{MI(U,V)-E\{MI(U,V)\}} {\max{\{H(U),H(V)\}}-E\{MI(U,V)\}}
</math>.
 
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.
 
==References==
<references />
 
==External links==
* [http://sites.google.com/site/vinhnguyenx/softwares Matlab code for computing the adjusted mutual information]
 
[[Category:Information theory]]
[[Category:Machine learning]]

Latest revision as of 03:07, 29 April 2014

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