Evaluation method in cluster analysis
The Fowlkes–Mallows index is an external evaluation method that is used to determine the similarity between two clusterings (clusters obtained after a clustering algorithm), and also a metric to measure confusion matrices. This measure of similarity could be either between two hierarchical clusterings or a clustering and a benchmark classification. A higher value for the Fowlkes–Mallows index indicates a greater similarity between the clusters and the benchmark classifications. It was invented by Bell Labs statisticians Edward Fowlkes and Collin Mallows in 1983.
Preliminaries.
The Fowlkes–Mallows index, when results of two clustering algorithms are used to evaluate the results, is defined as
formula_1
where formula_2 is the number of true positives, formula_3 is the number of false positives, and formula_4 is the number of false negatives. formula_5 is the "true positive rate", also called "sensitivity" or "recall", and formula_6 is the "positive predictive rate", also known as "precision".
The minimum possible value of the Fowlkes–Mallows index is 0, which corresponds to the worst binary classification possible, where all the elements have been misclassified. And the maximum possible value of the Fowlkes–Mallows index is 1, which corresponds to the best binary classification possible, where all the elements have been perfectly classified.
Definition.
Consider two hierarchical clusterings of formula_7 objects labeled formula_8 and formula_9. The trees formula_8 and formula_9 can be cut to produce formula_12 clusters for each tree (by either selecting clusters at a particular height of the tree or setting different strength of the hierarchical clustering). For each value of formula_13, the following table can then be created
formula_14
where formula_15 is of objects common between the formula_16th cluster of formula_8 and formula_18th cluster of formula_9. The Fowlkes–Mallows index for the specific value of formula_13 is then defined as
 formula_21
where
formula_22
formula_23
formula_24
formula_25 can then be calculated for every value of formula_13 and the similarity between the two clusterings can be shown by plotting formula_25 versus formula_13. For each formula_13 we have formula_30.
Fowlkes–Mallows index can also be defined based on the number of points that are common or uncommon in the two hierarchical clusterings. If we define
formula_2 as the number of pairs of points that are present in the same cluster in both formula_8 and formula_9.
formula_3 as the number of pairs of points that are present in the same cluster in formula_8 but not in formula_9.
formula_4 as the number of pairs of points that are present in the same cluster in formula_9 but not in formula_8.
formula_40 as the number of pairs of points that are in different clusters in both formula_8 and formula_9.
Each pair of points is counted in exactly one of formula_2, formula_3, formula_4, or formula_40, so the sum of these equals the total number of pairs: 
formula_47
The Fowlkes–Mallows index for two clusterings can be defined as
formula_48
where formula_2 is the number of true positives, formula_3 is the number of false positives, and formula_4 is the number of false negatives.
formula_5 is the "true positive rate", also called "sensitivity" or "recall", and formula_6 is the "positive predictive rate", also known as "precision".
The Fowlkes–Mallows index is the geometric mean of precision and recall.
Discussion.
Since the index is directly proportional to the number of true positives, a higher index means greater similarity between the two clusterings used to determine the index. One basic way to test the validity of this index is to compare two clusterings that are unrelated to each other. Fowlkes and Mallows showed that on using two unrelated clusterings, the value of this index approaches zero as the number of total data points chosen for clustering increase; whereas the value for the Rand index for the same data quickly approaches formula_54 making Fowlkes–Mallows index a much more accurate representation for unrelated data. This index also performs well if noise is added to an existing dataset and their similarity compared. Fowlkes and Mallows showed that the value of the index decreases as the component of the noise increases. The index also showed similarity even when the noisy dataset had a different number of clusters than the clusters of the original dataset. Thus making it a reliable tool for measuring similarity between two clusters.
References.
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