Rank Correlation

Definition

Rank correlation measures the correspondence between two rankings τ and τ′ of a set of m objects. Various proposals for such measures have been made, especially in the field of statistics. Two of the best-known measures are Spearman’s Rank Correlation and Kendall’s tau:

Spearman’s Rank correlation calculates the sum of squared rank distances and is normalized such that it evaluates to − 1 for reversed and to + 1 for identical rankings. Formally, it is defined as follows:

$$\begin{array}{rlrlrl} (\tau ,\tau ') &\mapsto 1 -\frac{6{\sum }_{i=1}^{m}{(\tau (i) - \tau '(i))}^{2}} {m({m}^{2} - 1)} & \end{array}$$

(1)

Kendall’s tau is the number of pairwise rank inversions between τ and τ′, again normalized to the range [ − 1, + 1]:

$$\begin{array}{rlrlrl} (\tau ,\tau ') &\mapsto 1 -\frac{4\left \vert \{(i,j)\mid i <j,\,\tau (i) <\tau (j)\, \wedge \, \tau '(i)> \tau '(j)\}\right \vert } {m(m - 1)} & \end{array}$$

(2)

Cross References

Preference Learning

ROC Analysis