Kendall tau distance

The Kendall tau distance is a metric that counts the number of pairwise disagreements between two lists. The larger the distance, the more dissimilar the two lists are. Kendall tau distance is also called bubble-sort distance since it is equivalent to the number of swaps that the bubble sort algorithm would make to place one list in the same order as the other list. The Kendall tau distance was created by Maurice Kendall.

Definition
The Kendall tau distance between two lists $$\tau_1$$ and $$\tau_2$$ is


 * $$K(\tau_1,\tau_2) = |(i,j): i < j, ( \tau_1(i) < \tau_1(j) \wedge \tau_2(i) > \tau_2(j) ) \vee ( \tau_1(i) > \tau_1(j) \wedge \tau_2(i) < \tau_2(j) )|.$$

$$K(\tau_1,\tau_2)$$ will be equal to 0 if the two lists are identical and $$n(n-1)/2$$ (where $$n$$ is the list size) if one list is the reverse of the other. Often Kendall tau distance is normalized by dividing by $$n(n-1)/2$$ so a value of 1 indicates maximum disagreement. The normalized Kendall tau distance therefore lies in the interval [0,1].

Kendall tau distance may also be defined as


 * $$K(\tau_1,\tau_2) = \begin{matrix} \sum_{\{i,j\}\in P} \bar{K}_{i,j}(\tau_1,\tau_2) \end{matrix}$$

where
 * P is the set of unordered pairs of distinct elements in $$\tau_1$$ and $$\tau_2$$
 * $$\bar{K}_{i,j}(\tau_1,\tau_2)$$ = 0 if i and j are in the same order in $$\tau_1$$ and $$\tau_2$$
 * $$\bar{K}_{i,j}(\tau_1,\tau_2)$$ = 1 if i and j are in the opposite order in $$\tau_1$$ and $$\tau_2.$$

Kendall tau distance can also be defined as the total number of discordant pairs.

Kendall tau distance in Rankings: A permutation (or ranking) is an array of N integers where each of the integers between 0 and N-1 appears exactly once. The Kendall tau distance between two rankings is the number of pairs that are in different order in the two rankings. For example the Kendall tau distance between 0 3 1 6 2 5 4 and 1 0 3 6 4 2 5 is four because the pairs 0-1, 3-1, 2-4, 5-4 are in different order in the two rankings, but all other pairs are in the same order.

Example
Suppose we rank a group of five people by height and by weight:

Here person A is tallest and third-heaviest, and so on.

In order to calculate the Kendall tau distance, pair each person with every other person and count the number of times the values in list 1 are in the opposite order of the values in list 2.

Since there are 4 pairs whose values are in opposite order, the Kendall tau distance is 4. The normalized Kendall tau distance is


 * $$\frac{4}{5(5 - 1)/2} = 0.4.$$

A value of 0.4 indicates a somewhat low agreement in the rankings.