Kendall tau distance

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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.

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[edit] Definition

The Kendall tau distance between two lists τ1 and τ2 is

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

K(τ1,τ2) 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 τ1 and τ2
  • \bar{K}_{i,j}(\tau_1,\tau_2) = 0 if i and j are in the same order in τ1 and τ2
  • \bar{K}_{i,j}(\tau_1,\tau_2) = 1 if i and j are in the opposite order in τ1 and τ2

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

[edit] Example

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

Person A B C D E
Rank by Height 1 2 3 4 5
Rank by Weight 3 4 1 2 5

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.

Pair Height Weight Count
(A,B) 1 < 2 3 < 4
(A,C) 1 < 3 3 > 1 X
(A,D) 1 < 4 3 > 2 X
(A,E) 1 < 5 3 < 5
(B,C) 2 < 3 4 > 1 X
(B,D) 2 < 4 4 > 2 X
(B,E) 2 < 5 4 < 5
(C,D) 3 < 4 1 < 2
(C,E) 3 < 5 1 < 5
(D,E) 4 < 5 2 < 5

Since there are 4 pairs that the 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.

[edit] See also

[edit] References

  • Fagin, R., Kumar, R., and Sivakumar, D. (2003). "Comparing top k lists". SIAM Journal on Discrete Mathematics: 134-160. 
  • Kendall, M. (1948) Rank Correlation Methods, Charles Griffin & Company Limited
  • Kendall, M. (1938) "A New Measure of Rank Correlation", Biometrica, 30, 81-89.

[edit] External links

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