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| The '''Kendall tau rank distance''' is a [[Metric (mathematics)|metric]] that counts the number of pairwise disagreements between two ranking 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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| ==Definition==
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| The Kendall tau ranking distance between two lists <math>L1</math> and <math>L2</math> is
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| : <math>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) )\}|.</math>
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| where
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| * <math>\tau_1</math> and <math>\tau_2</math> are the rankings of the elements in <math>L1</math> and <math>L2</math>
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| <math>K(\tau_1,\tau_2)</math> will be equal to 0 if the two lists are identical and <math>n(n-1)/2</math> (where <math>n</math> is the list size) if one list is the reverse of the other. Often Kendall tau distance is normalized by dividing by <math>n(n-1)/2</math> so a value of 1 indicates maximum disagreement. The normalized Kendall tau distance therefore lies in the interval [0,1].
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| Kendall tau distance may also be defined as
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| : <math>K(\tau_1,\tau_2) = \begin{matrix} \sum_{\{i,j\}\in P} \bar{K}_{i,j}(\tau_1,\tau_2) \end{matrix}</math>
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| where
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| * ''P'' is the set of unordered pairs of distinct elements in <math>\tau_1</math> and <math>\tau_2</math>
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| * <math>\bar{K}_{i,j}(\tau_1,\tau_2)</math> = 0 if ''i'' and ''j'' are in the same order in <math>\tau_1</math> and <math>\tau_2</math>
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| * <math>\bar{K}_{i,j}(\tau_1,\tau_2)</math> = 1 if ''i'' and ''j'' are in the opposite order in <math>\tau_1</math> and <math>\tau_2.</math>
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| Kendall tau distance can also be defined as the total number of [[discordant pairs]].
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| 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.
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| 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. <ref>http://algs4.cs.princeton.edu/25applications/</ref>
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| If Kendall tau function is performed as <math>K(L1,L2)</math> instead of <math>K(\tau_1,\tau_2)</math> (where <math>\tau_1</math> and <math>\tau_2</math> are the rankings of <math>L1</math> and <math>L2</math> elements respectively), then triangular inequality is not guaranteed. The triangular inequality fail in cases where there are repetitions in the lists. So then we are not any more dealing with a metric.
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| ==Example==
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| Suppose we rank a group of five people by height and by weight:
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| {| border="1" cellpadding="2"
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| ! Person !! A !! B !! C !! D !! E
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| ! Rank by Height
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| | 1 || 2 || 3 || 4 || 5
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| |-
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| ! Rank by Weight
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| | 3 || 4 || 1 || 2 || 5
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| |}
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| Here person A is tallest and third-heaviest, and so on. | |
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| 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.
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| {| border="1" cellpadding="2"
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| ! Pair !! Height !! Weight !! Count
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| ! (A,B)
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| | 1 < 2 || 3 < 4 ||
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| |-
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| ! (A,C)
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| | 1 < 3 || 3 > 1 || '''X'''
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| |-
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| ! (A,D)
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| | 1 < 4 || 3 > 2 || '''X'''
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| |-
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| ! (A,E)
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| | 1 < 5 || 3 < 5 ||
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| |-
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| ! (B,C)
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| | 2 < 3 || 4 > 1 || '''X'''
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| |-
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| ! (B,D)
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| | 2 < 4 || 4 > 2 || '''X'''
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| |-
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| ! (B,E)
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| | 2 < 5 || 4 < 5 ||
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| |-
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| ! (C,D)
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| | 3 < 4 || 1 < 2 ||
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| |-
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| ! (C,E)
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| | 3 < 5 || 1 < 5 ||
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| |-
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| ! (D,E)
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| | 4 < 5 || 2 < 5 ||
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| |}
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| Since there are 4 pairs whose values are in opposite order, the Kendall tau distance is 4. The normalized Kendall tau distance is
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| : <math>\frac{6-4}{5(5 - 1)/2} = 0.2.</math>
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| A value of 0.2 indicates a somewhat low agreement in the rankings.
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| ==See also==
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| * [[Kendall tau rank correlation coefficient]]
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| * [[Spearman's rank correlation coefficient]]
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| * [[Kemeny-Young method|Kemeny-Young (`maximum likelihood') voting rule]]
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| ==References==
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| {{reflist}}
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| * {{cite journal | author = Fagin, R., Kumar, R., and Sivakumar, D. | year = 2003 | title = Comparing top k lists | journal = [[SIAM Journal on Discrete Mathematics]] | pages = 134–160 | url = http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.86.3234&rep=rep1&type=pdf | issue = 1 | doi = 10.1137/S0895480102412856 | volume = 17}}
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| * Kendall, M. (1948) ''Rank Correlation Methods'', Charles Griffin & Company Limited
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| * Kendall, M. (1938) "A New Measure of Rank Correlation", [[Biometrika]], 30, 81-89.
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| ==External links==
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| * [http://www.rsscse-edu.org.uk/tsj/bts/noether/text.html Why Kendall tau?]
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| * [http://www.wessa.net/rwasp_kendall.wasp Online software: computes Kendall's tau rank correlation]
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| [[Category:Covariance and correlation]]
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| [[Category:Statistical distance measures]]
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| [[Category:Comparison of assessments]]
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Friends call him Royal. Playing croquet is something I will never give up. Interviewing is how I make a living and it's something I truly appreciate. Her family life in Idaho.
Here is my web page 81.30.156.125