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From "devl (JIRA)" <j...@apache.org>
Subject [jira] [Created] (MATH-814) Kendalls Tau Implementation
Date Tue, 10 Jul 2012 19:00:41 GMT
```devl created MATH-814:
-------------------------

Summary: Kendalls Tau Implementation
Key: MATH-814
URL: https://issues.apache.org/jira/browse/MATH-814
Project: Commons Math
Issue Type: New Feature
Affects Versions: 4.0
Environment: All
Reporter: devl
Fix For: 4.0

Implement the Kendall's Tau which is a measure of Association/Correlation between ranked ordinal
data.

A basic description is available at http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient
however the test implementation will follow that defined by "Handbook of Parametric and Nonparametric
Statistical Procedures, Fifth Edition, Page 1393 Test 30, ISBN-10: 1439858012 | ISBN-13: 978-1439858011."

The algorithm is proposed as follows.

Given two rankings or permutations represented by a 2D matrix; columns indicate rankings (e.g.
by an individual) and row are observations of each rank. The algorithm is to calculate the
total number of concordant pairs of ranks (between columns), discordant pairs of ranks  (between
columns) and calculate the Tau defined as

tau= (Number of concordant - number of discordant)/(n(n-1)/2)
where n(n-1)/2 is the total number of possible pairs of ranks.

The method will then output the tau value between -1 and 1 where 1 signifies a "perfect" correlation
between the two ranked lists.

Where ties exist within a ranking it is marked as neither concordant nor discordant in the
calculation. An optional merge sort can be used to speed up the implementation. Details are
in the wiki page.

Although this implementation is not particularly complex it would be useful to have it in
a consistent format in the commons math package in addition to existing correlation tests.
Kendall's Tau is used effectively in comparing ranks for products, rankings from search engines
or measurements from engineering equipment.

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