NAME
Statistics::RankCorrelation - Compute the rank correlation between two vectors
SYNOPSIS
use Statistics::RankCorrelation;
$x = [ 8, 7, 6, 5, 4, 3, 2, 1 ];
$y = [ 2, 1, 5, 3, 4, 7, 8, 6 ];
$c = Statistics::RankCorrelation->new( $x, $y, sorted => 1 );
$s = $c->size;
$xd = $c->x_data;
$yd = $c->y_data;
$xr = $c->x_rank;
$yr = $c->y_rank;
$n = $c->spearman;
$m = $c->csim;
$t = $c->kendall_tau;DESCRIPTION
This module computes rank correlation coefficient measures between two sample vectors.
Working examples may be found in the distribution eg/ directory and the module test file.
Also the HANDY FUNCTIONS section below has some ..handy functions to use when computing sorted rank coefficients by hand.
METHODS
new
$c = Statistics::RankCorrelation->new( \@u, \@v );This method constructs a new Statistics::RankCorrelation object.
If given two numeric vectors (as array references), the object is initialized by computing the statistical ranks of the vectors. If they are of different cardinality the shorter vector is first padded with trailing zeros.
x_data
$c->x_data( $y );
$x = $c->x_data;Return and set the one dimensional array reference data. This is the "unit" array, used as a reference for size and iteration.
y_data
$c->y_data( $y );
$x = $c->y_data;Return and set the one dimensional array reference data. This vector is dependent on the unit vector.
size
$c->size( $s );
$s = $c->size;Return and set the number of elements in the unit array.
x_rank
$c->x_rank( $rx );
$rx = $c->x_rank;Return and set the ranked "unit" data as an array reference.
y_rank
$ry = $c->y_rank;
$c->y_rank( $ry );Return (and optionally set) the ranked data as array references.
spearman
$n = $c->spearman;Spearman's rho rank-order correlation is a nonparametric measure of association based on the rank of the data values and is a special case of the Pearson product-moment correlation.
The formula is:
6 * sum( (Xi - Yi)^2 )
1 - --------------------------
n (n^2 - 1)Where X and Y are the two rank vectors and i is an index from one to n number of samples.
kendall_tau
$t = $c->kendall_tau;
4P
t = --------- - 1
n (n - 1)csim
$n = $c->csim;Return the contour similarity index measure. This is a single dimensional measure of the similarity between two vectors.
This returns a measure in the (inclusive) range [-1..1] and is computed using matrices of binary data representing "higher or lower" values in the original vectors.
This measure has been studied in musical contour analysis.
FUNCTIONS
rank
$ranks = rank( [ 1.0, 3.2, 2.1, 3.2, 3.2, 4.3 ] );
# [1, 4, 2, 4, 4, 6]Return an array reference of the ordinal ranks of the given data.
Note that the data must be sorted as measurement pairs prior to computing the statistical rank. This is done automatically by the object initialization method.
In the case of a tie in the data (identical values) the rank numbers are averaged. An example will elucidate:
sorted data: [ 1.0, 2.1, 3.2, 3.2, 3.2, 4.3 ]
ranks: [ 1, 2, 3, 4, 5, 6 ]
tied ranks: 3, 4, and 5
tied average: (3 + 4 + 5) / 3 == 4
averaged ranks: [ 1, 2, 4, 4, 4, 6 ]pad_vectors
( $u, $v ) = pad_vectors( [ 1, 2, 3, 4 ], [ 9, 8 ] );
# [1, 2, 3, 4], [9, 8, 0, 0]Append zeros to either input vector for all values in the other that do not have a corresponding value. That is, "pad" the tail of the shorter vector with zero values.
co_sort
( $u, $v ) = co_sort( $u, $v );Sort the vectors as two dimensional data-point pairs with u values sorted first.
correlation_matrix
$matrix = correlation_matrix( $u );Return the correlation matrix for a single vector.
This function builds a square, binary matrix that represents "higher or lower" value within the vector itself.
TO DO
Figure out what "> 0.5 discrepency" that Vladimir Babenko <Vl_Babenko@softhome.net> saw.
Handle any number of vectors instead of just two.
Implement other rank correlation measures that are out there...
SEE ALSO
For the csim method:
http://www2.mdanderson.org/app/ilya/Publications/JNMRcontour.pdf
For the spearman and kendall_tau methods:
http://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html
http://en.wikipedia.org/wiki/Kendall's_tau
THANK YOU
Thomas Breslin <thomas@thep.lu.se> for unsorted rank code.
Jerome <jerome.hert@free.fr> for Kendall's tau
AUTHOR
Gene Boggs <gene@cpan.org>
COPYRIGHT
Copyright 2003, Gene Boggs
LICENSE
This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.