NAME
Algorithm::RandomMatrixGeneration - Perl module to generate matrix given
the marginals.
SYNOPSIS
use Algorithm::RandomMatrixGeneration;
generateMatrix(\@row_marginals, \@col_marginals, 10, 3);
OR
generateMatrix(\@row_marginals, \@col_marginals, 10);
Example: If @row_marginals = (3,2,3,2) and @col_marginals = (2,2,2,2,2);
The following output matrix could be generated by the module:
0 0 0 1 2
1 0 1 0 0
1 0 1 1 0
0 2 0 0 0
DESCRIPTION
This module generates matrix given the row and the column marginals in
such a way that the row and the column marginals of the resultant matrix
are same as the given marginals. For example, given the following
marginals this module would generate the appropriate values for "x"s
such that the row and the column marginals are held fixed.
x x x x x | 3
x x x x x | 2
x x x x x | 3
x x x x x | 2
---------------
2 2 2 2 2 | 10
The algorithm we have used here: For each cell while traversing the
matrix in in row-major interpretation. 1. Generate random number using
the steps given below. 2. Reduce row and column marginals by value of
generated value. End for. Done.
Random number generation algorithm: for each cell C(i,j) { # Find the
range (min, max) for the random number generation max = MIN(row_marg[i],
col_marg[j])
# If max !=0 then decide the min.
# To decide min value sum together the col_marginals for all
# the columns past the current column - this sum gives the total
# of the column marginals yet to be satisfied beyond the current col.
# Subtract this sum from the current row_marginal to compute
# the lower bound on the random number. We do this because if we
# do not set this lower bound and thus a number smaller than this
# bound is generated then we will have a situation where satisfying
# both row_marginal and column marginals will be impossible.
if(max != 0)
{
2_term = 0
for each col k > j
{
2_term = 2_term + col_marg[k]
}
min = row_marg[i] - 2_term
if(min < 0)
{
min = 0
}
}
else
{
min = max = 0 # If max = 0 then min = 0
}
# Generate random number between the range
random_num = rand(min, max)
}
Example:
Cell 0:
max = MIN(3,2) = 2
2_term = 2 + 2 + 2 + 2 = 8
min = 3 - 8 = -5
therefore: min = 0
(min, max) = (0,2) = 0
0 x x x x | 3
x x x x x | 2
x x x x x | 3
x x x x x | 2
--------------
2 2 2 2 2 |
Cell 1:
max = MIN(3,2) = 2
2_term = 2 + 2 + 2 = 6
min = 3 - 6 = -3
therefore: min = 0
(min, max) = (0,2) = 0
0 0 x x x | 3
x x x x x | 2
x x x x x | 3
x x x x x | 2
----------------------------------------------
2 2 2 2 2 |
Cell 2:
max = MIN(3,2) = 2
2_term = 2 + 2 = 4
min = 3 - 4 = -1
therefore: min = 0
(min, max) = (0,2) = 0
0 0 0 x x | 3
x x x x x | 2
x x x x x | 3
x x x x x | 2
----------------------------------------------
2 2 2 2 2 |
Cell 3:
max = MIN(3,2) = 2
2_term = 2
min = 3 - 2 = 1
(min, max) = (1,2) = 1
0 0 0 1 x | 2
x x x x x | 2
x x x x x | 3
x x x x x | 2
----------------------------------------------
2 2 2 1 2 |
Cell 4:
max = MIN(2,2) = 2
2_term = 0
min = 2 - 0 = 2
(min, max) = (2,2) = 2
0 0 0 1 2 | 0
x x x x x | 2
x x x x x | 3
x x x x x | 2
----------------------------------------------
2 2 2 1 0 |
Cell 5:
max = MIN(2,2) = 2
2_term = 2 + 2 + 1 + 0 = 5
min = 3 - 5 = -2
therefore, min = 0
(min, max) = (0,2) = 1
0 0 0 1 2 | 0
1 x x x x | 1
x x x x x | 3
x x x x x | 2
----------------------------------------------
1 2 2 1 0 |
Cell 6:
max = MIN(1,2) = 1
2_term = 2 + 1 + 0 = 3
min = 1 - 3 = -2
therefore, min = 0
(min, max) = (0,1) = 0
0 0 0 1 2 | 0
1 0 x x x | 1
x x x x x | 3
x x x x x | 2
----------------------------------------------
1 2 2 1 0 |
Cell 7:
max = MIN(1,2) = 1
2_term = 1 + 0 = 1
min = 1 - 1 = 0
(min, max) = (0,1) = 1
0 0 0 1 2 | 0
1 0 1 x x | 0
x x x x x | 3
x x x x x | 2
----------------------------------------------
1 2 1 1 0 |
Cell 8:
max = MIN(0,1) = 0
min = 0
(min, max) = (0,0) = 0
0 0 0 1 2 | 0
1 0 1 0 x | 0
x x x x x | 3
x x x x x | 2
----------------------------------------------
1 2 1 1 0 |
Cell 9:
max = MIN(0,0) = 0
min = 0
(min, max) = (0,0) = 0
0 0 0 1 2 | 0
1 0 1 0 0 | 0
x x x x x | 3
x x x x x | 2
----------------------------------------------
1 2 1 1 0 |
Cell 10:
max = MIN(3,1) = 1
2_term = 2 + 1 + 1 + 0 = 4
min = 3 - 4 = -1
therefore, min = 0
(min, max) = (0,1) = 1
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 x x x x | 2
x x x x x | 2
----------------------------------------------
0 2 1 1 0 |
Cell 11:
max = MIN(2,2) = 2
2_term = 1 + 1 + 0 = 2
min = 2 - 2 = 0
(min, max) = (0,2) = 0
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 x x x | 2
x x x x x | 2
----------------------------------------------
0 2 1 1 0 |
Cell 12:
max = MIN(2,1) = 1
2_term = 1 + 0 = 1
min = 2 - 1 = 1
(min, max) = (1,1) = 1
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 1 x x | 1
x x x x x | 2
----------------------------------------------
0 2 0 1 0 |
Cell 13:
max = MIN(1,1) = 1
2_term = 0 = 0
min = 1 - 0 = 1
(min, max) = (1,1) = 1
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 1 1 x | 0
x x x x x | 2
----------------------------------------------
0 2 0 0 0 |
Cell 14:
max = MIN(0,0) = 0
min = 0
(min, max) = (0,0) = 0
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 1 1 0 | 0
x x x x x | 2
----------------------------------------------
0 2 0 0 0 |
Cell 15:
max = MIN(2,0) = 0
min = 0
(min, max) = (0,0) = 0
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 1 1 0 | 0
0 x x x x | 2
----------------------------------------------
0 2 0 0 0 |
Cell 16:
max = MIN(2,2) = 2
2_term = 0 + ... = 0
min = 2 - 0 = 2
(min, max) = (2,2) = 2
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 1 1 0 | 0
0 2 x x x | 0
----------------------------------------------
0 0 0 0 0 |
Cell 17:
max = MIN(0,0) = 0
min = 0
(min, max) = (0,0) = 0
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 1 1 0 | 0
0 2 0 x x | 0
----------------------------------------------
0 0 0 0 0 |
Cell 18:
max = MIN(0,0) = 0
min = 0
(min, max) = (0,0) = 0
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 1 1 0 | 0
0 2 0 0 x | 0
----------------------------------------------
0 0 0 0 0 |
Cell 19:
max = MIN(0,0) = 0
min = 0
(min, max) = (0,0) = 0
0 0 0 1 2 | 0
1 0 1 0 0 | 0
1 0 1 1 0 | 0
0 2 0 0 0 | 0
----------------------------------------------
0 0 0 0 0 |
Done!!
EXPORT
generateMatrix
AUTHOR
Anagha Kulkarni, E<lt>kulka020@d.umn.eduE<gt>
Ted Pedersen, E<lt.tpederse@d.umn.eduE<gt>
COPYRIGHT AND LICENSE
Copyright (C) 2006 by Anagha Kulkarni, Ted Pedersen
This library is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation; either version 2 of the License, or (at your
option) any later version. This program is distributed in the hope that
it will be useful, but WITHOUT ANY WARRANTY; without even the implied
warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License along
with this program; if not, write to the Free Software Foundation, Inc.,
59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.