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NAME

Algorithm::RandomMatrixGeneration - Generate internal cell values for a matrix given fixed marginal totals.

SYNOPSIS

use Algorithm::RandomMatrixGeneration;
my @result = generateMatrix(\@row_marginals, \@col_marginals);

Example: Negative Integer Valued Marginals:

use Algorithm::RandomMatrixGeneration;
my @rmar = ('-5','5','-3');
my @cmar = ('2','3','-2','-6');
my @result = generateMatrix(\@rmargs, \@cmargs, "-");

Output matrix could be:
0 -1 1 3 2 -5 3 -2
0 5
0 -2 2 3 3 -4

Example: Positive Real Valued Marginals:

use Algorithm::RandomMatrixGeneration;
my @rmargs = (13.01,11,13,13,12,13);
my @cmargs = (23.005,32.005,10,10);
my @result = generateMatrix(\@rmargs, \@cmargs, 3);

Output matrix could be:
0 2.694  1 9.665  2 0.393  3 0.258
0 6.539  1 0.910  2 2.209  3 1.342
0 8.469  1 3.565  2 0.839  3 0.127
0 2.719  1 2.748  2 0.604  3 6.929
0 0.946  1 3.771  2 5.939  3 1.344
0 1.638  1 11.346  2 0.016

INPUTS

The generateMatrix function can take 4 parameters:

1. Single dimensional array containing row marginals (Can be real valued or integers) =item 2. Single dimensional array containing column marginals (Can be real valued or integers) =item 3. Precision: For the integer valued marginal specifying "-". For real valued marginals specify the required precision for the generated matrix values. (Recommended Precision = 4) =item 4. Seed: Seed for the random number generator (Default: None) (Optional parameter) =back

OUTPUT

The generateMatrix function returns a two dimensional array containing the generated random matrix. The generated matrix is stored in sparse format in this returned array. That is, only non-zero values are stored in this matrix. Thus to access the values in the returned matrix one can use:

for(my \$row=0; \$i<=\$num_rows; \$i++)
{
for(my \$col=0; \$j<=\$num_cols; \$j++)
{
if(defined \$returned_matrix[\$row][\$col])
{
print "\$col \$returned_matrix[\$row][\$col]  ";
}
}
print "\n";
}

DESCRIPTION

This module generates a random matrix given the row and column marginals in such a way that the row and column marginals of the resultant matrix are same as the given marginals.

If the given marginals are real valued then the generated cell values are real too. If the given marginals are integer valued then the generated cell values are integers. If any of the marginals are negative then few/all of the generated cell values would be negative too.

FURTHER DETAILS

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(marginals positive)
{
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!!

generateMatrix

AUTHOR

Anagha Kulkarni, Carnegie-Mellon University
anaghak at cs.cmu.edu

Ted Pedersen, University of Minnesota, Duluth
tpederse at d.umn.edu

Copyright (C) 2006-2008 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.

1 POD Error

The following errors were encountered while parsing the POD:

Around line 327:

You forgot a '=back' before '=head1'