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, Ekulka020@d.umn.eduE Ted Pedersen, E 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.