getting a submatrix of all true

John Hunter jdhunter at
Tue Jul 8 07:11:39 CEST 2003

>>>>> "John" == John Hunter <jdhunter at> writes:
    John>Using this approach, I was able to find a 893x67 submatrix
    John>with a score of 59831 versus the 462x81 submatrix with a
    John>score of 37422.

Onwards and upwards....  Since this is beginning to look like a
gradient descent algorithm, it is natural to worry about local minima.
So I followed the lead of simulated annealing algorithms and added a
little noise to the step where the decision to delete the row or
column is made, eg,

  from MLab import randn  
  def trim_missing(a, thresh=(0.05,0.05), sigma=0.005 ):
          if fracRow-vthresh+sigma*randn()<fracCol-othresh:
              #delete the column
              #delete the row

where sigma is like the temperature parameter of a simulated annealing

Repeatedly stepping over a variety of sigmas in search of a minimum
has yielded minor improvement for the dataset of interest. Best

  Deterministic: 59831 893 67
  Random:        59928 908 66 for sigma=0.0017 

For random arrays the results are more striking.  For random matrices
500x500, comparing the scores of the deterministic (sigma=0) versus
random (best sigma in [0,0.01]) searches:
            score        score   
 % ones   mean determ  mean random    mean(ratio)  std(ratio) 
    1%     41753.2       42864.6         1.03         0.0093
    5%      5525.8        6174.2         1.12         0.06
   10%      1968.3        2193.9         1.12         0.05

For a matrix of the same number of total elements, but unbalanced in
the dimensions, eg, 1000x250, the results are more pronounced, in
favor of the random search algorithm

             score        score   
 % ones   mean determ  mean random    mean(ratio)  std(ratio) 
    1%     48724.0       51451.8         1.06         0.018
    5%      6340.5        7542.3         1.19         0.038
   10%      2092.5        2458.6         1.180        0.091

Below is a script that performs a random search of a random matrix for
a variety of sigmas.


from __future__ import division
import sys, time
from urllib import urlopen
from Numeric import *
from MLab import randn, rand, mean, std

def trim_missing(a, thresh=(0.05,0.05), sigma=0.005 ):
    a is a numObservations by numVariables boolean matrix

    thresh = (othresh, vthresh) are the observation and variable
    thresholds that trigger deletion

      remove any rows where percent missing vars > vthresh
      remove any cols where percent missing obs > othresh

    This is done iteratively rechecking the percent missing on each
    iteration or row or col removal, with the largest offender removed
    on each iteration

    return value is score, rowInd, colInd where the two ind arrays
    are the indices of the rows and cols that were kept

    othresh, vthresh = thresh # theshold for deletion
    dr=zeros(a.shape[0])   # Arrays to note deleted rows...
    dc=zeros(a.shape[1])   # ... and columns
    sizeleft=list(a.shape) # Remaining dimensions
    sr=sum(a,1)         # Column scores = ij's to remove 
    sc=sum(a,0)         # Row scores
    while 1:               # Keep deleting till none left    
        mr=argmax(sr)       # index highest row score
        mc=argmax(sc)       # index highest column score
        fracRow = sr[mr]/sizeleft[1]
        fracCol = sc[mc]/sizeleft[0]

        if fracRow<=vthresh and fracCol<=othresh: 
            break            # stop when missing criteria are satisfied
        if fracRow-vthresh+sigma*randn()<fracCol-othresh:
            sr=sr-a[:,mc]    # rows reduced by contents of column
            sc[mc]=0         # sum of column now zero
            dc[mc]=1         # tags column as deleted
            # deleting the row
            sc=sc-a[mr,:]    # columns reduced by contents of row
            sr[mr]=0         # sum of row is now zero
            dr[mr]=1         # tags row as deleted

    dr=compress(dr==0,range(a.shape[0]))  # gives return format of  
    dc=compress(dc==0,range(a.shape[1]))  # optimrcs posted by Bengt

    return score, dr, dc

results = []
for trial in range(10):

    score0, goodRows, goodCols = trim_missing(a, thresh=(0,0), sigma=0)
    maxScore = 0, score0, len(goodRows), len(goodCols)
    print 'Deterministic:', score0, len(goodRows), len(goodCols)
    for sigma in arange(0.0001, 0.01, 0.001):
        for i in range(10):
            score, goodRows, goodCols = trim_missing(
                a, thresh=(0,0), sigma=sigma)
            if score>maxScore[1]:
                #print '\t', sigma, score, len(goodRows), len(goodCols)
                maxScore = sigma, score, len(goodRows), len(goodCols)
    print '\t', maxScore

    results.append((score0, maxScore[1], maxScore[1]/score0))
results = array(results)
print mean(results)
print std(results)

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