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Re^5: Algorithm for cancelling common factors between two lists of multiplicands

by tmoertel (Chaplain)
on Aug 08, 2005 at 23:31 UTC ( #482064=note: print w/replies, xml ) Need Help??


in reply to Re^4: Algorithm for cancelling common factors between two lists of multiplicands
in thread Algorithm for cancelling common factors between two lists of multiplicands

Quick question: If the sample size is so large, is there a reason you aren't using the Chi-square test? My understanding is Fisher's Exact Test may be preferred when the sample size isn't large enough to reasonably support the large-sample approximation for the Chi-square test. Since you have a large sample, why not take the easy road?
  • Comment on Re^5: Algorithm for cancelling common factors between two lists of multiplicands

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Re^6: Algorithm for cancelling common factors between two lists of multiplicands
by BrowserUk (Pope) on Aug 09, 2005 at 00:03 UTC
    ... is there a reason you aren't using the Chi-square test?

    I'm not trying to solve the sample problem. That was just an example I found on the web and used as a test.

    I set out to solve the problem of performing the FET using Perl. First I did it with Math::Pari, but which gives a result of 8.070604647867604097576877668E-7030 in 26ms, but I was unsure about the accuracy. It also imposes a binary dependancy.

    #! perl -slw use strict; use Benchmark::Timer; use List::Util qw[ sum reduce ]; use Math::Pari qw[ factorial ]; $a=$b; sub product{ reduce{ $a *= $b } 1, @_ } sub FishersExactTest { my @data = @_; return unless @data == 4; my @C = ( sum( @data[ 0, 2 ] ), sum( @data[ 1, 3 ] ) ); my @R = ( sum( @data[ 0, 1 ] ), sum( @data[ 2, 3 ] ) ); my $N = sum @C; my $dividend = product map{ factorial $_ } grep $_, @R, @C; my $divisor = product map{ factorial $_ } grep $_, $N, @data; return $dividend / $divisor; } my $T = new Benchmark::Timer; $T->start( '' ); print FishersExactTest 989, 9400, 43300, 2400;; $T->stop( '' ); $T->report; __END__ P:\test>MP-FET.pl 8.070604647867604097576877668E-7030 1 trial of _default ( 25.852ms total), 25.852ms/trial

    So, then I coded it using Math::BigFloat

    #! perl -slw use strict; use Benchmark::Timer; use List::Util qw[ reduce ]; use Math::BigFloat; $a=$b; sub product{ reduce{ $a *= $b } 1, @_ } sub sum{ reduce{ $a += $b } 0, @_ } sub FishersExactTest { my @data = map{ Math::BigFloat->new( $_ ) } @_; return unless @data == 4; my @C = ( sum( @data[ 0, 2 ] ), sum( @data[ 1, 3 ] ) ); my @R = ( sum( @data[ 0, 1 ] ), sum( @data[ 2, 3 ] ) ); my $N = sum @C; my $dividend = product map{ $_->bfac } grep $_, @R, @C; my $divisor = product map{ $_->bfac } grep $_, $N, @data; return $dividend / $divisor; } my $T = new Benchmark::Timer; $T->start( '' ); print FishersExactTest 989, 9400, 43300, 2400;; $T->stop( '' ); $T->report;

    But that ran for 20 minutes without producing any output before I killed it (I've set it running again now, and my machines fan has been thrashing at full speed for the last 25 minutes).

    Whilst I was waiting for the BigFloat version, I coded this version which attempts to reduce the size of the problem by eliminating (exactly common) factors:

    sub FishersExactTest2 { my @data = @_; return unless @data == 4; my @C = ( sum( @data[ 0, 2 ] ), sum( @data[ 1, 3 ] ) ); my @R = ( sum( @data[ 0, 1 ] ), sum( @data[ 2, 3 ] ) ); my $N = sum @C; my %dividends; $dividends{ $_ }++ for map{ factors $_ } grep $_, @ +R, @C; my %divisors; $divisors { $_ }++ for map{ factors $_ } grep $_, $ +N, @data; for my $i ( keys %divisors ) { if( exists $dividends{ $i } ) { $divisors{ $i }--, $dividends{ $i }-- while $divisors{ $i } and $dividends{ $i }; delete $divisors { $i } unless $divisors { $i }; delete $dividends{ $i } unless $dividends{ $i }; } } my $dividend = product( map{ ( $_ ) x $dividends{ $_ } } keys %div +idends ); my $divisor = product( map{ ( $_ ) x $divisors { $_ } } keys %div +isors ); return $dividend / $divisor; }

    This works well for values smallish values, but cannot handle the example I gave above (NV overflow).

    It was then I started thinking about how to eliminate more factors from the equation so as to reduce the size of the intermediate terms, and posted my SoPW. I think that hv's solution of expanding all terms to their prime factorizations before performing the cancelling out will be a winner--but I haven't finished coding that yet.


    Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
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    The "good enough" maybe good enough for the now, and perfection maybe unobtainable, but that should not preclude us from striving for perfection, when time, circumstance or desire allow.
      My Haskell implementation represents numbers as the ratio of products of ordered integer streams. For example, I represent 3!/(4*5) as (R numerator=[1,2,3] denominator=[4,5]). In this representation, multiplication becomes merging the numerator and denominator streams and then canceling the first stream by the second. In this way I can remove all cancelable original terms in the Pcutoff formula before finally multiplying the terms that remain.
      *FishersExactTest> fac 6 R {numer = [2,3,4,5,6], denom = []} *FishersExactTest> fac 3 R {numer = [2,3], denom = []} *FishersExactTest> fac 6 `rdivide` fac 3 R {numer = [4,5,6], denom = []}
      Here's the example from the MathWorld page:
      *FishersExactTest> rpCutoff [ [5,0], [1,4] ] R {numer = [2,3,4,5], denom = [7,8,9,10]} *FishersExactTest> fromRational . toRatio $ it 2.3809523809523808e-2
      The code:
      module FishersExactTest (pCutoff) where import Data.Ratio import Data.List (transpose) pCutoff = toRatio . rpCutoff rpCutoff rows = facproduct (rs ++ cs) `rdivide` facproduct (n:xs) where rs = map sum rows cs = map sum (transpose rows) n = sum rs xs = concat rows -- cells facproduct = rproduct . map fac fac n | n < 2 = runit | otherwise = R [2..n] [] -- I represent numbers as ratios of products of integer streams -- R [1,2,3] [4,5] === (1 * 2 * 3) / (4 * 5) data Rops = R { numer :: [Int], denom :: [Int] } deriving Show runit = R [] [] -- the number 1 toRatio (R ns ds) = bigProduct ns % bigProduct ds bigProduct = product . map toInteger -- multiplication is merging numerator and denominator streams -- and then canceling the first by the second rtimes (R xns xds) (R yns yds) = uncurry R $ (merge xns yns) `cancel` (merge xds yds) rproduct = foldr rtimes runit -- division is multiplication by the inverse rdivide x (R yns yds) = rtimes x (R yds yns) -- helpers merge (x:xs) (y:ys) | x < y = x : merge xs (y:ys) | otherwise = y : merge (x:xs) ys merge [] ys = ys merge xs [] = xs cancel (x:xs) (y:ys) | x == y = cancel xs ys | x < y = let (xs', ys') = cancel xs (y:ys) in (x:xs', ys') | otherwise = let (xs', ys') = cancel (x:xs) ys in (xs', y:ys') cancel xs ys = (xs, ys)

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