/ 
Sample-1.2
/ Algorithm::Numerical::Sample

NAME

Algorithms::Numerical::Sample - Draw samples from a set

SYNOPSIS

use Algorithms::Numerical::Sample  qw /sample/;

@sample = sample (-set         => [1 .. 10000],
                  -sample_size => 100);

$sampler = Algorithms::Numerical::Sample::Stream -> new;
while (<>) {$sampler -> data ($_)}
$random_line = $sampler -> extract;

DESCRIPTION

This package gives two methods to draw fair, random samples from a set. There is a procedural interface for the case the entire set is known, and an object oriented interface when the a set with unknown size has to be processed.

A: sample (set => ARRAYREF [,sample_size => EXPR])

The sample function takes a set and a sample size as arguments. If the sample size is omitted, a sample of 1 is taken. The keywords set and sample_size may be preceeded with an optional -. The function returns the sample list, or a reference to the sample list, depending on the context.

B: Algorithms::Numerical::Sample::Stream

The class Algorithms::Numerical::Sample::Stream has the following methods:

new

This function returns an object of the Algorithms::Numerical::Sample::Stream class. It will take an optional argument of the form sample_size => EXPR, where EXPR evaluates to the sample size to be taken. If this argument is missing, a sample of size 1 will be taken. The keyword sample_size may be preceeded by an optional dash.

data (LIST)

The method data takes a list of parameters which are elements of the set we are sampling. Any number of arguments can be given.

extract

This method will extract the sample from the object, and reset it to a fresh state, such that a sample of the same size but from a different set, can be taken. extract will return a list in list context, or the first element of the sample in scalar context.

CORRECTNESS PROOFS

Algorithm A.

Crucial to see that the sample algorithm is correct is the fact that when we sample n elements from a set of size N that the t + 1st element is choosen with probability (n - m)/(N - t), when already m elements have been choosen. We can immediately see that we will never pick too many elements (as the probability is 0 as soon as n == m), nor too few, as the probability will be 1 if we have k elements to choose from the remaining k elements, for some k. For the proof that the sampling is unbiased, we refer to [3]. (Section 3.4.2, Exercise 3).

Algorithm B.

It is easy to see that the second algorithm returns the correct number of elements. For a sample of size n, the first n elements go into the reservoir, and after that, the reservoir never grows or shrinks in size; elements only get replaced. A detailed proof of the fairness of the algorithm appears in [3]. (Section 3.4.2, Exercise 7).

LITERATURE

Both algorithms are discussed by Knuth [3] (Section 3.4.2). The first algoritm, Selection sampling technique, was discovered by Fan, Muller and Rezucha [1], and independently by Jones [2]. The second algorithm, Reservoir sampling, is due to Waterman.

REFERENCES

[1]

C. T. Fan, M. E. Muller and I. Rezucha, J. Amer. Stat. Assoc. 57 (1962), pp 387 - 402.

[2]

T. G. Jones, CACM 5 (1962), pp 343.

[3]

D. E. Knuth: The Art of Computer Programming, Volume 2, Third edition. Reading: Addison-Wesley, 1997. ISBN: 0-201-89684-2.

HISTORY

$Date: 1999/03/01 21:06:07 $

$Log: Sample.pm,v $
Revision 1.2  1999/03/01 21:06:07  abigail
Changed package to Algorithm::*

Revision 1.1  1998/04/29 03:05:57  abigail
Initial revision

AUTHOR

This package was written by Abigail.

COPYRIGHT

Copyright 1998, 1999 by Abigail.

LICENSE

This package is free and open software.

You may use, copy, modify, distribute and sell this package or any modifications there of in any form you wish, provided you do not do any of the following:

- claim that any of the original code was written by someone
  else than the original author(s).
- restrict someone in using, copying, modifying, distributing or
  selling this program or module or any modifications of it.

THIS PACKAGE IS PROVIDED "AS IS" AND WITHOUT ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.