```
NAME
Algorithm::Numerical::Sample - Draw samples from a set
SYNOPSIS
use Algorithm::Numerical::Sample qw /sample/;
@sample = sample (-set => [1 .. 10000],
-sample_size => 100);
$sampler = Algorithm::Numerical::Sample::Stream -> new;
while (<>) {$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: "Algorithm::Numerical::Sample::Stream"
The class "Algorithm::Numerical::Sample::Stream" has the following
methods:
"new"
This function returns an object of the
"Algorithm::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 + 1"st
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.
DEVELOPMENT
The current sources of this module are found on github,
<git://github.com/Abigail/algorithms--numerical--sample.git>.
AUTHOR
This package was written by Abigail, cpan@abigail.be.
COPYRIGHT and LICENSE
Copyright (C) 1998, 1999, 2009, Abigail.
Permission is hereby granted, free of charge, to any person obtaining a
copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:
The above copyright notice and this permission notice shall be included
in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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