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

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;

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.

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.

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.

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).

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).

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.

- [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.

The current sources of this module are found on github,
<git://github.com/Abigail/algorithm--numerical--sample.git>.

This package was written by Abigail, cpan@abigail.be.

Copyright (C) 1998, 1999, 2009, Abigail.

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