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/algorithm--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.