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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.