mWNChygepdf

mWNChygepdf returns Wallenius' multivariate non-central hypergeometric probability density function

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Syntax

  • Wpdf=mWNChygepdf(x, m, w)example
  • Wpdf=mWNChygepdf(x, m, w, accuracy)example

Description

This function calls function CMultiWalleniusNCHypergeometricpdf which is a translation into MATLAB of the corresponding C++ function of Fog (2008).

The notation which is used in WNChygepdf and the order of the arguments is the one of MATLAB hyge. The notation which is used inside CMultiWalleniusNCHypergeometricpdf is the original one of Fog.

To illustrate the meaning of Wallenius and Fisher' function parameters, let's use the classical biased urn example, with $K$ red balls and $M-K$ white balls, totalling $M$ balls. $n$ balls are drawn at random from the urn without replacement. Each red ball has the weight $\omega_{1}$, and each white ball has the weight $\omega_{2}$; the probability ratio of red over white balls is then given by $odds = \omega_{1} / \omega_{2}$. Note that the odds are fixed once and for all during the drawings.

If the balls are taken one by one, the probability (say $p_1$) that the first ball picked is red is equal to the weight fraction of red balls:

\[ p_1= \frac{K w_1}{K w_1 + (M-K) w_2} \]

In the Wallenius distribution the probability that the second ball picked is red depends on whether the first ball was red or white. If the first ball was red then the above formula is used with $K$ reduced by one. If the first ball was not red then the above formula is used with $M-K$ reduced by one. The number of red balls that we get in this experiment is a random variable with Wallenius' noncentral hypergeometric distribution.

The important fact that distinguishes Wallenius' distribution is that there is competition between the balls. The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. And the weight of the competing balls depends on the outcomes of all preceding draws. In the Fisher model, the fates of the balls are independent and there is no dependence between draws. One may as well take all $n$ balls at the same time. Each ball has no "knowledge" of what happens to the other balls.

More formally, if the total number $n$ of balls taken is known then the conditional distribution of the number of taken red balls for given $n$ is Fisher's noncentral hypergeometric distribution.

These two distributions have important applications in evolutionary biology and population genetics. If animals of a particular species are competing for a limited food resource so that individual animals are dying one by one until there is enough food for the remaining animals, and if there are different variants of animals with different chances of finding food, then we can expect the animals that die to follow a Wallenius noncentral hypergeometric distribution. Fisher’s noncentral hypergeometric distribution may be used instead of Wallenius distribution in cases where the fates of the individual animals are independent and the total number of survivors is known or controlled as part of a simulation experiment (Fog 2008).

Wallenius distribution is a general model of biased sampling. Fisher’s noncentral hypergeometric distribution is used for statistical tests on contingency tables where the marginals are fixed (McCullagh and Nelder, 1983).

The multivariate Fishers and Wallenius noncentral hypergeometric distribution, are referred to the case where the type of different balls is greater than 2 and each ball has a different probability.

The two distributions are both equal to the (multivariate) hypergeometric distribution when the odds ratio is 1.

example

Wpdf =mWNChygepdf(x, m, w) Problem description.

example

Wpdf =mWNChygepdf(x, m, w, accuracy) In presence of just two colors in the urn it is possible to use WNChygepdf or mWNChygepdf.

Examples

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  • Problem description.

    • We have 70 balls in a urn of 3 different colors. 

    % Initially in the urn we have 20 red balls, 30 white and 20 black balls.
m=[20,30,20];
% The weight of each color is 1, 2,5 and 1.8
w=[1, 2.5, 1.8];
% We want to know the probability of extracting without replacement exactly 8 red balls, 10
% white balls and 6 black balls (the total number of balls taken from
% the urn is 8+10+6=20);
x=[8, 10, 6];
% Function mWNChygepdf is called without specifying fourth optional argument and
% therefore the precision is set to 1e-08. 
wpdf = mWNChygepdf(x,m,w);
disp('We have an urn with 3 groups of balls; 20 red, 30 white and 20 black.');
disp('In total the urn contains 70 balls.');
disp('But the weight of getting the 3 balls is 1, 2.5 and 1.8;');
disp('The probability of getting 8 red balls, 10, white and 6 black balls');
disp('in an extraction without replacement is ');
disp(wpdf);
    We have an urn with 3 groups of balls; 20 red, 30 white and 20 black.
In total the urn contains 70 balls.
But the weight of getting the 3 balls is 1, 2.5 and 1.8;
The probability of getting 8 red balls, 10, white and 6 black balls
in an extraction without replacement is 
    0.0036

  • In presence of just two colors in the urn it is possible to use WNChygepdf or mWNChygepdf.

    • We have M balls in the urn. 

    M  = 200;
% We have K red balls
K  = 6;
%we extract n balls, one at a time, without replacement
n  = 5;
% red balls have a probability 100 times greater to be extracted than white balls
odds  = 100;
% We compute the probability of getting 0, 1, 2 or 3, ..., n red balls 
x = (0:n)';
% Densities using call to WNChygepdf
wpdf = WNChygepdf(x,M,K,n,odds);
% Densities using call to mWNChygepdf
XX=[x n-x];
m=[K M-K]; w=[100 1];
wpdfCHK=mWNChygepdf(XX,m,w);
assert(isequal(wpdf,wpdfCHK),'Call to WNChygepdf produces different results from call to mWNChygepdf')

    Input Arguments

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    x — Number of balls of each color sampled. Evaluation point.

    Vector or matrix.

    Evaluation points specified as 1-by-length(m) numeric vector or an ntrials-by-length(m) numeric matrix where ntrials is a positive scalar integer defining the number of evaluation points.

    Data Types: single|double

    m — Initial number of balls of each color in the urn. Vector.

    The length of vector M is the number of colors in the urn. M=sum(m) is the the total number of balls.

    If the number of colors is equal to 2, m(1)+m(2) corresponds to M and m(1) corresponds to K (number of red balls in the urn).

    Data Types: single|double

    w — Weights for each color. Vector.

    The length of vector w must be equal to the length of vector m. If the number of colors is equal to 2, odds=w(1)/w(2). If all the elements of w are equal we have the central hypergeometric multivariate distribution.

    Data Types: single|double

    Optional Arguments

    accuracy — accuracy of the calculations. Scalar.

    The default value of accuracy is 1e-08.

    Example: 1e-06

    Data Types: single|double

    Output Arguments

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    Wpdf —Wallenius' multivariate pdf values. Scalar or column vector of length(ntrials)

    Wpdf(1) is the probability of drawing exactly x(1,1) balls of a possible m(1) items, x(1,2) balls of a possible m(2) items, .., x(1,end) balls of a possible m(end) items in n drawings (where n=sum(1,:)) without replacement from a group of sum(m) objects, when objects are from length(m) weighted groups. The size of Wpdf is a scalar if x is a row vector. The size of Wpdf is a column vector of length ntrials if x is a matrix of size ntrials-by-length(m).

    References

    Fog, A. (2008), Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution, "Communications in Statistics - Simulation and Computation", Vol. 37, pp. 258-273.

    See Also

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