| BiasedUrn-Multivariate {BiasedUrn} | R Documentation |
Biased urn models: Multivariate distributions
Description
Statistical models of biased sampling in the form of multivariate noncentral hypergeometric distributions, including Wallenius' noncentral hypergeometric distribution and Fisher's noncentral hypergeometric distribution (also called extended hypergeometric distribution).
These are distributions that you can get when taking colored balls from an urn without replacement, with bias. The univariate distributions are used when there are two colors of balls. The multivariate distributions are used when there are more than two colors of balls.
Please see vignette("UrnTheory")
for a definition of these distributions and how
to decide which distribution to use in a specific case.
Usage
dMWNCHypergeo(x, m, n, odds, precision = 1E-7)
dMFNCHypergeo(x, m, n, odds, precision = 1E-7)
rMWNCHypergeo(nran, m, n, odds, precision = 1E-7)
rMFNCHypergeo(nran, m, n, odds, precision = 1E-7)
meanMWNCHypergeo(m, n, odds, precision = 0.1)
meanMFNCHypergeo(m, n, odds, precision = 0.1)
varMWNCHypergeo(m, n, odds, precision = 0.1)
varMFNCHypergeo(m, n, odds, precision = 0.1)
momentsMWNCHypergeo(m, n, odds, precision = 0.1)
momentsMFNCHypergeo(m, n, odds, precision = 0.1)
oddsMWNCHypergeo(mu, m, n, precision = 0.1)
oddsMFNCHypergeo(mu, m, n, precision = 0.1)
numMWNCHypergeo(mu, n, N, odds, precision = 0.1)
numMFNCHypergeo(mu, n, N, odds, precision = 0.1)
minMHypergeo(m, n)
maxMHypergeo(m, n)
Arguments
x |
Number of balls of each color sampled. Vector with length = number of colors, or matrix with nrows = number of colors. |
m |
Initial number of balls of each color in the urn. Length of vector = number of colors. |
n |
Total number of balls sampled. Scalar. |
N |
Total number of balls in urn before sampling. Scalar. |
odds |
Odds or weight for each color, arbitrarily scaled. Length of vector = number of colors. Gives the (central) multivariate hypergeometric distribution if all odds are equal. |
nran |
Number of random variates to generate. Scalar. |
mu |
Mean x for each color. Length of vector = number of colors. |
precision |
Desired precision of calculation. Scalar. |
Details
Allowed parameter values
x, m, odds and mu are all vectors with one
element for each color. These vectors must have the same length.
x can also be a matrix with one column for each observation.
The number of rows in this matrix must be equal to the number of colors.
The maximum number of colors is currently set to 32.
All parameters must be non-negative.
n cannot exceed N = sum(m).
The odds may be arbitrarily scaled.
The code has been tested with odds ratios in the range
10^{-9} \ldots 10^9 and zero.
The code may work with odds ratios
outside this range, but errors or NAN can occur for extreme values of odds.
A ball with odds = 0 is equivalent to no ball.
mu must be within the possible range of x.
Calculation time
The calculation time depends on the specified precision and the number of colors.
The calculation time can be high for rMWNCHypergeo and rMFNCHypergeo when nran
is high.
The calculation time can be extremely high for dMFNCHypergeo when n is high and
the number of colors is high.
The calculation time can be extremely high for the mean... var... and moments...
functions when precision < 0.1 and n is high and the
number of colors is high.
Value
dMWNCHypergeo and dMFNCHypergeo return the probability mass
function for the multivariate Wallenius' and Fisher's noncentral hypergeometric
distribution, respectively.
A single value is returned if x is a vector with length = number of colors.
Multiple values are returned if x is a matrix with one column for each
observation. The number of rows must be equal to the number of colors.
rMWNCHypergeo and rMFNCHypergeo return random vectors with
the multivariate Wallenius' and Fisher's noncentral hypergeometric
distribution, respectively.
A vector is returned when nran = 1. A matrix with one column for each
observation is returned when nran > 1.
meanMWNCHypergeo and meanMFNCHypergeo return the mean
of the multivariate Wallenius' and Fisher's noncentral hypergeometric
distribution, respectively. A simple and fast approximation is used when
precision >= 0.1. A full calculation of all
possible x combinations is used when precision < 0.1.
This can take extremely long time when the number of colors is high.
varMWNCHypergeo and varMFNCHypergeo return the variance
of the multivariate Wallenius' and Fisher's noncentral hypergeometric
distribution, respectively. A simple and fast approximation is used when
precision >= 0.1. A full calculation of all
possible x combinations is used when precision < 0.1.
This can take extremely long time when the number of colors is high.
momentsMWNCHypergeo and momentsMFNCHypergeo return a data
frame with the mean and variance of the multivariate Wallenius' and
Fisher's noncentral hypergeometric distribution, respectively.
Calculating the mean and variance in the same operation saves time when
precision < 0.1.
oddsMWNCHypergeo and oddsMFNCHypergeo estimate the odds
from an observed mean for the multivariate Wallenius' and
Fisher's noncentral hypergeometric distribution, respectively.
A vector of odds is returned if mu is a vector.
A matrix is returned if mu is a matrix with one row for each color.
A simple and fast approximation is used regardless of the specified precision.
Exact calculation is not supported.
See demo(OddsPrecision).
numMWNCHypergeo and numMFNCHypergeo estimate the
number of balls of each color in the urn before sampling from
experimental mean and known odds ratios for
Wallenius' and Fisher's noncentral hypergeometric distributions.
The returned m values are not integers.
A vector of m is returned if mu is a vector.
A matrix of m is returned if mu is a matrix with one row for each color.
A simple and fast approximation is used regardless of the specified precision.
Exact calculation is not supported.
The precision of calculation is indicated by demo(OddsPrecision).
minMHypergeo and maxMHypergeo calculate the
minimum and maximum value of x for the multivariate distributions.
The values are valid for the multivariate Wallenius' and Fisher's noncentral
hypergeometric distributions as well as for the multivariate (central)
hypergeometric distribution.
References
Fog, A. 2008a. Calculation methods for Wallenius’ noncentral hypergeometric distribution. Communications in Statistics—Simulation and Computation 37, 2 doi:10.1080/03610910701790269
Fog, A. 2008b. Sampling methods for Wallenius’ and Fisher’s noncentral hypergeometric distributions. Communications in Statistics—Simulation and Computation 37, 2 doi:10.1080/03610910701790236
See Also
vignette("UrnTheory")
BiasedUrn-Univariate.
BiasedUrn.
Examples
# get probability
dMWNCHypergeo(c(8,10,6), c(20,30,20), 24, c(1.,2.5,1.8))