dist.Multivariate.Polya {LaplacesDemon}

Multivariate Polya Distribution

Description

These functions provide the density and random number generation for the multivariate Polya distribution.

Usage

dmvpolya(x, alpha, log=FALSE)
rmvpolya(n, alpha)

Arguments

x

This is data or parameters in the form of a vector of length k.

n

This is the number of random draws to take from the distribution.

alpha

This is shape vector \alpha with length k.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Details

• Application: Discrete Multivariate

• Density:

  p(\theta) = \frac{N!}{\prod_k N_k!} \frac{(\sum_k \alpha_k - 1)!}{(\sum_k \theta_k + \sum_k \alpha_k - 1)!} \frac{\prod (\theta + \alpha - 1)!}{(\alpha - 1)!}

• Inventor: George Polya (1887-1985)

• Notation 1: \theta \sim \mathcal{MPO}(\alpha)

• Notation 3: p(\theta) = \mathcal{MPO}(\theta | \alpha)

• Parameter 1: shape parameter vector \alpha

• Mean: E(\theta) =

• Variance: var(\theta) =

• Mode: mode(\theta) =

The multivariate Polya distribution is named after George Polya (1887-1985). It is also called the Dirichlet compound multinomial distribution or the Dirichlet-multinomial distribution. The multivariate Polya distribution is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector \alpha, and a set of N discrete samples is drawn from the categorical distribution with probability vector p and having K discrete categories. The compounding corresponds to a Polya urn scheme. In document classification, for example, the distribution is used to represent probabilities over word counts for different document types. The multivariate Polya distribution is a multivariate extension of the univariate Beta-binomial distribution.

Value

dmvpolya gives the density and rmvpolya generates random deviates.

Author(s)

Statisticat, LLC [e-mail]

See Also

dcat, ddirichlet, and dmultinom.

Examples

library(LaplacesDemon)
dmvpolya(x=1:3, alpha=1:3, log=TRUE)
x <- rmvpolya(1000, c(0.1,0.3,0.6))