mvpvii: The Multivariate Pearson Type VII (PVII) Distribution

Description Usage Arguments Details Value Author(s) References See Also Examples

Description

Density and random generation functions for the multivariate Pearson Type VII (PVII) distribution.

Usage

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dmvpvii(x, mean, scale, shape, log=FALSE)
rmvpvii(n, mean, scale, shape)

Arguments

x

a numeric matrix of which each row represents an observation.

mean

a vector of mean parameters for the columns of x. Let D = ncol(x), and length(mean) should be equal to D.

scale

a positive definite D-by-D matrix.

shape

a positive scalar.

log

logical; if TRUE, density is given as the log-density.

n

number of vectors to simulate.

Details

The multivariate PVII distribution, a generalization of the multivariate t-distribution, arises when the inverse of the covariance of a multivariate normal random variable is itself a random variable with the Wishart distribution. See Sun et al. (2010) for details. As parameterized here, the density of a multivariate PVII random variable with mean mu, scale Sigma, and shape alpha is

f(x) = (2 pi)^(-D/2) |Sigma|^(-1/2) gamma(alpha)^(-1) gamma(alpha + D/2) {1 + (1/2) t(x-mu) Sigma^(-1) (x-mu)}^(alpha + D/2)

where \dQuote(gamma) denotes the gamma function.

Value

For dmvpvii, a vector of densities. For rmvpvii, a vector with n rows and D columns representing a sample from the multivariate PVII distribution with the specified parameters.

Author(s)

Daniel Dvorkin

References

Sun, J., Kaban, A., and Garibaldi, J.M. (2010) Robust mixture clustering using Pearson Type VII distribution. Pattern Recognition Letters 31(16), 2447–2454.

See Also

pvii for the univariate version; mvnorm for a related distribution; thetahat for parameter estimation.

Examples

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set.seed(123)

mu <- -1:1

Sigma <- mleCov(matrix(rnorm(30), ncol=3))
Sigma
#            [,1]       [,2]       [,3]
# [1,]  0.8187336  0.5147059 -0.3243663
# [2,]  0.5147059  0.9698367 -0.4933797
# [3,] -0.3243663 -0.4933797  0.7797652

alpha <- 2

x <- rmvpvii(5, mu, Sigma, alpha)
x
#            [,1]       [,2]     [,3]
# [1,] -0.7774939 -0.5824543 2.139000
# [2,] -0.3941455  0.5651861 1.157972
# [3,] -0.3595201  0.2209538 0.588348
# [4,] -1.4053874 -0.5759132 1.055372
# [5,] -1.6673451  1.3083343 1.625087

dmvpvii(x, mu, Sigma, alpha)
# [1] 0.024770962 0.103843268 0.147672666 0.189416293 0.001638714


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