# mvpvii: The Multivariate Pearson Type VII (PVII) Distribution In lcmix: Layered and chained mixture models

## Description

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

## Usage

 ```1 2``` ```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.

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.

`pvii` for the univariate version; `mvnorm` for a related distribution; `thetahat` for parameter estimation.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24``` ```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 ```