VGAM (version 1.0-4)

# dirichlet: Fitting a Dirichlet Distribution

## Description

Fits a Dirichlet distribution to a matrix of compositions.

## Usage

dirichlet(link = "loge", parallel = FALSE, zero = NULL, imethod = 1)

## Arguments

Link function applied to each of the $$M$$ (positive) shape parameters $$\alpha_j$$. See Links for more choices. The default gives $$\eta_j=\log(\alpha_j)$$.

parallel, zero, imethod

See CommonVGAMffArguments for more information.

## Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, rrvglm and vgam.

When fitted, the fitted.values slot of the object contains the $$M$$-column matrix of means.

## Details

In this help file the response is assumed to be a $$M$$-column matrix with positive values and whose rows each sum to unity. Such data can be thought of as compositional data. There are $$M$$ linear/additive predictors $$\eta_j$$.

The Dirichlet distribution is commonly used to model compositional data, including applications in genetics. Suppose $$(Y_1,\ldots,Y_{M})^T$$ is the response. Then it has a Dirichlet distribution if $$(Y_1,\ldots,Y_{M-1})^T$$ has density $$\frac{\Gamma(\alpha_{+})} {\prod_{j=1}^{M} \Gamma(\alpha_{j})} \prod_{j=1}^{M} y_j^{\alpha_{j} -1}$$ where $$\alpha_+=\alpha_1+\cdots+\alpha_M$$, $$\alpha_j > 0$$, and the density is defined on the unit simplex $$\Delta_{M} = \left\{ (y_1,\ldots,y_{M})^T : y_1 > 0, \ldots, y_{M} > 0, \sum_{j=1}^{M} y_j = 1 \right\}.$$ One has $$E(Y_j) = \alpha_j / \alpha_{+}$$, which are returned as the fitted values. For this distribution Fisher scoring corresponds to Newton-Raphson.

The Dirichlet distribution can be motivated by considering the random variables $$(G_1,\ldots,G_{M})^T$$ which are each independent and identically distributed as a gamma distribution with density $$f(g_j)=g_j^{\alpha_j - 1} e^{-g_j} / \Gamma(\alpha_j)$$. Then the Dirichlet distribution arises when $$Y_j=G_j / (G_1 + \cdots + G_M)$$.

## References

Lange, K. (2002) Mathematical and Statistical Methods for Genetic Analysis, 2nd ed. New York: Springer-Verlag.

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011) Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

rdiric, dirmultinomial, multinomial, simplex.

## Examples

Run this code
# NOT RUN {
ddata <- data.frame(rdiric(n = 1000,
shape = exp(c(y1 = -1, y2 = 1, y3 = 0))))
fit <- vglm(cbind(y1, y2, y3)  ~ 1, dirichlet,
data = ddata, trace = TRUE, crit = "coef")
Coef(fit)
coef(fit, matrix = TRUE)