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

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 $(Y_1,\dots ,Y_M{)}^T$ is the response. Then it has a Dirichlet distribution if $(Y_1,\dots ,Y_{M-1}{)}^T$ has density $$\frac{\mathrm{\Gamma }({\alpha }_{+})}{\prod _{j=1}^M\mathrm{\Gamma }({\alpha }_j)}\prod _{j=1}^M{y}_j^{{\alpha }_j-1}$$ where ${\alpha }_{+}={\alpha }_1+\cdots +{\alpha }_M$, ${\alpha }_j>0$, and the density is defined on the unit simplex $${\mathrm{\Delta }}_M=\{(y_1,\dots ,y_M{)}^T:y_1>0,\dots ,y_M>0,\sum _{j=1}^M{y}_j=1\}.$$ 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 $(G_1,\dots ,G_M{)}^T$ which are each independent and identically distributed as a gamma distribution with density $f(g_j)=g_j^{{\alpha }_j-1}{e}^{-g_j}/\mathrm{\Gamma }({\alpha }_j)$. 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.

See Also

rdiric, dirmultinomial, multinomial, simplex.

Examples

# 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)
head(fitted(fit))
# }