dirichlet {VGAM}R Documentation

Fitting a Dirichlet Distribution

Description

Fits a Dirichlet distribution to a matrix of compositions.

Usage

dirichlet(link = "loge", earg=list(), zero=NULL)

Arguments

In the following, 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.

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).
earg List. Extra argument for the link. See earg in Links for general information.
zero An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The default is none of them. If used, choose values from the set {1,2,...,M}.

Details

The Dirichlet distribution is commonly used to model compositional data, including applications in genetics. Suppose (Y_1,...,Y_M)^T is the response. Then it has a Dirichlet distribution if (Y_1,...,Y_{M-1})^T has density

(Gamma(alpha_+) / prod_{j=1}^M gamma(alpha_j)) prod_{j=1}^M y_j^(alpha_j -1)

where alpha_+= alpha_1 + ... + alpha_M, alpha_j > 0, and the density is defined on the unit simplex

Delta_M = { (y_1,...,y_M)^T : y_1 > 0, ..., 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,...,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 + ... + G_M).

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.

Note

The response should be a matrix of positive values whose rows each sum to unity. Similar to this is count data, where probably a multinomial logit model (multinomial) may be appropriate. Another similar distribution to the Dirichlet is the Dirichlet-multinomial (see dirmultinomial).

Author(s)

Thomas W. Yee

References

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

Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third edition.

See Also

rdiric, dirmultinomial, multinomial.

Examples

y = rdiric(n=1000, shape=exp(c(-1,1,0)))
fit = vglm(y ~ 1, dirichlet, trace = TRUE, crit="c")
Coef(fit)
coef(fit, matrix=TRUE)
fitted(fit)[1:2,]

[Package VGAM version 0.7-7 Index]