R: Fitting a Dirichlet Distribution

dirichlet {VGAM} | R Documentation

## 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.

### 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,…,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, ∑_{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.

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

    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))

[Package VGAM version 1.0-3 Index]