| dirichlet {VGAM} | R Documentation |
Fitting a Dirichlet Distribution
Description
Fits a Dirichlet distribution to a matrix of compositions.
Usage
dirichlet(link = "loglink", parallel = FALSE, zero = NULL,
imethod = 1)
Arguments
link |
Link function applied to each of the |
parallel, zero, imethod |
See |
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).
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(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))