View source: R/family.univariate.R

dirichlet | R Documentation |

Fits a Dirichlet distribution to a matrix of compositions.

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

`link` |
Link function applied to each of the |

`parallel, zero, imethod` |
See |

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)`

.

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.

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`

).

Thomas W. Yee

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`

.

```
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))
```

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