sirt (version 1.5-0)

dirichlet.mle: Maximum Likelihood Estimation of the Dirichlet Distribution

Description

Maximum likelihood estimation of the parameters of the Dirichlet distribution

Usage

dirichlet.mle(x, weights=NULL , eps = 10^(-5), convcrit = 1e-05 , maxit=1000, 
     oldfac = .3 , progress=FALSE)

Arguments

x

Data frame with $N$ observations and $K$ variables of a Dirichlet distribution

weights

Optional vector of frequency weights

eps

Tolerance number which is added to prevent from logarithms of zero

convcrit

Convergence criterion

maxit

Maximum number of iterations

oldfac

Covergence acceleration factor. It must be a parameter between 0 and 1.

progress

Display iteration progress?

Value

• A list with following entries
• alpha
  Vector of $\alpha$ parameters
• alpha0
  The concentration parameter $\alpha_0 = \sum_k \alpha_k$
• xsi
  Vector of proportions $\xi_k = \alpha_k / \alpha_0$

References

Minka, T. P. (2012). Estimating a Dirichlet distribution. Technical Report.

See Also

For simulating Dirichlet vectors with matrix-wise $\bold{\alpha}$ parameters see dirichlet.simul. For a variety of functions concerning the Dirichlet distribution see the DirichletReg package.

Examples

#############################################################################
# SIMULATED EXAMPLE 1: Simulate and estimate Dirichlet distribution
#############################################################################	
	
# (1) simulate data
set.seed(789)
N <- 200
probs <- c(.5 , .3 , .2 )
alpha0 <- .5
alpha <- alpha0*probs
alpha <- matrix( alpha , nrow=N , ncol=length(alpha) , byrow=TRUE  )
x <- dirichlet.simul( alpha )

# (2) estimate Dirichlet parameters
dirichlet.mle(x)
  ##   $alpha
  ##   [1] 0.24507708 0.14470944 0.09590745
  ##   $alpha0
  ##   [1] 0.485694
  ##   $xsi
  ##   [1] 0.5045916 0.2979437 0.1974648

#############################################################################
# SIMULATED EXAMPLE 2: Fitting Dirichlet distribution with frequency weights
#############################################################################

# define observed data
x <- scan( nlines=1)
    1 0   0 1   .5 .5
x <- matrix( x , nrow=3 , ncol=2 , byrow=TRUE)    

# transform observations x into (0,1)
eps <- .01
x <- ( x + eps ) / ( 1 + 2 * eps )

# compare results with likelihood fitting package maxLik
library(maxLik)
# define likelihood function
dirichlet.ll <- function(param) {
    ll <- sum( weights * log( ddirichlet( x , param ) ) )
    ll
}

#*** weights 10-10-1
weights <- c(10, 10 , 1 )
mod1a <- dirichlet.mle( x , weights= weights )
mod1a
# estimation in maxLik
mod1b <- maxLik::maxLik(loglik, start=c(.5,.5)) 
print( mod1b )
coef( mod1b )

#*** weights 10-10-10
weights <- c(10, 10 , 10 )
mod2a <- dirichlet.mle( x , weights= weights )
mod2a
# estimation in maxLik
mod2b <- maxLik::maxLik(loglik, start=c(.5,.5)) 
print( mod2b )
coef( mod2b )

#*** weights 30-10-2
weights <- c(30, 10 , 2 )
mod3a <- dirichlet.mle( x , weights= weights )
mod3a
# estimation in maxLik
mod3b <- maxLik::maxLik(loglik, start=c(.25,.25)) 
print( mod3b )
coef( mod3b )