dirichlet.mle: Maximum Likelihood Estimation of the Dirichlet Distribution

In sirt: Supplementary Item Response Theory Models

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	Convergence 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

#############################################################################
# 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 <- sirt::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

## Not run: 
#############################################################################
# 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
miceadds::library_install("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 <- sirt::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 <- sirt::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 <- sirt::dirichlet.mle( x, weights=weights )
mod3a
# estimation in maxLik
mod3b <- maxLik::maxLik(loglik, start=c(.25,.25))
print( mod3b )
coef( mod3b )

## End(Not run)

sirt documentation built on May 29, 2024, 8:43 a.m.