bmixgamma {bmixture} R Documentation

## Sampling algorithm for mixture of gamma distributions

### Description

This function consists of several sampling algorithms for Bayesian estimation for a mixture of Gamma distributions.

### Usage

bmixgamma( data, k = "unknown", iter = 1000, burnin = iter / 2, lambda = 1,
mu = NULL, nu = NULL, kesi = NULL, tau = NULL, k.start = NULL,
alpha.start = NULL, beta.start = NULL, pi.start = NULL,
k.max = 30, trace = TRUE )


### Arguments

 data vector of data with size n. k number of components of mixture distribution. It can take an integer values. iter number of iteration for the sampling algorithm. burnin number of burn-in iteration for the sampling algorithm. lambda For the case k = "unknown", it is the parameter of the prior distribution of number of components k. mu parameter of alpha in mixture distribution. nu parameter of alpha in mixture distribution. kesi parameter of beta in mixture distribution. tau parameter of beta in mixture distribution. k.start For the case k = "unknown", initial value for number of components of mixture distribution. alpha.start Initial value for parameter of mixture distribution. beta.start Initial value for parameter of mixture distribution. pi.start Initial value for parameter of mixture distribution. k.max For the case k = "unknown", maximum value for the number of components of mixture distribution. trace Logical: if TRUE (default), tracing information is printed.

### Details

Sampling from finite mixture of Gamma distribution, with density:

Pr(x|k, \underline{\pi}, \underline{\alpha}, \underline{\beta}) = \sum_{i=1}^{k} \pi_{i} Gamma(x|\alpha_{i}, \beta_{i}),

where k is the number of components of mixture distribution (as a defult we assume is unknown) and

Gamma(x|\alpha_{i}, \beta_{i})=\frac{(\beta_{i})^{\alpha_{i}}}{\Gamma(\alpha_{i})} x^{\alpha_{i}-1} e^{-\beta_{i}x}.

The prior distributions are defined as below

 P(K=k) \propto \frac{\lambda^k}{k!}, \ \ \ k=1,...,k_{max},

 \pi_{i} | k \sim Dirichlet( 1,..., 1 ),

 \alpha_{i} | k \sim Gamma(\nu, \upsilon),

 \beta_i | k \sim Gamma(\eta, \tau),

for more details see Mohammadi et al. (2013), doi: 10.1007/s00180-012-0323-3.

### Value

An object with S3 class "bmixgamma" is returned:

 all_k a vector which includes the waiting times for all iterations. It is needed for monitoring the convergence of the BD-MCMC algorithm. all_weights a vector which includes the waiting times for all iterations. It is needed for monitoring the convergence of the BD-MCMC algorithm. pi_sample a vector which includes the MCMC samples after burn-in from parameter pi of mixture distribution. alpha_sample a vector which includes the MCMC samples after burn-in from parameter alpha of mixture distribution. beta_sample a vector which includes the MCMC samples after burn-in from parameter beta of mixture distribution. data original data.

### References

Mohammadi, A., Salehi-Rad, M. R., and Wit, E. C. (2013) Using mixture of Gamma distributions for Bayesian analysis in an M/G/1 queue with optional second service. Computational Statistics, 28(2):683-700, doi: 10.1007/s00180-012-0323-3

Mohammadi, A., and Salehi-Rad, M. R. (2012) Bayesian inference and prediction in an M/G/1 with optional second service. Communications in Statistics-Simulation and Computation, 41(3):419-435, doi: 10.1080/03610918.2011.588358

Stephens, M. (2000) Bayesian analysis of mixture models with an unknown number of components-an alternative to reversible jump methods. Annals of statistics, 28(1):40-74, doi: 10.1214/aos/1016120364

Richardson, S. and Green, P. J. (1997) On Bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society: series B, 59(4):731-792, doi: 10.1111/1467-9868.00095

Green, P. J. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4):711-732, doi: 10.1093/biomet/82.4.711

Cappe, O., Christian P. R., and Tobias, R. (2003) Reversible jump, birth and death and more general continuous time Markov chain Monte Carlo samplers. Journal of the Royal Statistical Society: Series B, 65(3):679-700

Wade, S. and Ghahramani, Z. (2018) Bayesian Cluster Analysis: Point Estimation and Credible Balls (with Discussion). Bayesian Analysis, 13(2):559-626, doi: 10.1214/17-BA1073

bmixnorm, bmixt, bmixgamma

### Examples

## Not run:
set.seed( 70 )

# simulating data from mixture of gamma with two components
n      = 1000    # number of observations
weight = c( 0.6, 0.4 )
alpha  = c( 12 , 1   )
beta   = c( 3  , 2   )

data = rmixgamma( n = n, weight = weight, alpha = alpha, beta = beta )

# plot for simulation data
hist( data, prob = TRUE, nclass = 50, col = "gray" )

x     = seq( 0, 10, 0.05 )
truth = dmixgamma( x, weight, alpha, beta )

lines( x, truth, lwd = 2 )

# Runing bdmcmc algorithm for the above simulation data set
bmixgamma.obj = bmixgamma( data, iter = 1000 )

summary( bmixgamma.obj )

plot( bmixgamma.obj )

## End(Not run)


[Package bmixture version 1.7 Index]