bmixture (version 1.7)

bmixt: Sampling algorithm for mixture of t-distributions

Description

This function consists of several sampling algorithms for Bayesian estimation for finite mixture of t-distributions.

Usage

bmixt( data, k = "unknown", iter = 1000, burnin = iter / 2, lambda = 1, df = 1, 
       k.start = NULL, mu.start = NULL, sig.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. Defult is "unknown". 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.

df

Degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.

k.start

For the case k = "unknown", initial value for number of components of mixture distribution.

mu.start

Initial value for parameter of mixture distribution.

sig.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.

Value

An object with S3 class "bmixt" 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.

mu_sample

a vector which includes the MCMC samples after burn-in from parameter mu of mixture distribution.

sig_sample

a vector which includes the MCMC samples after burn-in from parameter sig of mixture distribution.

data

original data.

Details

Sampling from finite mixture of t-distribution, with density:

$$Pr(x|k, \underline{\pi}, \underline{\mu}, \underline{\sigma}) = \sum_{i=1}^{k} \pi_{i} t_p(x|\mu_{i}, \sigma^2_{i}),$$

where k is the number of components of mixture distribution (as a defult we assume is unknown). The prior distributions are defined as below

$$ P(K=k) \propto \frac{\lambda^k}{k!}, \ \ \ k=1,...,k_{max},$$ $$ \pi_{i} | k \sim Dirichlet( 1,..., 1 ), $$ $$ \mu_{i} | k \sim N( \epsilon, \kappa ), $$ $$ \sigma_i | k \sim IG( g, h ), $$

where IG denotes an inverted gamma distribution. For more details see Stephens, M. (2000), 10.1214/aos/1016120364.

References

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, 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, 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, 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

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, 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, 10.1080/03610918.2011.588358

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

See Also

bmixnorm, bmixgamma, rmixt

Examples

# NOT RUN {
set.seed( 20 )

# simulating data from mixture of Normal with 3 components
n      = 2000
weight = c( 0.3, 0.5, 0.2 )
mean   = c( 0  , 10 , 3   )
sd     = c( 1  , 1  , 1   )
    
data = rmixnorm( n = n, weight = weight, mean = mean, sd = sd )
   
# plot for simulation data      
hist( data, prob = TRUE, nclass = 30, col = "gray" )
    
x           = seq( -20, 20, 0.05 )
densmixnorm = dmixnorm( x, weight, mean, sd )
      
lines( x, densmixnorm, lwd = 2 )  
     
# Runing bdmcmc algorithm for the above simulation data set      
bmixt.obj = bmixt( data, k = 3, iter = 5000 )
    
summary( bmixt.obj ) 
# }