bmixnorm function

Sampling algorithm for mixture of Normal distributions

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

bmixnorm( data, k = "unknown", iter = 1000, burnin = iter / 2, lambda = 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. 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.
• 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.

Details

Sampling from finite mixture of Normal distribution, with density:

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

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 for more details see Stephens, M. (2000), tools:::Rd_expr_doi("10.1214/aos/1016120364") .

Returns

An object with S3 class "bmixnorm" 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.

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, tools:::Rd_expr_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, tools:::Rd_expr_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, tools:::Rd_expr_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

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, tools:::Rd_expr_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, tools:::Rd_expr_doi("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, tools:::Rd_expr_doi("10.1214/17-BA1073")

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixt, bmixgamma, rmixnorm

Examples

## Not run:

data( galaxy )

set.seed( 70 )
# Runing bdmcmc algorithm for the galaxy dataset      
mcmc_sample = bmixnorm( data = galaxy )

summary( mcmc_sample ) 
plot( mcmc_sample )
print( mcmc_sample)

# simulating data from mixture of Normal with 3 components
n      = 500
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      
bmixnorm.obj = bmixnorm( data, k = 3, iter = 1000 )
    
summary( bmixnorm.obj ) 
## End(Not run)