Title: Bayesian Estimation for Finite Mixture of Distributions
Version: 1.7
Description: Provides statistical tools for Bayesian estimation of mixture distributions, mainly a mixture of Gamma, Normal, and t-distributions. The package is implemented based on the Bayesian literature for the finite mixture of distributions, including Mohammadi and et al. (2013) <doi:10.1007/s00180-012-0323-3> and Mohammadi and Salehi-Rad (2012) <doi:10.1080/03610918.2011.588358>.
URL: https://www.uva.nl/profile/a.mohammadi
Depends: R (≥ 3.0.0)
Imports: BDgraph
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Repository: CRAN
NeedsCompilation: yes
Author: Reza Mohammadi ORCID iD [aut, cre]
Maintainer: Reza Mohammadi <a.mohammadi@uva.nl>
Packaged: 2021-05-11 09:14:57 UTC; reza
Date/Publication: 2021-05-11 11:42:19 UTC

Bayesian Estimation for Finite Mixture of Distributions

Description

The R package bmixture provides statistical tools for Bayesian estimation in finite mixture of distributions. The package implemented the improvements in the Bayesian literature, including Mohammadi and Salehi-Rad (2012) and Mohammadi et al. (2013). Besides, the package contains several functions for simulation and visualization, as well as a real dataset taken from the literature.

How to cite this package

Whenever using this package, please cite as Mohammadi R. (2019). bmixture: Bayesian Estimation for Finite Mixture of Distributions, R package version 1.5, https://CRAN.R-project.org/package=bmixture

Author(s)

Reza Mohammadi <a.mohammadi@uva.nl>

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

Examples

## Not run: 

require( bmixture )

data( galaxy )

# 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
mean   = c( 0  , 10 , 3   )
sd     = c( 1  , 1  , 1   )
weight = c( 0.3, 0.5, 0.2 )
    
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)

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.

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

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

See Also

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)

Sampling algorithm for mixture of Normal distributions

Description

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

Usage

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), doi: 10.1214/aos/1016120364.

Value

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.

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

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, 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

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

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

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)

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.

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), doi: 10.1214/aos/1016120364.

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.

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

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, 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

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

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

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 ) 

## End(Not run)

Internal bmixture functions and datasets

Description

Nothing to see here move along

Author(s)

Reza Mohammadi <a.mohammadi@uva.nl>

Galaxy data

Description

This dataset considers of 82 observatons of the velocities (in 1000 km/second) of distant galaxies diverging from our own, from six well-separated conic sections of the Corona Borealis. The dataset has been analyzed under a variety of mixture models; See e.g. Stephens (2000).

Usage

 data( galaxy )

Format

A data frame with 82 observations on the following variable.

speed

a numeric vector giving the speed of galaxies (in 1000 km/second).

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, doi: 10.1214/aos/1016120364

Examples

data( galaxy )
  
hist( galaxy, prob = TRUE, xlim = c( 0, 40 ), ylim = c( 0, 0.3 ), nclass = 20, 
      col = "gray", border = "white" )

lines( density( galaxy ), col = "black", lwd = 2 )

Mixture of Gamma distribution

Description

Random generation and density function for a finite mixture of Gamma distribution.

Usage

rmixgamma( n = 10, weight = 1, alpha = 1, beta = 1 )

dmixgamma( x, weight = 1, alpha = 1, beta = 1 )

Arguments

n

number of observations.

x

vector of quantiles.

weight

vector of probability weights, with length equal to number of components (k). This is assumed to sum to 1; if not, it is normalized.

alpha

vector of non-negative parameters of the Gamma distribution.

beta

vector of non-negative parameters of the Gamma distribution.

Details

Sampling from finite mixture of Gamma distribution, with density:

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

where

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

Value

Generated data as an vector with size n.

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

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

See Also

rgamma, rmixnorm, rmixt

Examples

## Not run: 
n      = 10000   
weight = c( 0.6  , 0.3  , 0.1   )
alpha  = c( 100  , 200  , 300   )
beta   = c( 100/3, 200/4, 300/5 )
    
data = rmixgamma( n = n, weight = weight, alpha = alpha, beta = beta )
  
hist( data, prob = TRUE, nclass = 30, col = "gray" )
  
x            = seq( -20, 20, 0.05 )
densmixgamma = dmixnorm( x, weight, alpha, beta )
      
lines( x, densmixgamma, lwd = 2 )

## End(Not run)

Mixture of Normal distribution

Description

Random generation and density function for a finite mixture of univariate Normal distribution.

Usage

 
rmixnorm( n = 10, weight = 1, mean = 0, sd = 1 ) 

dmixnorm( x, weight = 1, mean = 0, sd = 1 )

Arguments

n

number of observations.

x

vector of quantiles.

weight

vector of probability weights, with length equal to number of components (k). This is assumed to sum to 1; if not, it is normalized.

mean

vector of means.

sd

vector of standard deviations.

Details

Sampling from finite mixture of Normal distribution, with density:

Pr(x|\underline{w}, \underline{\mu}, \underline{\sigma}) = \sum_{i=1}^{k} w_{i} N(x|\mu_{i}, \sigma_{i}).

Value

Generated data as an vector with size n.

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

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

See Also

rnorm, rmixt, rmixgamma

Examples

## Not run: 
n      = 10000   
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 )
  
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 )

## End(Not run)

Mixture of t-distribution

Description

Random generation and density function for a finite mixture of univariate t-distribution.

Usage

 
rmixt( n = 10, weight = 1, df = 1, mean = 0, sd = 1 ) 

dmixt( x, weight = 1, df = 1, mean = 0, sd = 1 )

Arguments

n

number of observations.

x

vector of quantiles.

weight

vector of probability weights, with length equal to number of components (k). This is assumed to sum to 1; if not, it is normalized.

df

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

mean

vector of means.

sd

vector of standard deviations.

Details

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

Pr(x|\underline{w}, \underline{df}, \underline{\mu}, \underline{\sigma}) = \sum_{i=1}^{k} w_{i} t_{df}(x| \mu_{i}, \sigma_{i}),

where

t_{df}(x| \mu, \sigma) = \frac{ \Gamma( \frac{df+1}{2} ) }{ \Gamma( \frac{df}{2} ) \sqrt{\pi df} \sigma } \left( 1 + \frac{1}{df} \left( \frac{x-\mu}{\sigma} \right) ^2 \right) ^{- \frac{df+1}{2} }.

Value

Generated data as an vector with size n.

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

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

See Also

rt, rmixnorm, rmixgamma

Examples

## Not run: 
n      = 10000   
weight = c( 0.3, 0.5, 0.2 )
df     = c( 4  , 4  , 4   )
mean   = c( 0  , 10 , 3   )
sd     = c( 1  , 1  , 1   )
    
data = rmixt( n = n, weight = weight, df = df, mean = mean, sd = sd )
  
hist( data, prob = TRUE, nclass = 30, col = "gray" )
  
x           = seq( -20, 20, 0.05 )
densmixt = dmixt( x, weight, df, mean, sd )
      
lines( x, densmixt, lwd = 2 )

## End(Not run)

Plot function for S3 class "bmixgamma"

Description

Visualizes the results for function bmixgamma.

Usage

## S3 method for class 'bmixgamma'
plot( x, ... )

Arguments

x

An object of S3 class "bmixgamma", from function bmixgamma.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixgamma

Examples

## Not run: 
# simulating data from mixture of gamma with two components
n      = 500 # 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 )
    
plot( bmixgamma.obj )

## End(Not run)

Plot function for S3 class "bmixnorm"

Description

Visualizes the results for function bmixnorm.

Usage

## S3 method for class 'bmixnorm'
plot( x, ... )

Arguments

x

An object of S3 class "bmixnorm", from function bmixnorm.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixnorm

Examples

## Not run: 
# 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 )
    
plot( bmixnorm.obj ) 

## End(Not run)

Plot function for S3 class "bmixt"

Description

Visualizes the results for function bmixt.

Usage

## S3 method for class 'bmixt'
plot( x, ... )

Arguments

x

An object of S3 class "bmixt", from function bmixt.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixt

Examples

## Not run: 
# 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      
bmixt.obj = bmixt( data, k = 3 )
    
plot( bmixt.obj ) 

## End(Not run)

Print function for S3 class "bmixgamma"

Description

Prints the information about the output of function bmixgamma.

Usage

## S3 method for class 'bmixgamma'
print( x, ... )

Arguments

x

An object of S3 class "bmixgamma", from function bmixgamma.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixgamma

Examples

## Not run: 
# simulating data from mixture of gamma with two components
n      = 500 # 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 = 500 )
    
print( bmixgamma.obj )

## End(Not run)

Print function for S3 class "bmixnorm"

Description

Prints the information about the output of function bmixnorm.

Usage

## S3 method for class 'bmixnorm'
print( x, ... )

Arguments

x

An object of S3 class "bmixnorm", from function bmixnorm.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixnorm

Examples

## Not run: 
# 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 )
    
print( bmixnorm.obj ) 

## End(Not run)

Print function for S3 class "bmixt"

Description

Prints the information about the output of function bmixt.

Usage

## S3 method for class 'bmixt'
print( x, ... )

Arguments

x

object of S3 class "bmixt", from function bmixt.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixt

Examples

## Not run: 
# 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      
bmixt.obj = bmixt( data, k = 3, iter = 1000 )
    
print( bmixt.obj ) 

## End(Not run)

Random generation for the Dirichlet distribution

Description

Random generation from the Dirichlet distribution.

Usage

 rdirichlet( n = 10, alpha = c( 1, 1 ) )

Arguments

n

number of observations.

alpha

vector of shape parameters.

Details

The Dirichlet distribution is the multidimensional generalization of the beta distribution.

Value

A matrix with n rows, each containing a single Dirichlet random deviate.

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

Beta

Examples

  draws = rdirichlet( n = 500, alpha = c( 1, 1, 1 ) )
  
  boxplot( draws )

Summary function for S3 class "bmixgamma"

Description

Provides a summary of the results for function bmixgamma.

Usage

## S3 method for class 'bmixgamma'
summary( object, ... )

Arguments

object

An object of S3 class "bmixgamma", from function bmixgamma.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixgamma

Examples

## Not run: 
# simulating data from mixture of gamma with two components
n      = 500 # 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 = 500 )
    
summary( bmixgamma.obj )

## End(Not run)

Summary function for S3 class "bmixnorm"

Description

Provides a summary of the results for function bmixnorm.

Usage

## S3 method for class 'bmixnorm'
summary( object, ... )

Arguments

object

An object of S3 class "bmixnorm", from function bmixnorm.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixnorm

Examples

## Not run: 
# 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)

Summary function for S3 class "bmixt"

Description

Provides a summary of the results for function bmixt.

Usage

## S3 method for class 'bmixt'
summary( object, ... )

Arguments

object

An object of S3 class "bmixt", from function bmixt.

...

System reserved (no specific usage).

Author(s)

Reza Mohammadi a.mohammadi@uva.nl

See Also

bmixt

Examples

## Not run: 
# 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      
bmixt.obj = bmixt( data, k = 3, iter = 1000 )
    
summary( bmixt.obj ) 

## End(Not run)