Finite Mixture Modeling for Raw and Binned Data

Description

The package mixR performs maximum likelihood estimation for finite mixture models for families including Normal, Weibull, Gamma and Lognormal via EM algorithm. It also conducts model selection by using information criteria or bootstrap likelihood ratio test. The data used for mixture model fitting can be raw data or binned data. The model fitting is accelerated by using R package Rcpp.

Details

Finite mixture models can be represented by

f(x; \Phi) = \sum_{j = 1}^g \pi_j f_j(x; \theta_j)

where f(x; \Phi) is the probability density function (p.d.f.) or probability mass function (p.m.f.) of the mixture model, f_j(x; \theta_j) is the p.d.f. or p.m.f. of the jth component of the mixture model, \pi_j is the proportion of the jth component and \theta_j is the parameter of the jth component, which can be a scalar or a vector, \Phi is a vector of all the parameters of the mixture model. The maximum likelihood estimate of the parameter vector \Phi can be obtained by using the EM algorithm (Dempster et al, 1977). The binned data is present sometimes instead of the raw data, for the reason of storage convenience or necessity. The binned data is recorded in the form of (a_i, b_i, n_i) where a_i is the lower bound of the ith bin, b_i is the upper bound of the ith bin, and n_i is the number of observations that fall in the ith bin, for i = 1, \dots, r, and r is the total number of bins.

To obtain maximum likelihood estimate of the finite mixture model for binned data, we can introduce two types of latent variables x and z, wherex represents the value of the unknown raw data, and z is a vector of zeros and one indicating the component that x belongs to. To use the EM algorithm we first write the complete-data log-likelihood

Q(\Phi; \Phi^{(p)}) = \sum_{j = 1}^{g} \sum_{i = 1}^r n_i z^{(p)} [\log f(x^{(p)}; \theta_j) + \log \pi_j ]

where z^{(p)} is the expected value of z given the estimated value of \Phi and expected value x^{(p)} at pth iteration. The estimated value of \Phi can be updated iteratively via the E-step, in which we estimate \Phi by maximizing the complete-data loglikelihood, and M-step, in which we calculate the expected value of the latent variables x and z. The EM algorithm is terminated by using a stopping rule. The M-step of the EM algorithm may or may not have closed-form solution (e.g. the Weibull mixture model or Gamma mixture model). If not, an iterative approach like Newton's algorithm or bisection method may be used.

For a given data set, when we have no prior information about the number of components g, its value should be estimated from the data. Because mixture models don't satisfy the regularity condition for the likelihood ratio test (which requires that the true parameter under the null hypothesis should be in the interior of the parameter space of the full model under the alternative hypothesis), a bootstrap approach is usually used in the literature (see McLachlan (1987, 2004), Feng and McCulloch (1996)). The general step of bootstrap likelihood ratio test is as follows.

1. For the given data x, estimate \Phi under both the null and the alternative hypothesis to get \hat\Phi_0 and \hat\Phi_1. Calculate the observed log-likelihood \ell(x; \hat\Phi_0) and \ell(x; \hat\Phi_1). The likelihood ratio test statistic is defined as

   w_0 = -2(\ell(x; \hat\Phi_0) - \ell(x; \hat\Phi_1)).

2. Generate random data of the same size as the original data x from the model under the null hypothesis using estimated parameter \hat\Phi_0, then repeat step 1 using the simulated data. Repeat this process for B times to get a vector of the simulated likelihood ratio test statistics w_1^{1}, \dots, w_1^{B}.

3. Calculate the empirical p-value

   p = \frac{1}{B} \sum_{i=1}^B I(w_1^{(i)} > w_0)

   where I is the indicator function.

This package does the following three things.

1. Fitting finite mixture models for both raw data and binned data by using EM algorithm, together with Newton-Raphson algorithm and bisection method.

2. Do parametric bootstrap likelihood ratio test for two candidate models.

3. Do model selection by Bayesian information criterion.

To speed up computation, the EM algorithm is fulfilled in C++ by using Rcpp (Eddelbuettel and Francois (2011)).

Author(s)

Maintainer: Youjiao Yu jiaoisjiao@gmail.com

References

Dempster, A. P., Laird, N. M., and Rubin, D. B. Maximum likelihood from incomplete data via the EM algorithm. Journal of the royal statistical society. Series B (methodological), pages 1-38, 1977.

Dirk Eddelbuettel and Romain Francois (2011). Rcpp: Seamless R and C++ Integration. Journal of Statistical Software, 40(8), 1-18. URL http://www.jstatsoft.org/v40/i08/.

Efron, B. Bootstrap methods: Another look at the jackknife. Ann. Statist., 7(1):1-26, 01 1979.

Feng, Z. D. and McCulloch, C. E. Using bootstrap likelihood ratios in finite mixture models. Journal of the Royal Statistical Society. Series B (Methodological), pages 609-617, 1996.

Lo, Y., Mendell, N. R., and Rubin, D. B. Testing the number of components in a normal mixture. Biometrika, 88(3):767-778, 2001.

McLachlan, G. J. On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Applied statistics, pages 318-324, 1987.

McLachlan, G. and Jones, P. Fitting mixture models to grouped and truncated data via the EM algorithm. Biometrics, pages 571-578, 1988.

McLachlan, G. and Peel, D. Finite mixture models. John Wiley & Sons, 2004.

1872 Hidalgo Stamp Data

Description

A vector containing the 1872 Hidalgo stamp data

Usage

Stamp

Format

A vector with 485 measurements of the thickness (nm) of the stamps

References

Izenman, A. J. and Sommer, C. J. Philatelic mixtures and multimodal densities. Journal of the American Statistical association, 83(404):941-953, 1988.

1872 Hidalgo Stamp Data (Binned)

Description

A dataset containing the 1872 Hidalgo stamp data in the form of binned data

Usage

Stamp2

Format

A matrix with 62 rows and 3 columns:

lower

the lower bin values

upper

the upper bin values

freq

the number of observations in each bin

Binning the Raw Data

Description

This function creates a binned data set from the raw data

Usage

bin(x, brks)

Arguments

Details

Given a numeric vector, the function bin creates a binned data set with bin values provided by brks. Fitting mixture models with a large data set may be slow, especially when we want to fit non-normal mixture models. Binning the data with a relatively small bin width speeds up the computation of EM algorithm while at the same time keeps the precision of the estimation result.

Value

The function bin returns a matrix with three columns, representing the value of the left bin, the value of the right bin and the number of observations in x that falls in each bin.

See Also

reinstate

Examples

set.seed(99)
x <- rmixnormal(200, c(0.5, 0.5), c(2, 5), c(1, 1))
data <- bin(x, seq(-2, 10, 0.1))
fit1 <- mixfit(x, ncomp = 2)
fit2 <- mixfit(data, ncomp = 2)

Bootstrap Likelihood Ratio Test for Finite Mixture Models

Description

This function performs the likelihood ratio test by parametric bootstrapping for two mixture models with different number of components.

Usage

bs.test(
  x,
  ncomp = c(1, 2),
  family = c("normal", "weibull", "gamma", "lnorm"),
  B = 100,
  ev = FALSE,
  mstep.method = c("bisection", "newton"),
  init.method = c("kmeans", "hclust"),
  tol = 1e-06,
  max_iter = 500
)

Arguments

Details

For the given data x and the specified family, the function bs.test conducts a bootstrap likelihood ratio test for two mixture models with the number of components under the null and the alternative hypothesis specified in ncomp.

Value

The function bs.test returns an object of class bootEM which contains the following three items.

See Also

plot.bootEM, mixfit, select

Examples

## testing normal mixture models with 2 and 3 components
set.seed(100)
x <- rmixnormal(200, c(0.5, 0.5), c(2, 5), c(1, 0.7))
ret <- bs.test(x, ncomp = c(2, 3), B = 30)
ret

## (not run) testing Weibull mixture models with 2 and 3 components
## set.seed(101)
## x <- rmixweibull(200, c(0.3, 0.4, 0.3), c(2, 5, 8), c(1, 0.6, 0.8))
## ret <- bs.test(x, ncomp = c(2, 3), family = "weibull", B = 30)
## ret

## (not run) testing Gamma mixture models with 1 and 2 components
## set.seed(102)
## x <- rgamma(200, 2, 1)
## ret <- bs.test(x, ncomp = c(1, 2), family = "gamma", B = 30)
## ret

The Density of Finite Mixture Models

Description

This function calculates the probability density of a finite mixture model.

Usage

## S3 method for class 'mixfitEM'
density(x, at, smoothness = 512, cut = 3.8, ...)

Arguments

Details

The function density.mixfitEM is the method of the generic function density for the class mixfitEM.

Value

This function returns a list of class densityEM, which contains the following items.

See Also

mixfit

Examples

set.seed(102)
x <- rmixnormal(200, c(0.5, 0.5), c(2, 5), c(1, 0.7))
fit1 <- mixfit(x, ncomp = 2)
d1 = density(fit1)
d2 = density(fit1, at = 0)

Initialization of the EM Algorithm

Description

This function returns the mean and standard deviation of each component by using K-means clustering or hierarchical clustering.

Usage

initz(x, ncomp, init.method = c("kmeans", "hclust"))

Arguments

Details

The function initz returns the mean and standard deviation of each component of a mixture model by using K-means clustering algorithm, or hierarchical clustering method. It is used for automatically selecting initial values for the EM algorithm, so as to enable mixture model selection by bootstrapping likelihood ratio test or using information criteria.

Value

initz returns a list with three items

Examples

x <- rmixnormal(500, c(0.5, 0.5), c(2, 5), c(1, 0.7))
data <- bin(x, seq(-2, 8, 0.25))
par1 <- initz(x, 2)
par2 <- initz(data, 2)

Finite Mixture Modeling for Raw Data and Binned Data

Description

This function is used to perform the maximum likelihood estimation for a variety of finite mixture models for both raw and binned data by using the EM algorithm, together with Newton-Raphson algorithm or bisection method when necessary.

Usage

mixfit(
  x,
  ncomp = NULL,
  family = c("normal", "weibull", "gamma", "lnorm"),
  pi = NULL,
  mu = NULL,
  sd = NULL,
  ev = FALSE,
  mstep.method = c("bisection", "newton"),
  init.method = c("kmeans", "hclust"),
  tol = 1e-06,
  max_iter = 500
)

Arguments

Details

The function mixfit is the core function in this package. It is used to perform the maximum likelihood estimation for finite mixture models from the families of normal, weibull, gamma or lognormal by using the EM algorithm. When the family is weibull or gamma, the M-step of the EM algorithm has no closed-form solution and we can use Newton algorithm by specifying method = "newton" or use bisection method by specifying method = "bisection".

The initial values of the EM algorithm can be provided by specifying the proportion of each component pi, the mean of each component mu and the standard deviation of each component sd. If one or more of these initial values are not provided, then their values are estimated by using K-means clustering method or hierarchical clustering method. If all of pi, mu, and sd are not provided, then ncomp should be provided so initial values are automatically generated. For the normal mixture models, we can control whether each component has the same variance or not.

Value

the function mixfit return an object of class mixfitEM, which contains a list of different number of items when fitting different mixture models. The common items include

For the Weibull mixture model, the following extra items are returned.

For the Gamma mixture model, the following extra items are returned.

For the lognormal mixture model, the following extra items are returned.

See Also

plot.mixfitEM, density.mixfitEM, select, bs.test

Examples

## fitting the normal mixture models
set.seed(103)
x <- rmixnormal(200, c(0.3, 0.7), c(2, 5), c(1, 1))
data <- bin(x, seq(-1, 8, 0.25))
fit1 <- mixfit(x, ncomp = 2)  # raw data
fit2 <- mixfit(data, ncomp = 2)  # binned data
fit3 <- mixfit(x, pi = c(0.5, 0.5), mu = c(1, 4), sd = c(1, 1))  # providing the initial values
fit4 <- mixfit(x, ncomp = 2, ev = TRUE)  # setting the same variance

## (not run) fitting the weibull mixture models
## x <- rmixweibull(200, c(0.3, 0.7), c(2, 5), c(1, 1))
## data <- bin(x, seq(0, 8, 0.25))
## fit5 <- mixfit(x, ncomp = 2, family = "weibull")  # raw data
## fit6 <- mixfit(data, ncomp = 2, family = "weibull")  # binned data

## (not run) fitting the Gamma mixture models
## x <- rmixgamma(200, c(0.3, 0.7), c(2, 5), c(1, 1))
## data <- bin(x, seq(0, 8, 0.25))
## fit7 <- mixfit(x, ncomp = 2, family = "gamma")  # raw data
## fit8 <- mixfit(data, ncomp = 2, family = "gamma")  # binned data

## (not run) fitting the lognormal mixture models
## x <- rmixlnorm(200, c(0.3, 0.7), c(2, 5), c(1, 1))
## data <- bin(x, seq(0, 8, 0.25))
## fit9 <- mixfit(x, ncomp = 2, family = "lnorm")  # raw data
## fit10 <- mixfit(data, ncomp = 2, family = "lnorm")  # binned data

Plot Bootstrap Likelihood Ratio Test

Description

This function is the plot method for the class bootEM.

Usage

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

Arguments

Details

The histogram of the bootstrap LRT statistics w_1 is plotted, with the observed LRT statistic imposed in a red vertical line.

See Also

bs.test

Examples

## plotting the bootstrap LRT result
set.seed(100)
x <- rmixnormal(200, c(0.5, 0.5), c(2, 5), c(1, 0.7))
ret <- bs.test(x, ncomp = c(2, 3), B = 30)
plot(ret)

Plotting the Fitted Mixture Models

Description

This is the plot method for the class mixfitEM. It is used to plot the fitted mixture models by using base R plotting system or using the package ggplot2.

Usage

## S3 method for class 'mixfitEM'
plot(
  x,
  theme = NULL,
  add_hist = TRUE,
  add_poly = TRUE,
  add_legend = TRUE,
  smoothness = 512,
  trans = 0.5,
  cut = 3.8,
  xlab,
  ylab,
  title,
  breaks,
  plot.title = element_text(hjust = 0.5),
  axis.text.x = element_text(),
  axis.text.y = element_text(),
  axis.title.x = element_text(),
  axis.title.y = element_text(),
  legend.title = element_text(),
  legend.text = element_text(),
  legend.position = "right",
  legend.direction = ifelse(legend.position %in% c("top", "bottom"), "horizontal",
    "vertical"),
  ...
)

Arguments

Details

The function plot.mixfitEM is used for plotting an object of class mixfitEM, which is an output of the function mixfit. Users can choose base R plotting system or ggplot2 (the package ggplot2 needs to be installed). plotting system. The plot is a density plot of the fitted mixture model imposed on top of a histogram. The parameters that control the appearance of the histogram and the density curve can be changed. The density curve of each component can be shown or hidden.

See Also

mixfit

Examples

x <- rmixnormal(200, c(0.3, 0.7), c(2, 5), c(1, 0.7))
mod <- mixfit(x, ncomp = 2)
plot(mod)
plot(mod, theme = 'classic') 

Plot Method for Class selectEM

Description

This function plots the result of mixture model selection by BIC.

Usage

## S3 method for class 'selectEM'
plot(x, leg.loc = "topright", ...)

Arguments

Details

The function plot.selectEM is the plot method for the class selectEM. It plots the number of components against the corresponding value of BIC. It is used to visually display the mixture model selection result by BIC.

See Also

select

Examples

x <- rmixnormal(200, c(0.3, 0.7), c(2, 5), c(1, 1))
res <- select(x, ncomp = 1:3)
plot(res)

Print Method for Class mixfitEM

Description

This function is the print method for the mixfitEM class.

Usage

## S3 method for class 'mixfitEM'
print(x, digits = getOption("digits"), ...)

Arguments

Details

print.mixfitEM prints the value of the parameters of a fitted mixture model, together with some other information like the number of iterations of the EM algorithm, the loglikelihood, the value of AIC and BIC.

See Also

mixfit

Examples

x <- rmixnormal(200, c(0.5, 0.5), c(2, 5), c(1, 0.7))
fit <- mixfit(x, ncomp = 2)
print(x)

Print Method for Class selectEM

Description

The function prints the result of mixture model selection.

Usage

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

Arguments

Details

The function print.selectEM is the print method for the class selectEM, which is the output of the function select. It prints a data frame which contains the following information of each candidate mixture models: the number of components, whether the variance is the same for each component in a mixture model (only for normal), the value of BIC, and an indicator of the best model.

See Also

select

Reinstate the Binned Data to the Raw Data

Description

This function creates a numeric vector approximating the raw data from binned data

Usage

reinstate(data)

Arguments

Details

The function reinstate creates a numeric vector by generating n_i random data from the Uniform distribution U(a_i, b_i) for i = 1, \dots, r and then combine all random data together. a_i, b_i, n_i are the first, second and the third column of the matrix data and r is the number of bins. It is used for enabling parameter initialization for EM algorithm when we fit mixture models for binned data.

Value

The function returns a numeric vector.

See Also

bin

Examples

x <- rnorm(100)
data <- bin(x, seq(-3, 3, 0.25))
y <- reinstate(data)

Generating Random Data From A Gamma Mixture Model

Description

The function rmixgamma generates random data from a Gamma mixture model.

Usage

rmixgamma(n, pi, mu, sd)

Arguments

Details

The number of random data from each component n_0 (a vector) is generated from a multinomial distribution Multinom(n, pi). Then the random data from each component is generated with the sample sized specified in n_0 and parameters of Gamma distributions specified in mu and sd.

Value

The function rmixgamma returns a numeric vector of random data from the specified Gamma mixture model.

See Also

rmixnormal, rmixweibull, rmixlnorm

Examples

x <- rmixgamma(1000, c(0.4, 0.6), c(2, 5), c(1, 0.5))
hist(x, breaks = 40)

Generating Random Data From A Lognormal Mixture Model

Description

The function rmixlnorm generates random data from a lognormal mixture model.

Usage

rmixlnorm(n, pi, mu, sd)

Arguments

Details

The number of random data from each component n_0 (a vector) is generated from a multinomial distribution Multinom(n, pi). Then the random data from each component is generated with the sample sized specified in n_0 and parameters of lognormal distributions specified in mu and sd.

Value

The function rmixlnorm returns a numeric vector of random data from the specified lognormal mixture model.

See Also

rmixnormal, rmixweibull, rmixgamma

Examples

x <- rmixlnorm(1000, c(0.4, 0.6), c(2, 5), c(1, 0.5))
hist(x, breaks = 40)

Generating Random Data From A Normal Mixture Model

Description

The function rmixnormal generates random data from a normal mixture model.

Usage

rmixnormal(n, pi, mu, sd)

Arguments

Details

The number of random data from each component n_0 (a vector) is generated from a multinomial distribution Multinom(n, pi). Then the random data from each component is generated with the sample sized specified in n_0 and parameters of normal distributions specified in mu and sd.

Value

The function rmixnormal returns a numeric vector of random data from the specified normal mixture model.

See Also

rmixweibull, rmixgamma, rmixlnorm

Examples

x <- rmixnormal(1000, c(0.4, 0.6), c(2, 5), c(1, 0.5))
hist(x, breaks = 40)

Generating Random Data From A Weibull Mixture Model

Description

The function rmixweibull generates random data from a normal Weibull model.

Usage

rmixweibull(n, pi, mu, sd)

Arguments

Details

The number of random data from each component n_0 (a vector) is generated from a multinomial distribution Multinom(n, pi). Then the random data from each component is generated with the sample sized specified in n_0 and parameters of Weibull distributions specified in mu and sd.

Value

The function rmixweibull returns a numeric vector of random data from the specified Weibull mixture model.

See Also

rmixnormal, rmixgamma, rmixlnorm

Examples

x <- rmixweibull(1000, c(0.4, 0.6), c(2, 5), c(1, 0.5))
hist(x, breaks = 40)

Finite Mixture Model Selection by Information Criterion

Description

This function selects the best model from a candidate of mixture models based on the information criterion BIC.

Usage

select(
  x,
  ncomp,
  family = c("normal", "weibull", "gamma", "lnorm"),
  mstep.method = c("bisection", "newton"),
  init.method = c("kmeans", "hclust"),
  tol = 1e-06,
  max_iter = 500
)

Arguments

Details

By specifying different number of components, the function select fits a series of mixture models for a given family, and a mixture model with minimum value of BIC is regarded as the best.

Value

The function returns an object of class selectEM which contains the following items.

See Also

plot.selectEM, bs.test, mixfit

Examples

## selecting the optimal normal mixture model by BIC
set.seed(105)
x <- rmixnormal(1000, c(0.3, 0.4, 0.3), c(-4, 0, 4), c(1, 1, 1))
hist(x, breaks = 40)
ret <- select(x, ncomp = 2:5)
## [1] "The final model: normal mixture (equal variance) with 3 components"

## (not run) selecting the optimal Weibull mixture model by BIC
## set.seed(106)
## x <- rmixweibull(1000, c(0.3, 0.4, 0.3), c(2, 5, 8), c(0.7, 0.6, 1))
## ret <- select(x, ncomp = 2:5, family = "weibull")
## [1] "The final model: weibull mixture with 3 components"

## (not run) selecting the optimal Gamma mixture model by BIC
## set.seed(107)
## x <- rmixgamma(1000, c(0.3, 0.7), c(2, 5), c(0.7, 1))
## ret <- select(x, ncomp = 2:5, family = "gamma")
## [1] "The final model: gamma mixture with 2 components"


## (not run) selecting the optimal lognormal mixture model by BIC
## set.seed(108)
## x <- rmixlnorm(1000, c(0.2, 0.3, 0.2, 0.3), c(4, 7, 9, 12), c(1, 0.5, 0.7, 1))
## ret <- select(x, ncomp = 2:6, family = "lnorm")
## [1] "The final model: lnorm mixture with 4 components"

Parameter Conversion for Weibull Distribution

Description

The function to_k_lambda_weibull converts the mean and standard deviation to the shape and scale for the Weibull distributions.

Usage

to_k_lambda_weibull(mu, sd)

Arguments

Details

The purpose of this function is to convert the parameterization of Weibull distribution in the form of mean and standard deviation to the form of shape and scale. It can be used for specifying the initial values for the EM algorithm when the first-hand initial values are in the form of mean and standard deviation from K-means clustering algorithm.

Value

a list of two items

See Also

to_mu_sd_weibull

Examples

to_k_lambda_weibull(2, 1)
to_k_lambda_weibull(c(2, 5), c(1, 0.7))

Parameter Conversion for Gamma Distribution

Description

The function to_mu_sd_gamma converts the shape and rate to the mean and standard deviation

Usage

to_mu_sd_gamma(alpha, lambda)

Arguments

Details

The purpose of this function is to convert the parameterization of gamma distribution in the form of shape and rate to the form of mean and standard deviation.

Value

a list of two items

See Also

to_shape_rate_gamma

Examples

to_mu_sd_gamma(2, 1)
to_mu_sd_gamma(c(2, 4), c(1, 1))

Parameter Conversion for Lognormal Distribution

Description

The function to_mu_sd_lnorm converts the logarithm mean and logarithm standard deviation to the mean and standard deviation

Usage

to_mu_sd_lnorm(mulog, sdlog)

Arguments

Details

The purpose of this function is to convert the parameterization of lognormal distribution in the form of logarithm mean and logarithm standard deviation to the form of mean and standard deviation.

Value

a list of two items

See Also

to_mulog_sdlog_lnorm

Examples

to_mu_sd_lnorm(2, 1)
to_mu_sd_lnorm(c(2, 4), c(1, 1))

Parameter Conversion for Weibull Distribution

Description

The function to_mu_sd_weibull converts the parameters of shape and scale of weibull distributions to the parameters of the mean and standard deviation.

Usage

to_mu_sd_weibull(k, lambda)

Arguments

Details

The purpose of this function is to convert the parameterization of Weibull distribution in the form of shape and scale to the form of mean and standard deviation.

Value

a list of two items

See Also

to_k_lambda_weibull

Examples

to_mu_sd_weibull(2, 1)
to_mu_sd_weibull(c(2, 4), c(1, 1))

Parameter Conversion for Lognormal Distribution

Description

The function to_mulog_sdlog_lnorm converts the mean and standard deviation to the logarithm mean and logarithm standard deviation

Usage

to_mulog_sdlog_lnorm(mu, sd)

Arguments

Details

The purpose of this function is to convert the parameterization of lognormal distribution in the form of mean and standard deviation to the form of logarithm mean and logarithm standard deviation. It can be used for specifying the initial values for the EM algorithm when the first-hand initial values are in the form of mean and standard deviation from K-means clustering algorithm.

Value

a list of two items

See Also

to_mu_sd_lnorm

Examples

to_mulog_sdlog_lnorm(2, 1)
to_mulog_sdlog_lnorm(c(2, 4), c(1, 1))

Parameter Conversion for Gamma Distribution

Description

The function to_shape_rate_gamma converts the mean and standard deviation to the shape and rate

Usage

to_shape_rate_gamma(mu, sd)

Arguments

Details

The purpose of this function is to convert the parameterization of gamma distribution in the form of mean and standard deviation to the form of shape and rate. It can be used for specifying the initial values for the EM algorithm when the first-hand initial values are in the form of mean and standard deviation from K-means clustering algorithm.

Value

a list of two items

See Also

to_mu_sd_gamma

Examples

to_shape_rate_gamma(2, 1)
to_shape_rate_gamma(c(2, 4), c(1, 1))